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complexOTHERcomputational(3) LAPACK complexOTHERcomputational(3)

NAME

complexOTHERcomputational

SYNOPSIS

Functions


subroutine cbbcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, RWORK, LRWORK, INFO)
CBBCSD subroutine cbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO)
CBDSQR subroutine cgghd3 (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
CGGHD3 subroutine cgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
CGGHRD subroutine cggqrf (N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
CGGQRF subroutine cggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
CGGRQF subroutine cggsvp3 (JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, LWORK, INFO)
CGGSVP3 subroutine cgsvj0 (JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
CGSVJ0 pre-processor for the routine cgesvj. subroutine cgsvj1 (JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
CGSVJ1 pre-processor for the routine cgesvj, applies Jacobi rotations targeting only particular pivots. subroutine chbgst (VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO)
CHBGST subroutine chbtrd (VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
CHBTRD subroutine chfrk (TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C)
CHFRK performs a Hermitian rank-k operation for matrix in RFP format. subroutine chpcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO)
CHPCON subroutine chpgst (ITYPE, UPLO, N, AP, BP, INFO)
CHPGST subroutine chprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CHPRFS subroutine chptrd (UPLO, N, AP, D, E, TAU, INFO)
CHPTRD subroutine chptrf (UPLO, N, AP, IPIV, INFO)
CHPTRF subroutine chptri (UPLO, N, AP, IPIV, WORK, INFO)
CHPTRI subroutine chptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
CHPTRS subroutine chsein (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO)
CHSEIN subroutine chseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
CHSEQR subroutine cla_lin_berr (N, NZ, NRHS, RES, AYB, BERR)
CLA_LIN_BERR computes a component-wise relative backward error. subroutine cla_wwaddw (N, X, Y, W)
CLA_WWADDW adds a vector into a doubled-single vector. subroutine claed0 (QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS, RWORK, IWORK, INFO)
CLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. subroutine claed7 (N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, INFO)
CLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense. subroutine claed8 (K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO)
CLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. subroutine clals0 (ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, RWORK, INFO)
CLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd. subroutine clalsa (ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, RWORK, IWORK, INFO)
CLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd. subroutine clalsd (UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, RWORK, IWORK, INFO)
CLALSD uses the singular value decomposition of A to solve the least squares problem. real function clanhf (NORM, TRANSR, UPLO, N, A, WORK)
CLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format. subroutine clarscl2 (M, N, D, X, LDX)
CLARSCL2 performs reciprocal diagonal scaling on a vector. subroutine clarz (SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
CLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix. subroutine clarzb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
CLARZB applies a block reflector or its conjugate-transpose to a general matrix. subroutine clarzt (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
CLARZT forms the triangular factor T of a block reflector H = I - vtvH. subroutine clascl2 (M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a vector. subroutine clatrz (M, N, L, A, LDA, TAU, WORK)
CLATRZ factors an upper trapezoidal matrix by means of unitary transformations. subroutine cpbcon (UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO)
CPBCON subroutine cpbequ (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
CPBEQU subroutine cpbrfs (UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPBRFS subroutine cpbstf (UPLO, N, KD, AB, LDAB, INFO)
CPBSTF subroutine cpbtf2 (UPLO, N, KD, AB, LDAB, INFO)
CPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm). subroutine cpbtrf (UPLO, N, KD, AB, LDAB, INFO)
CPBTRF subroutine cpbtrs (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
CPBTRS subroutine cpftrf (TRANSR, UPLO, N, A, INFO)
CPFTRF subroutine cpftri (TRANSR, UPLO, N, A, INFO)
CPFTRI subroutine cpftrs (TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
CPFTRS subroutine cppcon (UPLO, N, AP, ANORM, RCOND, WORK, RWORK, INFO)
CPPCON subroutine cppequ (UPLO, N, AP, S, SCOND, AMAX, INFO)
CPPEQU subroutine cpprfs (UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPPRFS subroutine cpptrf (UPLO, N, AP, INFO)
CPPTRF subroutine cpptri (UPLO, N, AP, INFO)
CPPTRI subroutine cpptrs (UPLO, N, NRHS, AP, B, LDB, INFO)
CPPTRS subroutine cpstf2 (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
CPSTF2 computes the Cholesky factorization with complete pivoting of complex Hermitian positive semidefinite matrix. subroutine cpstrf (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
CPSTRF computes the Cholesky factorization with complete pivoting of complex Hermitian positive semidefinite matrix. subroutine cspcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO)
CSPCON subroutine csprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CSPRFS subroutine csptrf (UPLO, N, AP, IPIV, INFO)
CSPTRF subroutine csptri (UPLO, N, AP, IPIV, WORK, INFO)
CSPTRI subroutine csptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
CSPTRS subroutine cstedc (COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
CSTEDC subroutine cstegr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
CSTEGR subroutine cstein (N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
CSTEIN subroutine cstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
CSTEMR subroutine csteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CSTEQR subroutine ctbcon (NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, RWORK, INFO)
CTBCON subroutine ctbrfs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CTBRFS subroutine ctbtrs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
CTBTRS subroutine ctfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
CTFSM solves a matrix equation (one operand is a triangular matrix in RFP format). subroutine ctftri (TRANSR, UPLO, DIAG, N, A, INFO)
CTFTRI subroutine ctfttp (TRANSR, UPLO, N, ARF, AP, INFO)
CTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP). subroutine ctfttr (TRANSR, UPLO, N, ARF, A, LDA, INFO)
CTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR). subroutine ctgsen (IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
CTGSEN subroutine ctgsja (JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
CTGSJA subroutine ctgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CTGSNA subroutine ctpcon (NORM, UPLO, DIAG, N, AP, RCOND, WORK, RWORK, INFO)
CTPCON subroutine ctpmqrt (SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
CTPMQRT subroutine ctpqrt (M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
CTPQRT subroutine ctpqrt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
CTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. subroutine ctprfs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CTPRFS subroutine ctptri (UPLO, DIAG, N, AP, INFO)
CTPTRI subroutine ctptrs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO)
CTPTRS subroutine ctpttf (TRANSR, UPLO, N, AP, ARF, INFO)
CTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF). subroutine ctpttr (UPLO, N, AP, A, LDA, INFO)
CTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR). subroutine ctrcon (NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, RWORK, INFO)
CTRCON subroutine ctrevc (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO)
CTREVC subroutine ctrevc3 (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, RWORK, LRWORK, INFO)
CTREVC3 subroutine ctrexc (COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
CTREXC subroutine ctrrfs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CTRRFS subroutine ctrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO)
CTRSEN subroutine ctrsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO)
CTRSNA subroutine ctrti2 (UPLO, DIAG, N, A, LDA, INFO)
CTRTI2 computes the inverse of a triangular matrix (unblocked algorithm). subroutine ctrtri (UPLO, DIAG, N, A, LDA, INFO)
CTRTRI subroutine ctrtrs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
CTRTRS subroutine ctrttf (TRANSR, UPLO, N, A, LDA, ARF, INFO)
CTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF). subroutine ctrttp (UPLO, N, A, LDA, AP, INFO)
CTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP). subroutine ctzrzf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
CTZRZF subroutine cunbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
CUNBDB subroutine cunbdb1 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
CUNBDB1 subroutine cunbdb2 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
CUNBDB2 subroutine cunbdb3 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
CUNBDB3 subroutine cunbdb4 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO)
CUNBDB4 subroutine cunbdb5 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
CUNBDB5 subroutine cunbdb6 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
CUNBDB6 recursive subroutine cuncsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK, RWORK, LRWORK, IWORK, INFO)
CUNCSD subroutine cuncsd2by1 (JOBU1, JOBU2, JOBV1T, M, P, Q, X11, LDX11, X21, LDX21, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, WORK, LWORK, RWORK, LRWORK, IWORK, INFO)
CUNCSD2BY1 subroutine cung2l (M, N, K, A, LDA, TAU, WORK, INFO)
CUNG2L generates all or part of the unitary matrix Q from a QL factorization determined by cgeqlf (unblocked algorithm). subroutine cung2r (M, N, K, A, LDA, TAU, WORK, INFO)
CUNG2R subroutine cunghr (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CUNGHR subroutine cungl2 (M, N, K, A, LDA, TAU, WORK, INFO)
CUNGL2 generates all or part of the unitary matrix Q from an LQ factorization determined by cgelqf (unblocked algorithm). subroutine cunglq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGLQ subroutine cungql (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGQL subroutine cungqr (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGQR subroutine cungr2 (M, N, K, A, LDA, TAU, WORK, INFO)
CUNGR2 generates all or part of the unitary matrix Q from an RQ factorization determined by cgerqf (unblocked algorithm). subroutine cungrq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGRQ subroutine cungtr (UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
CUNGTR subroutine cunm22 (SIDE, TRANS, M, N, N1, N2, Q, LDQ, C, LDC, WORK, LWORK, INFO)
CUNM22 multiplies a general matrix by a banded unitary matrix. subroutine cunm2l (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
CUNM2L multiplies a general matrix by the unitary matrix from a QL factorization determined by cgeqlf (unblocked algorithm). subroutine cunm2r (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
CUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf (unblocked algorithm). subroutine cunmbr (VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMBR subroutine cunmhr (SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMHR subroutine cunml2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
CUNML2 multiplies a general matrix by the unitary matrix from a LQ factorization determined by cgelqf (unblocked algorithm). subroutine cunmlq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMLQ subroutine cunmql (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMQL subroutine cunmqr (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMQR subroutine cunmr2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
CUNMR2 multiplies a general matrix by the unitary matrix from a RQ factorization determined by cgerqf (unblocked algorithm). subroutine cunmr3 (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, INFO)
CUNMR3 multiplies a general matrix by the unitary matrix from a RZ factorization determined by ctzrzf (unblocked algorithm). subroutine cunmrq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMRQ subroutine cunmrz (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMRZ subroutine cunmtr (SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMTR subroutine cupgtr (UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
CUPGTR subroutine cupmtr (SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
CUPMTR subroutine dorm22 (SIDE, TRANS, M, N, N1, N2, Q, LDQ, C, LDC, WORK, LWORK, INFO)
DORM22 multiplies a general matrix by a banded orthogonal matrix. subroutine sorm22 (SIDE, TRANS, M, N, N1, N2, Q, LDQ, C, LDC, WORK, LWORK, INFO)
SORM22 multiplies a general matrix by a banded orthogonal matrix. subroutine zunm22 (SIDE, TRANS, M, N, N1, N2, Q, LDQ, C, LDC, WORK, LWORK, INFO)
ZUNM22 multiplies a general matrix by a banded unitary matrix.

Detailed Description

This is the group of complex other Computational routines

Function Documentation

subroutine cbbcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, integer M, integer P, integer Q, real, dimension( * ) THETA, real, dimension( * ) PHI, complex, dimension( ldu1, * ) U1, integer LDU1, complex, dimension( ldu2, * ) U2, integer LDU2, complex, dimension( ldv1t, * ) V1T, integer LDV1T, complex, dimension( ldv2t, * ) V2T, integer LDV2T, real, dimension( * ) B11D, real, dimension( * ) B11E, real, dimension( * ) B12D, real, dimension( * ) B12E, real, dimension( * ) B21D, real, dimension( * ) B21E, real, dimension( * ) B22D, real, dimension( * ) B22E, real, dimension( * ) RWORK, integer LRWORK, integer INFO)

CBBCSD

Purpose:

 CBBCSD computes the CS decomposition of a unitary matrix in
 bidiagonal-block form,
     [ B11 | B12 0  0 ]
     [  0  |  0 -I  0 ]
 X = [----------------]
     [ B21 | B22 0  0 ]
     [  0  |  0  0  I ]
                               [  C | -S  0  0 ]
                   [ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**H
                 = [---------] [---------------] [---------]   .
                   [    | U2 ] [  S |  C  0  0 ] [    | V2 ]
                               [  0 |  0  0  I ]
 X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
 than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
 transposed and/or permuted. This can be done in constant time using
 the TRANS and SIGNS options. See CUNCSD for details.)
 The bidiagonal matrices B11, B12, B21, and B22 are represented
 implicitly by angles THETA(1:Q) and PHI(1:Q-1).
 The unitary matrices U1, U2, V1T, and V2T are input/output.
 The input matrices are pre- or post-multiplied by the appropriate
 singular vector matrices.

Parameters:

JOBU1

          JOBU1 is CHARACTER
          = 'Y':      U1 is updated;
          otherwise:  U1 is not updated.

JOBU2

          JOBU2 is CHARACTER
          = 'Y':      U2 is updated;
          otherwise:  U2 is not updated.

JOBV1T

          JOBV1T is CHARACTER
          = 'Y':      V1T is updated;
          otherwise:  V1T is not updated.

JOBV2T

          JOBV2T is CHARACTER
          = 'Y':      V2T is updated;
          otherwise:  V2T is not updated.

TRANS

          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.

M

          M is INTEGER
          The number of rows and columns in X, the unitary matrix in
          bidiagonal-block form.

P

          P is INTEGER
          The number of rows in the top-left block of X. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in the top-left block of X.
          0 <= Q <= MIN(P,M-P,M-Q).

THETA

          THETA is REAL array, dimension (Q)
          On entry, the angles THETA(1),...,THETA(Q) that, along with
          PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
          form. On exit, the angles whose cosines and sines define the
          diagonal blocks in the CS decomposition.

PHI

          PHI is REAL array, dimension (Q-1)
          The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
          THETA(Q), define the matrix in bidiagonal-block form.

U1

          U1 is COMPLEX array, dimension (LDU1,P)
          On entry, a P-by-P matrix. On exit, U1 is postmultiplied
          by the left singular vector matrix common to [ B11 ; 0 ] and
          [ B12 0 0 ; 0 -I 0 0 ].

LDU1

          LDU1 is INTEGER
          The leading dimension of the array U1, LDU1 >= MAX(1,P).

U2

          U2 is COMPLEX array, dimension (LDU2,M-P)
          On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
          postmultiplied by the left singular vector matrix common to
          [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].

LDU2

          LDU2 is INTEGER
          The leading dimension of the array U2, LDU2 >= MAX(1,M-P).

V1T

          V1T is COMPLEX array, dimension (LDV1T,Q)
          On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
          by the conjugate transpose of the right singular vector
          matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].

LDV1T

          LDV1T is INTEGER
          The leading dimension of the array V1T, LDV1T >= MAX(1,Q).

V2T

          V2T is COMPLEX array, dimension (LDV2T,M-Q)
          On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
          premultiplied by the conjugate transpose of the right
          singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
          [ B22 0 0 ; 0 0 I ].

LDV2T

          LDV2T is INTEGER
          The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).

B11D

          B11D is REAL array, dimension (Q)
          When CBBCSD converges, B11D contains the cosines of THETA(1),
          ..., THETA(Q). If CBBCSD fails to converge, then B11D
          contains the diagonal of the partially reduced top-left
          block.

B11E

          B11E is REAL array, dimension (Q-1)
          When CBBCSD converges, B11E contains zeros. If CBBCSD fails
          to converge, then B11E contains the superdiagonal of the
          partially reduced top-left block.

B12D

          B12D is REAL array, dimension (Q)
          When CBBCSD converges, B12D contains the negative sines of
          THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
          B12D contains the diagonal of the partially reduced top-right
          block.

B12E

          B12E is REAL array, dimension (Q-1)
          When CBBCSD converges, B12E contains zeros. If CBBCSD fails
          to converge, then B12E contains the subdiagonal of the
          partially reduced top-right block.

B21D

          B21D is REAL array, dimension (Q)
          When CBBCSD converges, B21D contains the negative sines of
          THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
          B21D contains the diagonal of the partially reduced bottom-left
          block.

B21E

          B21E is REAL array, dimension (Q-1)
          When CBBCSD converges, B21E contains zeros. If CBBCSD fails
          to converge, then B21E contains the subdiagonal of the
          partially reduced bottom-left block.

B22D

          B22D is REAL array, dimension (Q)
          When CBBCSD converges, B22D contains the negative sines of
          THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
          B22D contains the diagonal of the partially reduced bottom-right
          block.

B22E

          B22E is REAL array, dimension (Q-1)
          When CBBCSD converges, B22E contains zeros. If CBBCSD fails
          to converge, then B22E contains the subdiagonal of the
          partially reduced bottom-right block.

RWORK

          RWORK is REAL array, dimension (MAX(1,LRWORK))
          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

LRWORK

          LRWORK is INTEGER
          The dimension of the array RWORK. LRWORK >= MAX(1,8*Q).
          If LRWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the RWORK array,
          returns this value as the first entry of the work array, and
          no error message related to LRWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if CBBCSD did not converge, INFO specifies the number
                of nonzero entries in PHI, and B11D, B11E, etc.,
                contain the partially reduced matrix.

Internal Parameters:

  TOLMUL  REAL, default = MAX(10,MIN(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
          are within TOLMUL*EPS of either bound.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine cbdsqr (character UPLO, integer N, integer NCVT, integer NRU, integer NCC, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldvt, * ) VT, integer LDVT, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldc, * ) C, integer LDC, real, dimension( * ) RWORK, integer INFO)

CBDSQR

Purpose:

 CBDSQR computes the singular values and, optionally, the right and/or
 left singular vectors from the singular value decomposition (SVD) of
 a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
 zero-shift QR algorithm.  The SVD of B has the form
    B = Q * S * P**H
 where S is the diagonal matrix of singular values, Q is an orthogonal
 matrix of left singular vectors, and P is an orthogonal matrix of
 right singular vectors.  If left singular vectors are requested, this
 subroutine actually returns U*Q instead of Q, and, if right singular
 vectors are requested, this subroutine returns P**H*VT instead of
 P**H, for given complex input matrices U and VT.  When U and VT are
 the unitary matrices that reduce a general matrix A to bidiagonal
 form: A = U*B*VT, as computed by CGEBRD, then
    A = (U*Q) * S * (P**H*VT)
 is the SVD of A.  Optionally, the subroutine may also compute Q**H*C
 for a given complex input matrix C.
 See "Computing  Small Singular Values of Bidiagonal Matrices With
 Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
 LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
 no. 5, pp. 873-912, Sept 1990) and
 "Accurate singular values and differential qd algorithms," by
 B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
 Department, University of California at Berkeley, July 1992
 for a detailed description of the algorithm.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.

N

          N is INTEGER
          The order of the matrix B.  N >= 0.

NCVT

          NCVT is INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

NRU

          NRU is INTEGER
          The number of rows of the matrix U. NRU >= 0.

NCC

          NCC is INTEGER
          The number of columns of the matrix C. NCC >= 0.

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B in decreasing
          order.

E

          E is REAL array, dimension (N-1)
          On entry, the N-1 offdiagonal elements of the bidiagonal
          matrix B.
          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
          will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given
          as input.

VT

          VT is COMPLEX array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**H * VT.
          Not referenced if NCVT = 0.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

U

          U is COMPLEX array, dimension (LDU, N)
          On entry, an NRU-by-N matrix U.
          On exit, U is overwritten by U * Q.
          Not referenced if NRU = 0.

LDU

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= max(1,NRU).

C

          C is COMPLEX array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**H * C.
          Not referenced if NCC = 0.

LDC

          LDC is INTEGER
          The leading dimension of the array C.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

RWORK

          RWORK is REAL array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm did not converge; D and E contain the
                elements of a bidiagonal matrix which is orthogonally
                similar to the input matrix B;  if INFO = i, i
                elements of E have not converged to zero.

Internal Parameters:

  TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          If it is positive, TOLMUL*EPS is the desired relative
             precision in the computed singular values.
          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
             desired absolute accuracy in the computed singular
             values (corresponds to relative accuracy
             abs(TOLMUL*EPS) in the largest singular value.
          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
             between 10 (for fast convergence) and .1/EPS
             (for there to be some accuracy in the results).
          Default is to lose at either one eighth or 2 of the
             available decimal digits in each computed singular value
             (whichever is smaller).
  MAXITR  INTEGER, default = 6
          MAXITR controls the maximum number of passes of the
          algorithm through its inner loop. The algorithms stops
          (and so fails to converge) if the number of passes
          through the inner loop exceeds MAXITR*N**2.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cgghd3 (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGGHD3

Purpose:

 CGGHD3 reduces a pair of complex matrices (A,B) to generalized upper
 Hessenberg form using unitary transformations, where A is a
 general matrix and B is upper triangular.  The form of the
 generalized eigenvalue problem is
    A*x = lambda*B*x,
 and B is typically made upper triangular by computing its QR
 factorization and moving the unitary matrix Q to the left side
 of the equation.
 This subroutine simultaneously reduces A to a Hessenberg matrix H:
    Q**H*A*Z = H
 and transforms B to another upper triangular matrix T:
    Q**H*B*Z = T
 in order to reduce the problem to its standard form
    H*y = lambda*T*y
 where y = Z**H*x.
 The unitary matrices Q and Z are determined as products of Givens
 rotations.  They may either be formed explicitly, or they may be
 postmultiplied into input matrices Q1 and Z1, so that
      Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
      Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
 If Q1 is the unitary matrix from the QR factorization of B in the
 original equation A*x = lambda*B*x, then CGGHD3 reduces the original
 problem to generalized Hessenberg form.
 This is a blocked variant of CGGHRD, using matrix-matrix
 multiplications for parts of the computation to enhance performance.

Parameters:

COMPQ

          COMPQ is CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 unitary matrix Q is returned;
          = 'V': Q must contain a unitary matrix Q1 on entry,
                 and the product Q1*Q is returned.

COMPZ

          COMPZ is CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 unitary matrix Z is returned;
          = 'V': Z must contain a unitary matrix Z1 on entry,
                 and the product Z1*Z is returned.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to CGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is COMPLEX array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**H B Z.  The
          elements below the diagonal are set to zero.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

Q

          Q is COMPLEX array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the unitary matrix Q1, typically
          from the QR factorization of B.
          On exit, if COMPQ='I', the unitary matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

          Z is COMPLEX array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the unitary matrix Z1.
          On exit, if COMPZ='I', the unitary matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

WORK

          WORK is COMPLEX array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The length of the array WORK.  LWORK >= 1.
          For optimum performance LWORK >= 6*N*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

Further Details:

  This routine reduces A to Hessenberg form and maintains B in
  using a blocked variant of Moler and Stewart's original algorithm,
  as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
  (BIT 2008).

subroutine cgghrd (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldz, * ) Z, integer LDZ, integer INFO)

CGGHRD

Purpose:

 CGGHRD reduces a pair of complex matrices (A,B) to generalized upper
 Hessenberg form using unitary transformations, where A is a
 general matrix and B is upper triangular.  The form of the generalized
 eigenvalue problem is
    A*x = lambda*B*x,
 and B is typically made upper triangular by computing its QR
 factorization and moving the unitary matrix Q to the left side
 of the equation.
 This subroutine simultaneously reduces A to a Hessenberg matrix H:
    Q**H*A*Z = H
 and transforms B to another upper triangular matrix T:
    Q**H*B*Z = T
 in order to reduce the problem to its standard form
    H*y = lambda*T*y
 where y = Z**H*x.
 The unitary matrices Q and Z are determined as products of Givens
 rotations.  They may either be formed explicitly, or they may be
 postmultiplied into input matrices Q1 and Z1, so that
      Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
      Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
 If Q1 is the unitary matrix from the QR factorization of B in the
 original equation A*x = lambda*B*x, then CGGHRD reduces the original
 problem to generalized Hessenberg form.

Parameters:

COMPQ

          COMPQ is CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 unitary matrix Q is returned;
          = 'V': Q must contain a unitary matrix Q1 on entry,
                 and the product Q1*Q is returned.

COMPZ

          COMPZ is CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 unitary matrix Z is returned;
          = 'V': Z must contain a unitary matrix Z1 on entry,
                 and the product Z1*Z is returned.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to CGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is COMPLEX array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**H B Z.  The
          elements below the diagonal are set to zero.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

Q

          Q is COMPLEX array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the unitary matrix Q1, typically
          from the QR factorization of B.
          On exit, if COMPQ='I', the unitary matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

          Z is COMPLEX array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the unitary matrix Z1.
          On exit, if COMPZ='I', the unitary matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  This routine reduces A to Hessenberg and B to triangular form by
  an unblocked reduction, as described in _Matrix_Computations_,
  by Golub and van Loan (Johns Hopkins Press).

subroutine cggqrf (integer N, integer M, integer P, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAUA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) TAUB, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGGQRF

Purpose:

 CGGQRF computes a generalized QR factorization of an N-by-M matrix A
 and an N-by-P matrix B:
             A = Q*R,        B = Q*T*Z,
 where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix,
 and R and T assume one of the forms:
 if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                 (  0  ) N-M                         N   M-N
                    M
 where R11 is upper triangular, and
 if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                  P-N  N                           ( T21 ) P
                                                      P
 where T12 or T21 is upper triangular.
 In particular, if B is square and nonsingular, the GQR factorization
 of A and B implicitly gives the QR factorization of inv(B)*A:
              inv(B)*A = Z**H * (inv(T)*R)
 where inv(B) denotes the inverse of the matrix B, and Z' denotes the
 conjugate transpose of matrix Z.

Parameters:

N

          N is INTEGER
          The number of rows of the matrices A and B. N >= 0.

M

          M is INTEGER
          The number of columns of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of columns of the matrix B.  P >= 0.

A

          A is COMPLEX array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the elements on and above the diagonal of the array
          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
          upper triangular if N >= M); the elements below the diagonal,
          with the array TAUA, represent the unitary matrix Q as a
          product of min(N,M) elementary reflectors (see Further
          Details).

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

TAUA

          TAUA is COMPLEX array, dimension (min(N,M))
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q (see Further Details).

B

          B is COMPLEX array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)-th subdiagonal
          contain the N-by-P upper trapezoidal matrix T; the remaining
          elements, with the array TAUB, represent the unitary
          matrix Z as a product of elementary reflectors (see Further
          Details).

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

TAUB

          TAUB is COMPLEX array, dimension (min(N,P))
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Z (see Further Details).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the QR factorization
          of an N-by-M matrix, NB2 is the optimal blocksize for the
          RQ factorization of an N-by-P matrix, and NB3 is the optimal
          blocksize for a call of CUNMQR.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The matrix Q is represented as a product of elementary reflectors
     Q = H(1) H(2) . . . H(k), where k = min(n,m).
  Each H(i) has the form
     H(i) = I - taua * v * v**H
  where taua is a complex scalar, and v is a complex vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
  and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine CUNGQR.
  To use Q to update another matrix, use LAPACK subroutine CUNMQR.
  The matrix Z is represented as a product of elementary reflectors
     Z = H(1) H(2) . . . H(k), where k = min(n,p).
  Each H(i) has the form
     H(i) = I - taub * v * v**H
  where taub is a complex scalar, and v is a complex vector with
  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine CUNGRQ.
  To use Z to update another matrix, use LAPACK subroutine CUNMRQ.

subroutine cggrqf (integer M, integer P, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAUA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) TAUB, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGGRQF

Purpose:

 CGGRQF computes a generalized RQ factorization of an M-by-N matrix A
 and a P-by-N matrix B:
             A = R*Q,        B = Z*T*Q,
 where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
 matrix, and R and T assume one of the forms:
 if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                  N-M  M                           ( R21 ) N
                                                      N
 where R12 or R21 is upper triangular, and
 if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
                 (  0  ) P-N                         P   N-P
                    N
 where T11 is upper triangular.
 In particular, if B is square and nonsingular, the GRQ factorization
 of A and B implicitly gives the RQ factorization of A*inv(B):
              A*inv(B) = (R*inv(T))*Z**H
 where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
 conjugate transpose of the matrix Z.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B. N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, if M <= N, the upper triangle of the subarray
          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
          if M > N, the elements on and above the (M-N)-th subdiagonal
          contain the M-by-N upper trapezoidal matrix R; the remaining
          elements, with the array TAUA, represent the unitary
          matrix Q as a product of elementary reflectors (see Further
          Details).

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

TAUA

          TAUA is COMPLEX array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q (see Further Details).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, the elements on and above the diagonal of the array
          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
          upper triangular if P >= N); the elements below the diagonal,
          with the array TAUB, represent the unitary matrix Z as a
          product of elementary reflectors (see Further Details).

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

TAUB

          TAUB is COMPLEX array, dimension (min(P,N))
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Z (see Further Details).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the RQ factorization
          of an M-by-N matrix, NB2 is the optimal blocksize for the
          QR factorization of a P-by-N matrix, and NB3 is the optimal
          blocksize for a call of CUNMRQ.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO=-i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The matrix Q is represented as a product of elementary reflectors
     Q = H(1) H(2) . . . H(k), where k = min(m,n).
  Each H(i) has the form
     H(i) = I - taua * v * v**H
  where taua is a complex scalar, and v is a complex vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
  A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine CUNGRQ.
  To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
  The matrix Z is represented as a product of elementary reflectors
     Z = H(1) H(2) . . . H(k), where k = min(p,n).
  Each H(i) has the form
     H(i) = I - taub * v * v**H
  where taub is a complex scalar, and v is a complex vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
  and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine CUNGQR.
  To use Z to update another matrix, use LAPACK subroutine CUNMQR.

subroutine cggsvp3 (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, real TOLA, real TOLB, integer K, integer L, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldq, * ) Q, integer LDQ, integer, dimension( * ) IWORK, real, dimension( * ) RWORK, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGGSVP3

Purpose:

 CGGSVP3 computes unitary matrices U, V and Q such that
                    N-K-L  K    L
  U**H*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
                 L ( 0     0   A23 )
             M-K-L ( 0     0    0  )
                  N-K-L  K    L
         =     K ( 0    A12  A13 )  if M-K-L < 0;
             M-K ( 0     0   A23 )
                  N-K-L  K    L
  V**H*B*Q =   L ( 0     0   B13 )
             P-L ( 0     0    0  )
 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
 upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
 otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
 numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H.
 This decomposition is the preprocessing step for computing the
 Generalized Singular Value Decomposition (GSVD), see subroutine
 CGGSVD3.

Parameters:

JOBU

          JOBU is CHARACTER*1
          = 'U':  Unitary matrix U is computed;
          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1
          = 'V':  Unitary matrix V is computed;
          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1
          = 'Q':  Unitary matrix Q is computed;
          = 'N':  Q is not computed.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular (or trapezoidal) matrix
          described in the Purpose section.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains the triangular matrix described in
          the Purpose section.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

TOLA

          TOLA is REAL

TOLB

          TOLB is REAL
          TOLA and TOLB are the thresholds to determine the effective
          numerical rank of matrix B and a subblock of A. Generally,
          they are set to
             TOLA = MAX(M,N)*norm(A)*MACHEPS,
             TOLB = MAX(P,N)*norm(B)*MACHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.

K

          K is INTEGER

L

          L is INTEGER
          On exit, K and L specify the dimension of the subblocks
          described in Purpose section.
          K + L = effective numerical rank of (A**H,B**H)**H.

U

          U is COMPLEX array, dimension (LDU,M)
          If JOBU = 'U', U contains the unitary matrix U.
          If JOBU = 'N', U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

V

          V is COMPLEX array, dimension (LDV,P)
          If JOBV = 'V', V contains the unitary matrix V.
          If JOBV = 'N', V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

Q

          Q is COMPLEX array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the unitary matrix Q.
          If JOBQ = 'N', Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

IWORK

          IWORK is INTEGER array, dimension (N)

RWORK

          RWORK is REAL array, dimension (2*N)

TAU

          TAU is COMPLEX array, dimension (N)

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

August 2015

Further Details:

  The subroutine uses LAPACK subroutine CGEQP3 for the QR factorization
  with column pivoting to detect the effective numerical rank of the
  a matrix. It may be replaced by a better rank determination strategy.
  CGGSVP3 replaces the deprecated subroutine CGGSVP.

subroutine cgsvj0 (character*1 JOBV, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( n ) D, real, dimension( n ) SVA, integer MV, complex, dimension( ldv, * ) V, integer LDV, real EPS, real SFMIN, real TOL, integer NSWEEP, complex, dimension( lwork ) WORK, integer LWORK, integer INFO)

CGSVJ0 pre-processor for the routine cgesvj.

Purpose:

 CGSVJ0 is called from CGESVJ as a pre-processor and that is its main
 purpose. It applies Jacobi rotations in the same way as CGESVJ does, but
 it does not check convergence (stopping criterion). Few tuning
 parameters (marked by [TP]) are available for the implementer.

Parameters:

JOBV

          JOBV is CHARACTER*1
          Specifies whether the output from this procedure is used
          to compute the matrix V:
          = 'V': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the N-by-N array V.
                (See the description of V.)
          = 'A': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the MV-by-N array V.
                (See the descriptions of MV and V.)
          = 'N': the Jacobi rotations are not accumulated.

M

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the input matrix A.
          M >= N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, M-by-N matrix A, such that A*diag(D) represents
          the input matrix.
          On exit,
          A_onexit * diag(D_onexit) represents the input matrix A*diag(D)
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of D, TOL and NSWEEP.)

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is COMPLEX array, dimension (N)
          The array D accumulates the scaling factors from the complex scaled
          Jacobi rotations.
          On entry, A*diag(D) represents the input matrix.
          On exit, A_onexit*diag(D_onexit) represents the input matrix
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of A, TOL and NSWEEP.)

SVA

          SVA is REAL array, dimension (N)
          On entry, SVA contains the Euclidean norms of the columns of
          the matrix A*diag(D).
          On exit, SVA contains the Euclidean norms of the columns of
          the matrix A_onexit*diag(D_onexit).

MV

          MV is INTEGER
          If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'N',   then MV is not referenced.

V

          V is COMPLEX array, dimension (LDV,N)
          If JOBV .EQ. 'V' then N rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'N',   then V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V,  LDV >= 1.
          If JOBV = 'V', LDV .GE. N.
          If JOBV = 'A', LDV .GE. MV.

EPS

          EPS is REAL
          EPS = SLAMCH('Epsilon')

SFMIN

          SFMIN is REAL
          SFMIN = SLAMCH('Safe Minimum')

TOL

          TOL is REAL
          TOL is the threshold for Jacobi rotations. For a pair
          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
          applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.

NSWEEP

          NSWEEP is INTEGER
          NSWEEP is the number of sweeps of Jacobi rotations to be
          performed.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          LWORK is the dimension of WORK. LWORK .GE. M.

INFO

          INFO is INTEGER
          = 0 : successful exit.
          < 0 : if INFO = -i, then the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

CGSVJ0 is used just to enable CGESVJ to call a simplified version of itself to work on a submatrix of the original matrix.

Contributor:

Zlatko Drmac (Zagreb, Croatia)

Bugs, Examples and Comments:

Please report all bugs and send interesting test examples and comments to drmac@math.hr. Thank you.

subroutine cgsvj1 (character*1 JOBV, integer M, integer N, integer N1, complex, dimension( lda, * ) A, integer LDA, complex, dimension( n ) D, real, dimension( n ) SVA, integer MV, complex, dimension( ldv, * ) V, integer LDV, real EPS, real SFMIN, real TOL, integer NSWEEP, complex, dimension( lwork ) WORK, integer LWORK, integer INFO)

CGSVJ1 pre-processor for the routine cgesvj, applies Jacobi rotations targeting only particular pivots.

Purpose:

 CGSVJ1 is called from CGESVJ as a pre-processor and that is its main
 purpose. It applies Jacobi rotations in the same way as CGESVJ does, but
 it targets only particular pivots and it does not check convergence
 (stopping criterion). Few tunning parameters (marked by [TP]) are
 available for the implementer.
 Further Details
 ~~~~~~~~~~~~~~~
 CGSVJ1 applies few sweeps of Jacobi rotations in the column space of
 the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
 off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
 block-entries (tiles) of the (1,2) off-diagonal block are marked by the
 [x]'s in the following scheme:
    | *  *  * [x] [x] [x]|
    | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
    | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
    |[x] [x] [x] *  *  * |
    |[x] [x] [x] *  *  * |
    |[x] [x] [x] *  *  * |
 In terms of the columns of A, the first N1 columns are rotated 'against'
 the remaining N-N1 columns, trying to increase the angle between the
 corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
 tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
 The number of sweeps is given in NSWEEP and the orthogonality threshold
 is given in TOL.

Parameters:

JOBV

          JOBV is CHARACTER*1
          Specifies whether the output from this procedure is used
          to compute the matrix V:
          = 'V': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the N-by-N array V.
                (See the description of V.)
          = 'A': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the MV-by-N array V.
                (See the descriptions of MV and V.)
          = 'N': the Jacobi rotations are not accumulated.

M

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the input matrix A.
          M >= N >= 0.

N1

          N1 is INTEGER
          N1 specifies the 2 x 2 block partition, the first N1 columns are
          rotated 'against' the remaining N-N1 columns of A.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, M-by-N matrix A, such that A*diag(D) represents
          the input matrix.
          On exit,
          A_onexit * D_onexit represents the input matrix A*diag(D)
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of N1, D, TOL and NSWEEP.)

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is COMPLEX array, dimension (N)
          The array D accumulates the scaling factors from the fast scaled
          Jacobi rotations.
          On entry, A*diag(D) represents the input matrix.
          On exit, A_onexit*diag(D_onexit) represents the input matrix
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of N1, A, TOL and NSWEEP.)

SVA

          SVA is REAL array, dimension (N)
          On entry, SVA contains the Euclidean norms of the columns of
          the matrix A*diag(D).
          On exit, SVA contains the Euclidean norms of the columns of
          the matrix onexit*diag(D_onexit).

MV

          MV is INTEGER
          If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'N',   then MV is not referenced.

V

          V is COMPLEX array, dimension (LDV,N)
          If JOBV .EQ. 'V' then N rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'N',   then V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V,  LDV >= 1.
          If JOBV = 'V', LDV .GE. N.
          If JOBV = 'A', LDV .GE. MV.

EPS

          EPS is REAL
          EPS = SLAMCH('Epsilon')

SFMIN

          SFMIN is REAL
          SFMIN = SLAMCH('Safe Minimum')

TOL

          TOL is REAL
          TOL is the threshold for Jacobi rotations. For a pair
          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
          applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.

NSWEEP

          NSWEEP is INTEGER
          NSWEEP is the number of sweeps of Jacobi rotations to be
          performed.

WORK

         WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          LWORK is the dimension of WORK. LWORK .GE. M.

INFO

          INFO is INTEGER
          = 0 : successful exit.
          < 0 : if INFO = -i, then the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributor:

Zlatko Drmac (Zagreb, Croatia)

subroutine chbgst (character VECT, character UPLO, integer N, integer KA, integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB, complex, dimension( ldx, * ) X, integer LDX, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CHBGST

Purpose:

 CHBGST reduces a complex Hermitian-definite banded generalized
 eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
 such that C has the same bandwidth as A.
 B must have been previously factorized as S**H*S by CPBSTF, using a
 split Cholesky factorization. A is overwritten by C = X**H*A*X, where
 X = S**(-1)*Q and Q is a unitary matrix chosen to preserve the
 bandwidth of A.

Parameters:

VECT

          VECT is CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

KA

          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

KB

          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= KB >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
          On exit, the transformed matrix X**H*A*X, stored in the same
          format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

BB

          BB is COMPLEX array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of
          B, as returned by CPBSTF, stored in the first kb+1 rows of
          the array.

LDBB

          LDBB is INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

X

          X is COMPLEX array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X.
          If VECT = 'N', the array X is not referenced.

LDX

          LDX is INTEGER
          The leading dimension of the array X.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

WORK

          WORK is COMPLEX array, dimension (N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chbtrd (character VECT, character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( * ) WORK, integer INFO)

CHBTRD

Purpose:

 CHBTRD reduces a complex Hermitian band matrix A to real symmetric
 tridiagonal form T by a unitary similarity transformation:
 Q**H * A * Q = T.

Parameters:

VECT

          VECT is CHARACTER*1
          = 'N':  do not form Q;
          = 'V':  form Q;
          = 'U':  update a matrix X, by forming X*Q.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by
          values generated during the reduction.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

D

          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

Q

          Q is COMPLEX array, dimension (LDQ,N)
          On entry, if VECT = 'U', then Q must contain an N-by-N
          matrix X; if VECT = 'N' or 'V', then Q need not be set.
          On exit:
          if VECT = 'V', Q contains the N-by-N unitary matrix Q;
          if VECT = 'U', Q contains the product X*Q;
          if VECT = 'N', the array Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.

WORK

          WORK is COMPLEX array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  Modified by Linda Kaufman, Bell Labs.

subroutine chfrk (character TRANSR, character UPLO, character TRANS, integer N, integer K, real ALPHA, complex, dimension( lda, * ) A, integer LDA, real BETA, complex, dimension( * ) C)

CHFRK performs a Hermitian rank-k operation for matrix in RFP format.

Purpose:

 Level 3 BLAS like routine for C in RFP Format.
 CHFRK performs one of the Hermitian rank--k operations
    C := alpha*A*A**H + beta*C,
 or
    C := alpha*A**H*A + beta*C,
 where alpha and beta are real scalars, C is an n--by--n Hermitian
 matrix and A is an n--by--k matrix in the first case and a k--by--n
 matrix in the second case.

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal Form of RFP A is stored;
          = 'C':  The Conjugate-transpose Form of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
           On  entry,   UPLO  specifies  whether  the  upper  or  lower
           triangular  part  of the  array  C  is to be  referenced  as
           follows:
              UPLO = 'U' or 'u'   Only the  upper triangular part of  C
                                  is to be referenced.
              UPLO = 'L' or 'l'   Only the  lower triangular part of  C
                                  is to be referenced.
           Unchanged on exit.

TRANS

          TRANS is CHARACTER*1
           On entry,  TRANS  specifies the operation to be performed as
           follows:
              TRANS = 'N' or 'n'   C := alpha*A*A**H + beta*C.
              TRANS = 'C' or 'c'   C := alpha*A**H*A + beta*C.
           Unchanged on exit.

N

          N is INTEGER
           On entry,  N specifies the order of the matrix C.  N must be
           at least zero.
           Unchanged on exit.

K

          K is INTEGER
           On entry with  TRANS = 'N' or 'n',  K  specifies  the number
           of  columns   of  the   matrix   A,   and  on   entry   with
           TRANS = 'C' or 'c',  K  specifies  the number of rows of the
           matrix A.  K must be at least zero.
           Unchanged on exit.

ALPHA

          ALPHA is REAL
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.

A

          A is COMPLEX array, dimension (LDA,ka)
           where KA
           is K  when TRANS = 'N' or 'n', and is N otherwise. Before
           entry with TRANS = 'N' or 'n', the leading N--by--K part of
           the array A must contain the matrix A, otherwise the leading
           K--by--N part of the array A must contain the matrix A.
           Unchanged on exit.

LDA

          LDA is INTEGER
           On entry, LDA specifies the first dimension of A as declared
           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
           then  LDA must be at least  max( 1, n ), otherwise  LDA must
           be at least  max( 1, k ).
           Unchanged on exit.

BETA

          BETA is REAL
           On entry, BETA specifies the scalar beta.
           Unchanged on exit.

C

          C is COMPLEX array, dimension (N*(N+1)/2)
           On entry, the matrix A in RFP Format. RFP Format is
           described by TRANSR, UPLO and N. Note that the imaginary
           parts of the diagonal elements need not be set, they are
           assumed to be zero, and on exit they are set to zero.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chpcon (character UPLO, integer N, complex, dimension( * ) AP, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

CHPCON

Purpose:

 CHPCON estimates the reciprocal of the condition number of a complex
 Hermitian packed matrix A using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHPTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHPTRF, stored as a
          packed triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHPTRF.

ANORM

          ANORM is REAL
          The 1-norm of the original matrix A.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is COMPLEX array, dimension (2*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chpgst (integer ITYPE, character UPLO, integer N, complex, dimension( * ) AP, complex, dimension( * ) BP, integer INFO)

CHPGST

Purpose:

 CHPGST reduces a complex Hermitian-definite generalized
 eigenproblem to standard form, using packed storage.
 If ITYPE = 1, the problem is A*x = lambda*B*x,
 and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)
 If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
 B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H*A*L.
 B must have been previously factorized as U**H*U or L*L**H by CPPTRF.

Parameters:

ITYPE

          ITYPE is INTEGER
          = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
          = 2 or 3: compute U*A*U**H or L**H*A*L.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**H*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**H.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the Hermitian matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

BP

          BP is COMPLEX array, dimension (N*(N+1)/2)
          The triangular factor from the Cholesky factorization of B,
          stored in the same format as A, as returned by CPPTRF.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chprfs (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( * ) AFP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CHPRFS

Purpose:

 CHPRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is Hermitian indefinite
 and packed, and provides error bounds and backward error estimates
 for the solution.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the Hermitian matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

AFP

          AFP is COMPLEX array, dimension (N*(N+1)/2)
          The factored form of the matrix A.  AFP contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**H or
          A = L*D*L**H as computed by CHPTRF, stored as a packed
          triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHPTRF.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by CHPTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

  ITMAX is the maximum number of steps of iterative refinement.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chptrd (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, integer INFO)

CHPTRD

Purpose:

 CHPTRD reduces a complex Hermitian matrix A stored in packed form to
 real symmetric tridiagonal form T by a unitary similarity
 transformation: Q**H * A * Q = T.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the Hermitian matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the unitary
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the unitary matrix Q as a product
          of elementary reflectors. See Further Details.

D

          D is REAL array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

E

          E is REAL array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

TAU

          TAU is COMPLEX array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(n-1) . . . H(2) H(1).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(n-1).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
  overwriting A(i+2:n,i), and tau is stored in TAU(i).

subroutine chptrf (character UPLO, integer N, complex, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)

CHPTRF

Purpose:

 CHPTRF computes the factorization of a complex Hermitian packed
 matrix A using the Bunch-Kaufman diagonal pivoting method:
    A = U*D*U**H  or  A = L*D*L**H
 where U (or L) is a product of permutation and unit upper (lower)
 triangular matrices, and D is Hermitian and block diagonal with
 1-by-1 and 2-by-2 diagonal blocks.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the Hermitian matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L, stored as a packed triangular
          matrix overwriting A (see below for further details).

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  If UPLO = 'U', then A = U*D*U**H, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k
  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).
  If UPLO = 'L', then A = L*D*L**H, where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1
  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

J. Lewis, Boeing Computer Services Company

subroutine chptri (character UPLO, integer N, complex, dimension( * ) AP, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer INFO)

CHPTRI

Purpose:

 CHPTRI computes the inverse of a complex Hermitian indefinite matrix
 A in packed storage using the factorization A = U*D*U**H or
 A = L*D*L**H computed by CHPTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by CHPTRF,
          stored as a packed triangular matrix.
          On exit, if INFO = 0, the (Hermitian) inverse of the original
          matrix, stored as a packed triangular matrix. The j-th column
          of inv(A) is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
          if UPLO = 'L',
             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHPTRF.

WORK

          WORK is COMPLEX array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chptrs (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CHPTRS

Purpose:

 CHPTRS solves a system of linear equations A*X = B with a complex
 Hermitian matrix A stored in packed format using the factorization
 A = U*D*U**H or A = L*D*L**H computed by CHPTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**H;
          = 'L':  Lower triangular, form is A = L*D*L**H.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CHPTRF, stored as a
          packed triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CHPTRF.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine chsein (character SIDE, character EIGSRC, character INITV, logical, dimension( * ) SELECT, integer N, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IFAILL, integer, dimension( * ) IFAILR, integer INFO)

CHSEIN

Purpose:

 CHSEIN uses inverse iteration to find specified right and/or left
 eigenvectors of a complex upper Hessenberg matrix H.
 The right eigenvector x and the left eigenvector y of the matrix H
 corresponding to an eigenvalue w are defined by:
              H * x = w * x,     y**h * H = w * y**h
 where y**h denotes the conjugate transpose of the vector y.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'R': compute right eigenvectors only;
          = 'L': compute left eigenvectors only;
          = 'B': compute both right and left eigenvectors.

EIGSRC

          EIGSRC is CHARACTER*1
          Specifies the source of eigenvalues supplied in W:
          = 'Q': the eigenvalues were found using CHSEQR; thus, if
                 H has zero subdiagonal elements, and so is
                 block-triangular, then the j-th eigenvalue can be
                 assumed to be an eigenvalue of the block containing
                 the j-th row/column.  This property allows CHSEIN to
                 perform inverse iteration on just one diagonal block.
          = 'N': no assumptions are made on the correspondence
                 between eigenvalues and diagonal blocks.  In this
                 case, CHSEIN must always perform inverse iteration
                 using the whole matrix H.

INITV

          INITV is CHARACTER*1
          = 'N': no initial vectors are supplied;
          = 'U': user-supplied initial vectors are stored in the arrays
                 VL and/or VR.

SELECT

          SELECT is LOGICAL array, dimension (N)
          Specifies the eigenvectors to be computed. To select the
          eigenvector corresponding to the eigenvalue W(j),
          SELECT(j) must be set to .TRUE..

N

          N is INTEGER
          The order of the matrix H.  N >= 0.

H

          H is COMPLEX array, dimension (LDH,N)
          The upper Hessenberg matrix H.
          If a NaN is detected in H, the routine will return with INFO=-6.

LDH

          LDH is INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

W

          W is COMPLEX array, dimension (N)
          On entry, the eigenvalues of H.
          On exit, the real parts of W may have been altered since
          close eigenvalues are perturbed slightly in searching for
          independent eigenvectors.

VL

          VL is COMPLEX array, dimension (LDVL,MM)
          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
          contain starting vectors for the inverse iteration for the
          left eigenvectors; the starting vector for each eigenvector
          must be in the same column in which the eigenvector will be
          stored.
          On exit, if SIDE = 'L' or 'B', the left eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VL, in the same order as their eigenvalues.
          If SIDE = 'R', VL is not referenced.

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

VR

          VR is COMPLEX array, dimension (LDVR,MM)
          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
          contain starting vectors for the inverse iteration for the
          right eigenvectors; the starting vector for each eigenvector
          must be in the same column in which the eigenvector will be
          stored.
          On exit, if SIDE = 'R' or 'B', the right eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VR, in the same order as their eigenvalues.
          If SIDE = 'L', VR is not referenced.

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

MM

          MM is INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

M

          M is INTEGER
          The number of columns in the arrays VL and/or VR required to
          store the eigenvectors (= the number of .TRUE. elements in
          SELECT).

WORK

          WORK is COMPLEX array, dimension (N*N)

RWORK

          RWORK is REAL array, dimension (N)

IFAILL

          IFAILL is INTEGER array, dimension (MM)
          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
          eigenvector in the i-th column of VL (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
          eigenvector converged satisfactorily.
          If SIDE = 'R', IFAILL is not referenced.

IFAILR

          IFAILR is INTEGER array, dimension (MM)
          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
          eigenvector in the i-th column of VR (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
          eigenvector converged satisfactorily.
          If SIDE = 'L', IFAILR is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, i is the number of eigenvectors which
                failed to converge; see IFAILL and IFAILR for further
                details.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x|+|y|.

subroutine chseqr (character JOB, character COMPZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHSEQR

Purpose:

    CHSEQR computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**H, where T is an upper triangular matrix (the
    Schur form), and Z is the unitary matrix of Schur vectors.
    Optionally Z may be postmultiplied into an input unitary
    matrix Q so that this routine can give the Schur factorization
    of a matrix A which has been reduced to the Hessenberg form H
    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*T*(QZ)**H.

Parameters:

JOB

          JOB is CHARACTER*1
           = 'E':  compute eigenvalues only;
           = 'S':  compute eigenvalues and the Schur form T.

COMPZ

          COMPZ is CHARACTER*1
           = 'N':  no Schur vectors are computed;
           = 'I':  Z is initialized to the unit matrix and the matrix Z
                   of Schur vectors of H is returned;
           = 'V':  Z must contain an unitary matrix Q on entry, and
                   the product Q*Z is returned.

N

          N is INTEGER
           The order of the matrix H.  N .GE. 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
           set by a previous call to CGEBAL, and then passed to ZGEHRD
           when the matrix output by CGEBAL is reduced to Hessenberg
           form. Otherwise ILO and IHI should be set to 1 and N
           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
           If N = 0, then ILO = 1 and IHI = 0.

H

          H is COMPLEX array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and JOB = 'S', H contains the upper
           triangular matrix T from the Schur decomposition (the
           Schur form). If INFO = 0 and JOB = 'E', the contents of
           H are unspecified on exit.  (The output value of H when
           INFO.GT.0 is given under the description of INFO below.)
           Unlike earlier versions of CHSEQR, this subroutine may
           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
           or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER
           The leading dimension of the array H. LDH .GE. max(1,N).

W

          W is COMPLEX array, dimension (N)
           The computed eigenvalues. If JOB = 'S', the eigenvalues are
           stored in the same order as on the diagonal of the Schur
           form returned in H, with W(i) = H(i,i).

Z

          Z is COMPLEX array, dimension (LDZ,N)
           If COMPZ = 'N', Z is not referenced.
           If COMPZ = 'I', on entry Z need not be set and on exit,
           if INFO = 0, Z contains the unitary matrix Z of the Schur
           vectors of H.  If COMPZ = 'V', on entry Z must contain an
           N-by-N matrix Q, which is assumed to be equal to the unit
           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
           if INFO = 0, Z contains Q*Z.
           Normally Q is the unitary matrix generated by CUNGHR
           after the call to CGEHRD which formed the Hessenberg matrix
           H. (The output value of Z when INFO.GT.0 is given under
           the description of INFO below.)

LDZ

          LDZ is INTEGER
           The leading dimension of the array Z.  if COMPZ = 'I' or
           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.

WORK

          WORK is COMPLEX array, dimension (LWORK)
           On exit, if INFO = 0, WORK(1) returns an estimate of
           the optimal value for LWORK.

LWORK

          LWORK is INTEGER
           The dimension of the array WORK.  LWORK .GE. max(1,N)
           is sufficient and delivers very good and sometimes
           optimal performance.  However, LWORK as large as 11*N
           may be required for optimal performance.  A workspace
           query is recommended to determine the optimal workspace
           size.
           If LWORK = -1, then CHSEQR does a workspace query.
           In this case, CHSEQR checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER
             =  0:  successful exit
           .LT. 0:  if INFO = -i, the i-th argument had an illegal
                    value
           .GT. 0:  if INFO = i, CHSEQR failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)
                If INFO .GT. 0 and JOB = 'E', then on exit, the
                remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.
                If INFO .GT. 0 and JOB   = 'S', then on exit
           (*)  (initial value of H)*U  = U*(final value of H)
                where U is a unitary matrix.  The final
                value of  H is upper Hessenberg and triangular in
                rows and columns INFO+1 through IHI.
                If INFO .GT. 0 and COMPZ = 'V', then on exit
                  (final value of Z)  =  (initial value of Z)*U
                where U is the unitary matrix in (*) (regard-
                less of the value of JOB.)
                If INFO .GT. 0 and COMPZ = 'I', then on exit
                      (final value of Z)  = U
                where U is the unitary matrix in (*) (regard-
                less of the value of JOB.)
                If INFO .GT. 0 and COMPZ = 'N', then Z is not
                accessed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Further Details:

             Default values supplied by
             ILAENV(ISPEC,'CHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
             It is suggested that these defaults be adjusted in order
             to attain best performance in each particular
             computational environment.
            ISPEC=12: The CLAHQR vs CLAQR0 crossover point.
                      Default: 75. (Must be at least 11.)
            ISPEC=13: Recommended deflation window size.
                      This depends on ILO, IHI and NS.  NS is the
                      number of simultaneous shifts returned
                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
                      The default for (IHI-ILO+1).LE.500 is NS.
                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.
            ISPEC=14: Nibble crossover point. (See IPARMQ for
                      details.)  Default: 14% of deflation window
                      size.
            ISPEC=15: Number of simultaneous shifts in a multishift
                      QR iteration.
                      If IHI-ILO+1 is ...
                      greater than      ...but less    ... the
                      or equal to ...      than        default is
                           1               30          NS =   2(+)
                          30               60          NS =   4(+)
                          60              150          NS =  10(+)
                         150              590          NS =  **
                         590             3000          NS =  64
                        3000             6000          NS = 128
                        6000             infinity      NS = 256
                  (+)  By default some or all matrices of this order
                       are passed to the implicit double shift routine
                       CLAHQR and this parameter is ignored.  See
                       ISPEC=12 above and comments in IPARMQ for
                       details.
                 (**)  The asterisks (**) indicate an ad-hoc
                       function of N increasing from 10 to 64.
            ISPEC=16: Select structured matrix multiply.
                      If the number of simultaneous shifts (specified
                      by ISPEC=15) is less than 14, then the default
                      for ISPEC=16 is 0.  Otherwise the default for
                      ISPEC=16 is 2.

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

subroutine cla_lin_berr (integer N, integer NZ, integer NRHS, complex, dimension( n, nrhs ) RES, real, dimension( n, nrhs ) AYB, real, dimension( nrhs ) BERR)

CLA_LIN_BERR computes a component-wise relative backward error.

Purpose:

    CLA_LIN_BERR computes componentwise relative backward error from
    the formula
        max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
    where abs(Z) is the componentwise absolute value of the matrix
    or vector Z.

Parameters:

N

          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.

NZ

          NZ is INTEGER
     We add (NZ+1)*SLAMCH( 'Safe minimum' ) to R(i) in the numerator to
     guard against spuriously zero residuals. Default value is N.

NRHS

          NRHS is INTEGER
     The number of right hand sides, i.e., the number of columns
     of the matrices AYB, RES, and BERR.  NRHS >= 0.

RES

          RES is COMPLEX array, dimension (N,NRHS)
     The residual matrix, i.e., the matrix R in the relative backward
     error formula above.

AYB

          AYB is REAL array, dimension (N, NRHS)
     The denominator in the relative backward error formula above, i.e.,
     the matrix abs(op(A_s))*abs(Y) + abs(B_s). The matrices A, Y, and B
     are from iterative refinement (see cla_gerfsx_extended.f).

BERR

          BERR is REAL array, dimension (NRHS)
     The componentwise relative backward error from the formula above.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine cla_wwaddw (integer N, complex, dimension( * ) X, complex, dimension( * ) Y, complex, dimension( * ) W)

CLA_WWADDW adds a vector into a doubled-single vector.

Purpose:

    CLA_WWADDW adds a vector W into a doubled-single vector (X, Y).
    This works for all extant IBM's hex and binary floating point
    arithmetics, but not for decimal.

Parameters:

N

          N is INTEGER
            The length of vectors X, Y, and W.

X

          X is COMPLEX array, dimension (N)
            The first part of the doubled-single accumulation vector.

Y

          Y is COMPLEX array, dimension (N)
            The second part of the doubled-single accumulation vector.

W

          W is COMPLEX array, dimension (N)
            The vector to be added.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine claed0 (integer QSIZ, integer N, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldqs, * ) QSTORE, integer LDQS, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)

CLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.

Purpose:

 Using the divide and conquer method, CLAED0 computes all eigenvalues
 of a symmetric tridiagonal matrix which is one diagonal block of
 those from reducing a dense or band Hermitian matrix and
 corresponding eigenvectors of the dense or band matrix.

Parameters:

QSIZ

          QSIZ is INTEGER
         The dimension of the unitary matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

N

          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

D

          D is REAL array, dimension (N)
         On entry, the diagonal elements of the tridiagonal matrix.
         On exit, the eigenvalues in ascending order.

E

          E is REAL array, dimension (N-1)
         On entry, the off-diagonal elements of the tridiagonal matrix.
         On exit, E has been destroyed.

Q

          Q is COMPLEX array, dimension (LDQ,N)
         On entry, Q must contain an QSIZ x N matrix whose columns
         unitarily orthonormal. It is a part of the unitary matrix
         that reduces the full dense Hermitian matrix to a
         (reducible) symmetric tridiagonal matrix.

LDQ

          LDQ is INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

IWORK

          IWORK is INTEGER array,
         the dimension of IWORK must be at least
                      6 + 6*N + 5*N*lg N
                      ( lg( N ) = smallest integer k
                                  such that 2^k >= N )

RWORK

          RWORK is REAL array,
                               dimension (1 + 3*N + 2*N*lg N + 3*N**2)
                        ( lg( N ) = smallest integer k
                                    such that 2^k >= N )

QSTORE

          QSTORE is COMPLEX array, dimension (LDQS, N)
         Used to store parts of
         the eigenvector matrix when the updating matrix multiplies
         take place.

LDQS

          LDQS is INTEGER
         The leading dimension of the array QSTORE.
         LDQS >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine claed7 (integer N, integer CUTPNT, integer QSIZ, integer TLVLS, integer CURLVL, integer CURPBM, real, dimension( * ) D, complex, dimension( ldq, * ) Q, integer LDQ, real RHO, integer, dimension( * ) INDXQ, real, dimension( * ) QSTORE, integer, dimension( * ) QPTR, integer, dimension( * ) PRMPTR, integer, dimension( * ) PERM, integer, dimension( * ) GIVPTR, integer, dimension( 2, * ) GIVCOL, real, dimension( 2, * ) GIVNUM, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)

CLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

 CLAED7 computes the updated eigensystem of a diagonal
 matrix after modification by a rank-one symmetric matrix. This
 routine is used only for the eigenproblem which requires all
 eigenvalues and optionally eigenvectors of a dense or banded
 Hermitian matrix that has been reduced to tridiagonal form.
   T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)
   where Z = Q**Hu, u is a vector of length N with ones in the
   CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
    The eigenvectors of the original matrix are stored in Q, and the
    eigenvalues are in D.  The algorithm consists of three stages:
       The first stage consists of deflating the size of the problem
       when there are multiple eigenvalues or if there is a zero in
       the Z vector.  For each such occurrence the dimension of the
       secular equation problem is reduced by one.  This stage is
       performed by the routine SLAED2.
       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine SLAED4 (as called by SLAED3).
       This routine also calculates the eigenvectors of the current
       problem.
       The final stage consists of computing the updated eigenvectors
       directly using the updated eigenvalues.  The eigenvectors for
       the current problem are multiplied with the eigenvectors from
       the overall problem.

Parameters:

N

          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

CUTPNT

          CUTPNT is INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

QSIZ

          QSIZ is INTEGER
         The dimension of the unitary matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N.

TLVLS

          TLVLS is INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

CURLVL

          CURLVL is INTEGER
         The current level in the overall merge routine,
         0 <= curlvl <= tlvls.

CURPBM

          CURPBM is INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

D

          D is REAL array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

Q

          Q is COMPLEX array, dimension (LDQ,N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ

          LDQ is INTEGER
         The leading dimension of the array Q.  LDQ >= max(1,N).

RHO

          RHO is REAL
         Contains the subdiagonal element used to create the rank-1
         modification.

INDXQ

          INDXQ is INTEGER array, dimension (N)
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order,
         ie. D( INDXQ( I = 1, N ) ) will be in ascending order.

IWORK

          IWORK is INTEGER array, dimension (4*N)

RWORK

          RWORK is REAL array,
                                 dimension (3*N+2*QSIZ*N)

WORK

          WORK is COMPLEX array, dimension (QSIZ*N)

QSTORE

          QSTORE is REAL array, dimension (N**2+1)
         Stores eigenvectors of submatrices encountered during
         divide and conquer, packed together. QPTR points to
         beginning of the submatrices.

QPTR

          QPTR is INTEGER array, dimension (N+2)
         List of indices pointing to beginning of submatrices stored
         in QSTORE. The submatrices are numbered starting at the
         bottom left of the divide and conquer tree, from left to
         right and bottom to top.

PRMPTR

          PRMPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and also the size of
         the full, non-deflated problem.

PERM

          PERM is INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
         indicates the number of Givens rotations.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is REAL array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine claed8 (integer K, integer N, integer QSIZ, complex, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) D, real RHO, integer CUTPNT, real, dimension( * ) Z, real, dimension( * ) DLAMDA, complex, dimension( ldq2, * ) Q2, integer LDQ2, real, dimension( * ) W, integer, dimension( * ) INDXP, integer, dimension( * ) INDX, integer, dimension( * ) INDXQ, integer, dimension( * ) PERM, integer GIVPTR, integer, dimension( 2, * ) GIVCOL, real, dimension( 2, * ) GIVNUM, integer INFO)

CLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

 CLAED8 merges the two sets of eigenvalues together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 eigenvalues are close together or if there is a tiny element in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.

Parameters:

K

          K is INTEGER
         Contains the number of non-deflated eigenvalues.
         This is the order of the related secular equation.

N

          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.

QSIZ

          QSIZ is INTEGER
         The dimension of the unitary matrix used to reduce
         the dense or band matrix to tridiagonal form.
         QSIZ >= N if ICOMPQ = 1.

Q

          Q is COMPLEX array, dimension (LDQ,N)
         On entry, Q contains the eigenvectors of the partially solved
         system which has been previously updated in matrix
         multiplies with other partially solved eigensystems.
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

LDQ

          LDQ is INTEGER
         The leading dimension of the array Q.  LDQ >= max( 1, N ).

D

          D is REAL array, dimension (N)
         On entry, D contains the eigenvalues of the two submatrices to
         be combined.  On exit, D contains the trailing (N-K) updated
         eigenvalues (those which were deflated) sorted into increasing
         order.

RHO

          RHO is REAL
         Contains the off diagonal element associated with the rank-1
         cut which originally split the two submatrices which are now
         being recombined. RHO is modified during the computation to
         the value required by SLAED3.

CUTPNT

          CUTPNT is INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  MIN(1,N) <= CUTPNT <= N.

Z

          Z is REAL array, dimension (N)
         On input this vector contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).  The contents of Z are
         destroyed during the updating process.

DLAMDA

          DLAMDA is REAL array, dimension (N)
         Contains a copy of the first K eigenvalues which will be used
         by SLAED3 to form the secular equation.

Q2

          Q2 is COMPLEX array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
         Contains a copy of the first K eigenvectors which will be used
         by SLAED7 in a matrix multiply (SGEMM) to update the new
         eigenvectors.

LDQ2

          LDQ2 is INTEGER
         The leading dimension of the array Q2.  LDQ2 >= max( 1, N ).

W

          W is REAL array, dimension (N)
         This will hold the first k values of the final
         deflation-altered z-vector and will be passed to SLAED3.

INDXP

          INDXP is INTEGER array, dimension (N)
         This will contain the permutation used to place deflated
         values of D at the end of the array. On output INDXP(1:K)
         points to the nondeflated D-values and INDXP(K+1:N)
         points to the deflated eigenvalues.

INDX

          INDX is INTEGER array, dimension (N)
         This will contain the permutation used to sort the contents of
         D into ascending order.

INDXQ

          INDXQ is INTEGER array, dimension (N)
         This contains the permutation which separately sorts the two
         sub-problems in D into ascending order.  Note that elements in
         the second half of this permutation must first have CUTPNT
         added to their values in order to be accurate.

PERM

          PERM is INTEGER array, dimension (N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER
         Contains the number of Givens rotations which took place in
         this subproblem.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is REAL array, dimension (2, N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine clals0 (integer ICOMPQ, integer NL, integer NR, integer SQRE, integer NRHS, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldbx, * ) BX, integer LDBX, integer, dimension( * ) PERM, integer GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, real, dimension( ldgnum, * ) GIVNUM, integer LDGNUM, real, dimension( ldgnum, * ) POLES, real, dimension( * ) DIFL, real, dimension( ldgnum, * ) DIFR, real, dimension( * ) Z, integer K, real C, real S, real, dimension( * ) RWORK, integer INFO)

CLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

 CLALS0 applies back the multiplying factors of either the left or the
 right singular vector matrix of a diagonal matrix appended by a row
 to the right hand side matrix B in solving the least squares problem
 using the divide-and-conquer SVD approach.
 For the left singular vector matrix, three types of orthogonal
 matrices are involved:
 (1L) Givens rotations: the number of such rotations is GIVPTR; the
      pairs of columns/rows they were applied to are stored in GIVCOL;
      and the C- and S-values of these rotations are stored in GIVNUM.
 (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
      J-th row.
 (3L) The left singular vector matrix of the remaining matrix.
 For the right singular vector matrix, four types of orthogonal
 matrices are involved:
 (1R) The right singular vector matrix of the remaining matrix.
 (2R) If SQRE = 1, one extra Givens rotation to generate the right
      null space.
 (3R) The inverse transformation of (2L).
 (4R) The inverse transformation of (1L).

Parameters:

ICOMPQ

          ICOMPQ is INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Left singular vector matrix.
         = 1: Right singular vector matrix.

NL

          NL is INTEGER
         The row dimension of the upper block. NL >= 1.

NR

          NR is INTEGER
         The row dimension of the lower block. NR >= 1.

SQRE

          SQRE is INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

NRHS

          NRHS is INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

B

          B is COMPLEX array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M. On output, B contains
         the solution X in rows 1 through N.

LDB

          LDB is INTEGER
         The leading dimension of B. LDB must be at least
         max(1,MAX( M, N ) ).

BX

          BX is COMPLEX array, dimension ( LDBX, NRHS )

LDBX

          LDBX is INTEGER
         The leading dimension of BX.

PERM

          PERM is INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) applied
         to the two blocks.

GIVPTR

          GIVPTR is INTEGER
         The number of Givens rotations which took place in this
         subproblem.

GIVCOL

          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of rows/columns
         involved in a Givens rotation.

LDGCOL

          LDGCOL is INTEGER
         The leading dimension of GIVCOL, must be at least N.

GIVNUM

          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value used in the
         corresponding Givens rotation.

LDGNUM

          LDGNUM is INTEGER
         The leading dimension of arrays DIFR, POLES and
         GIVNUM, must be at least K.

POLES

          POLES is REAL array, dimension ( LDGNUM, 2 )
         On entry, POLES(1:K, 1) contains the new singular
         values obtained from solving the secular equation, and
         POLES(1:K, 2) is an array containing the poles in the secular
         equation.

DIFL

          DIFL is REAL array, dimension ( K ).
         On entry, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.

DIFR

          DIFR is REAL array, dimension ( LDGNUM, 2 ).
         On entry, DIFR(I, 1) contains the distances between I-th
         updated (undeflated) singular value and the I+1-th
         (undeflated) old singular value. And DIFR(I, 2) is the
         normalizing factor for the I-th right singular vector.

Z

          Z is REAL array, dimension ( K )
         Contain the components of the deflation-adjusted updating row
         vector.

K

          K is INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

C

          C is REAL
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

S

          S is REAL
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

RWORK

          RWORK is REAL array, dimension
         ( K*(1+NRHS) + 2*NRHS )

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA

subroutine clalsa (integer ICOMPQ, integer SMLSIZ, integer N, integer NRHS, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldbx, * ) BX, integer LDBX, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldu, * ) VT, integer, dimension( * ) K, real, dimension( ldu, * ) DIFL, real, dimension( ldu, * ) DIFR, real, dimension( ldu, * ) Z, real, dimension( ldu, * ) POLES, integer, dimension( * ) GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, integer, dimension( ldgcol, * ) PERM, real, dimension( ldu, * ) GIVNUM, real, dimension( * ) C, real, dimension( * ) S, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)

CLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:

 CLALSA is an itermediate step in solving the least squares problem
 by computing the SVD of the coefficient matrix in compact form (The
 singular vectors are computed as products of simple orthorgonal
 matrices.).
 If ICOMPQ = 0, CLALSA applies the inverse of the left singular vector
 matrix of an upper bidiagonal matrix to the right hand side; and if
 ICOMPQ = 1, CLALSA applies the right singular vector matrix to the
 right hand side. The singular vector matrices were generated in
 compact form by CLALSA.

Parameters:

ICOMPQ

          ICOMPQ is INTEGER
         Specifies whether the left or the right singular vector
         matrix is involved.
         = 0: Left singular vector matrix
         = 1: Right singular vector matrix

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The row and column dimensions of the upper bidiagonal matrix.

NRHS

          NRHS is INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

B

          B is COMPLEX array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M.
         On output, B contains the solution X in rows 1 through N.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,MAX( M, N ) ).

BX

          BX is COMPLEX array, dimension ( LDBX, NRHS )
         On exit, the result of applying the left or right singular
         vector matrix to B.

LDBX

          LDBX is INTEGER
         The leading dimension of BX.

U

          U is REAL array, dimension ( LDU, SMLSIZ ).
         On entry, U contains the left singular vector matrices of all
         subproblems at the bottom level.

LDU

          LDU is INTEGER, LDU = > N.
         The leading dimension of arrays U, VT, DIFL, DIFR,
         POLES, GIVNUM, and Z.

VT

          VT is REAL array, dimension ( LDU, SMLSIZ+1 ).
         On entry, VT**H contains the right singular vector matrices of
         all subproblems at the bottom level.

K

          K is INTEGER array, dimension ( N ).

DIFL

          DIFL is REAL array, dimension ( LDU, NLVL ).
         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

DIFR

          DIFR is REAL array, dimension ( LDU, 2 * NLVL ).
         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
         distances between singular values on the I-th level and
         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
         record the normalizing factors of the right singular vectors
         matrices of subproblems on I-th level.

Z

          Z is REAL array, dimension ( LDU, NLVL ).
         On entry, Z(1, I) contains the components of the deflation-
         adjusted updating row vector for subproblems on the I-th
         level.

POLES

          POLES is REAL array, dimension ( LDU, 2 * NLVL ).
         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
         singular values involved in the secular equations on the I-th
         level.

GIVPTR

          GIVPTR is INTEGER array, dimension ( N ).
         On entry, GIVPTR( I ) records the number of Givens
         rotations performed on the I-th problem on the computation
         tree.

GIVCOL

          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
         locations of Givens rotations performed on the I-th level on
         the computation tree.

LDGCOL

          LDGCOL is INTEGER, LDGCOL = > N.
         The leading dimension of arrays GIVCOL and PERM.

PERM

          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
         On entry, PERM(*, I) records permutations done on the I-th
         level of the computation tree.

GIVNUM

          GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ).
         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
         values of Givens rotations performed on the I-th level on the
         computation tree.

C

          C is REAL array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.

S

          S is REAL array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         S( I ) contains the S-value of a Givens rotation related to
         the right null space of the I-th subproblem.

RWORK

          RWORK is REAL array, dimension at least
         MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ).

IWORK

          IWORK is INTEGER array, dimension (3*N)

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA

subroutine clalsd (character UPLO, integer SMLSIZ, integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB, real RCOND, integer RANK, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)

CLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

 CLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.
 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.
 This code makes very mild assumptions about floating point
 arithmetic. It will work on machines with a guard digit in
 add/subtract, or on those binary machines without guard digits
 which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

Parameters:

UPLO

          UPLO is CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER
         The number of columns of B. NRHS must be at least 1.

D

          D is REAL array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is REAL array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

B

          B is COMPLEX array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

RCOND

          RCOND is REAL
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

RANK

          RANK is INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

WORK

          WORK is COMPLEX array, dimension (N * NRHS).

RWORK

          RWORK is REAL array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
         MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
         where
         NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

IWORK

          IWORK is INTEGER array, dimension (3*N*NLVL + 11*N).

INFO

          INFO is INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute a singular value while
               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA

real function clanhf (character NORM, character TRANSR, character UPLO, integer N, complex, dimension( 0: * ) A, real, dimension( 0: * ) WORK)

CLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

Purpose:

 CLANHF  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex Hermitian matrix A in RFP format.

Returns:

CLANHF

    CLANHF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.

Parameters:

NORM

          NORM is CHARACTER
            Specifies the value to be returned in CLANHF as described
            above.

TRANSR

          TRANSR is CHARACTER
            Specifies whether the RFP format of A is normal or
            conjugate-transposed format.
            = 'N':  RFP format is Normal
            = 'C':  RFP format is Conjugate-transposed

UPLO

          UPLO is CHARACTER
            On entry, UPLO specifies whether the RFP matrix A came from
            an upper or lower triangular matrix as follows:
            UPLO = 'U' or 'u' RFP A came from an upper triangular
            matrix
            UPLO = 'L' or 'l' RFP A came from a  lower triangular
            matrix

N

          N is INTEGER
            The order of the matrix A.  N >= 0.  When N = 0, CLANHF is
            set to zero.

A

          A is COMPLEX array, dimension ( N*(N+1)/2 );
            On entry, the matrix A in RFP Format.
            RFP Format is described by TRANSR, UPLO and N as follows:
            If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
            K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
            TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A
            as defined when TRANSR = 'N'. The contents of RFP A are
            defined by UPLO as follows: If UPLO = 'U' the RFP A
            contains the ( N*(N+1)/2 ) elements of upper packed A
            either in normal or conjugate-transpose Format. If
            UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) elements
            of lower packed A either in normal or conjugate-transpose
            Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When
            TRANSR is 'N' the LDA is N+1 when N is even and is N when
            is odd. See the Note below for more details.
            Unchanged on exit.

WORK

          WORK is REAL array, dimension (LWORK),
            where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
            WORK is not referenced.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine clarscl2 (integer M, integer N, real, dimension( * ) D, complex, dimension( ldx, * ) X, integer LDX)

CLARSCL2 performs reciprocal diagonal scaling on a vector.

Purpose:

 CLARSCL2 performs a reciprocal diagonal scaling on an vector:
   x <-- inv(D) * x
 where the REAL diagonal matrix D is stored as a vector.
 Eventually to be replaced by BLAS_cge_diag_scale in the new BLAS
 standard.

Parameters:

M

          M is INTEGER
     The number of rows of D and X. M >= 0.

N

          N is INTEGER
     The number of columns of X. N >= 0.

D

          D is REAL array, length M
     Diagonal matrix D, stored as a vector of length M.

X

          X is COMPLEX array, dimension (LDX,N)
     On entry, the vector X to be scaled by D.
     On exit, the scaled vector.

LDX

          LDX is INTEGER
     The leading dimension of the vector X. LDX >= M.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine clarz (character SIDE, integer M, integer N, integer L, complex, dimension( * ) V, integer INCV, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)

CLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.

Purpose:

 CLARZ applies a complex elementary reflector H to a complex
 M-by-N matrix C, from either the left or the right. H is represented
 in the form
       H = I - tau * v * v**H
 where tau is a complex scalar and v is a complex vector.
 If tau = 0, then H is taken to be the unit matrix.
 To apply H**H (the conjugate transpose of H), supply conjg(tau) instead
 tau.
 H is a product of k elementary reflectors as returned by CTZRZF.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

L

          L is INTEGER
          The number of entries of the vector V containing
          the meaningful part of the Householder vectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

V

          V is COMPLEX array, dimension (1+(L-1)*abs(INCV))
          The vector v in the representation of H as returned by
          CTZRZF. V is not used if TAU = 0.

INCV

          INCV is INTEGER
          The increment between elements of v. INCV <> 0.

TAU

          TAU is COMPLEX
          The value tau in the representation of H.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
          or C * H if SIDE = 'R'.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

 

subroutine clarzb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( ldwork, * ) WORK, integer LDWORK)

CLARZB applies a block reflector or its conjugate-transpose to a general matrix.

Purpose:

 CLARZB applies a complex block reflector H or its transpose H**H
 to a complex distributed M-by-N  C from the left or the right.
 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply H or H**H from the Left
          = 'R': apply H or H**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply H (No transpose)
          = 'C': apply H**H (Conjugate transpose)

DIRECT

          DIRECT is CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise                        (not supported yet)
          = 'R': Rowwise

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

K

          K is INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).

L

          L is INTEGER
          The number of columns of the matrix V containing the
          meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

V

          V is COMPLEX array, dimension (LDV,NV).
          If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.

T

          T is COMPLEX array, dimension (LDT,K)
          The triangular K-by-K matrix T in the representation of the
          block reflector.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (LDWORK,K)

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

 

subroutine clarzt (character DIRECT, character STOREV, integer N, integer K, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( * ) TAU, complex, dimension( ldt, * ) T, integer LDT)

CLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:

 CLARZT forms the triangular factor T of a complex block reflector
 H of order > n, which is defined as a product of k elementary
 reflectors.
 If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
 If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
 If STOREV = 'C', the vector which defines the elementary reflector
 H(i) is stored in the i-th column of the array V, and
    H  =  I - V * T * V**H
 If STOREV = 'R', the vector which defines the elementary reflector
 H(i) is stored in the i-th row of the array V, and
    H  =  I - V**H * T * V
 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters:

DIRECT

          DIRECT is CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise                        (not supported yet)
          = 'R': rowwise

N

          N is INTEGER
          The order of the block reflector H. N >= 0.

K

          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

V

          V is COMPLEX array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

T

          T is COMPLEX array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and
  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.
  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                              ______V_____
         ( v1 v2 v3 )                        /                     ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
     V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
         ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
         ( v1 v2 v3 )
            .  .  .
            .  .  .
            1  .  .
               1  .
                  1
  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                        ______V_____
            1                                          /                        .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
            .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
            .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
            .  .  .
         ( v1 v2 v3 )
         ( v1 v2 v3 )
     V = ( v1 v2 v3 )
         ( v1 v2 v3 )
         ( v1 v2 v3 )

subroutine clascl2 (integer M, integer N, real, dimension( * ) D, complex, dimension( ldx, * ) X, integer LDX)

CLASCL2 performs diagonal scaling on a vector.

Purpose:

 CLASCL2 performs a diagonal scaling on a vector:
   x <-- D * x
 where the diagonal REAL matrix D is stored as a vector.
 Eventually to be replaced by BLAS_cge_diag_scale in the new BLAS
 standard.

Parameters:

M

          M is INTEGER
     The number of rows of D and X. M >= 0.

N

          N is INTEGER
     The number of columns of X. N >= 0.

D

          D is REAL array, length M
     Diagonal matrix D, stored as a vector of length M.

X

          X is COMPLEX array, dimension (LDX,N)
     On entry, the vector X to be scaled by D.
     On exit, the scaled vector.

LDX

          LDX is INTEGER
     The leading dimension of the vector X. LDX >= M.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine clatrz (integer M, integer N, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK)

CLATRZ factors an upper trapezoidal matrix by means of unitary transformations.

Purpose:

 CLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
 [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
 of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
 matrix and, R and A1 are M-by-M upper triangular matrices.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

L

          L is INTEGER
          The number of columns of the matrix A containing the
          meaningful part of the Householder vectors. N-M >= L >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements N-L+1 to
          N of the first M rows of A, with the array TAU, represent the
          unitary matrix Z as a product of M elementary reflectors.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (M)
          The scalar factors of the elementary reflectors.

WORK

          WORK is COMPLEX array, dimension (M)

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into
  the ( m - k + 1 )th row of A, is given in the form
     Z( k ) = ( I     0   ),
              ( 0  T( k ) )
  where
     T( k ) = I - tau*u( k )*u( k )**H,   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )
  tau is a scalar and z( k ) is an l element vector. tau and z( k )
  are chosen to annihilate the elements of the kth row of A2.
  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A2, such that the elements of z( k ) are
  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
  the upper triangular part of A1.
  Z is given by
     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

subroutine cpbcon (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real ANORM, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CPBCON

Purpose:

 CPBCON estimates the reciprocal of the condition number (in the
 1-norm) of a complex Hermitian positive definite band matrix using
 the Cholesky factorization A = U**H*U or A = L*L**H computed by
 CPBTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization
          A = U**H*U or A = L*L**H of the band matrix A, stored in the
          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

ANORM

          ANORM is REAL
          The 1-norm (or infinity-norm) of the Hermitian band matrix A.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpbequ (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)

CPBEQU

Purpose:

 CPBEQU computes row and column scalings intended to equilibrate a
 Hermitian positive definite band matrix A and reduce its condition
 number (with respect to the two-norm).  S contains the scale factors,
 S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
 elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
 choice of S puts the condition number of B within a factor N of the
 smallest possible condition number over all possible diagonal
 scalings.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular of A is stored;
          = 'L':  Lower triangular of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangle of the Hermitian band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column
          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array A.  LDAB >= KD+1.

S

          S is REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

SCOND

          SCOND is REAL
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

AMAX

          AMAX is REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpbrfs (character UPLO, integer N, integer KD, integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldafb, * ) AFB, integer LDAFB, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CPBRFS

Purpose:

 CPBRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is Hermitian positive definite
 and banded, and provides error bounds and backward error estimates
 for the solution.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangle of the Hermitian band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column
          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

AFB

          AFB is COMPLEX array, dimension (LDAFB,N)
          The triangular factor U or L from the Cholesky factorization
          A = U**H*U or A = L*L**H of the band matrix A as computed by
          CPBTRF, in the same storage format as A (see AB).

LDAFB

          LDAFB is INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by CPBTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

  ITMAX is the maximum number of steps of iterative refinement.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpbstf (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, integer INFO)

CPBSTF

Purpose:

 CPBSTF computes a split Cholesky factorization of a complex
 Hermitian positive definite band matrix A.
 This routine is designed to be used in conjunction with CHBGST.
 The factorization has the form  A = S**H*S  where S is a band matrix
 of the same bandwidth as A and the following structure:
   S = ( U    )
       ( M  L )
 where U is upper triangular of order m = (n+kd)/2, and L is lower
 triangular of order n-m.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first kd+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, if INFO = 0, the factor S from the split Cholesky
          factorization A = S**H*S. See Further Details.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the factorization could not be completed,
               because the updated element a(i,i) was negative; the
               matrix A is not positive definite.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The band storage scheme is illustrated by the following example, when
  N = 7, KD = 2:
  S = ( s11  s12  s13                     )
      (      s22  s23  s24                )
      (           s33  s34                )
      (                s44                )
      (           s53  s54  s55           )
      (                s64  s65  s66      )
      (                     s75  s76  s77 )
  If UPLO = 'U', the array AB holds:
  on entry:                          on exit:
   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53**H s64**H s75**H
   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55    s66    s77
  If UPLO = 'L', the array AB holds:
  on entry:                          on exit:
  a11  a22  a33  a44  a55  a66  a77  s11    s22    s33    s44  s55  s66  s77
  a21  a32  a43  a54  a65  a76   *   s12**H s23**H s34**H s54  s65  s76   *
  a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *
  Array elements marked * are not used by the routine; s12**H denotes
  conjg(s12); the diagonal elements of S are real.

subroutine cpbtf2 (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, integer INFO)

CPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

Purpose:

 CPBTF2 computes the Cholesky factorization of a complex Hermitian
 positive definite band matrix A.
 The factorization has the form
    A = U**H * U ,  if UPLO = 'U', or
    A = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix, U**H is the conjugate transpose
 of U, and L is lower triangular.
 This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',
          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**H *U or A = L*L**H of the band
          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite, and the factorization could not be
               completed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':
  On entry:                       On exit:
      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
  Similarly, if UPLO = 'L' the format of A is as follows:
  On entry:                       On exit:
     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
  Array elements marked * are not used by the routine.

subroutine cpbtrf (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, integer INFO)

CPBTRF

Purpose:

 CPBTRF computes the Cholesky factorization of a complex Hermitian
 positive definite band matrix A.
 The factorization has the form
    A = U**H * U,  if UPLO = 'U', or
    A = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the Hermitian band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**H*U or A = L*L**H of the band
          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':
  On entry:                       On exit:
      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
  Similarly, if UPLO = 'L' the format of A is as follows:
  On entry:                       On exit:
     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
  Array elements marked * are not used by the routine.

Contributors:

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

subroutine cpbtrs (character UPLO, integer N, integer KD, integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CPBTRS

Purpose:

 CPBTRS solves a system of linear equations A*X = B with a Hermitian
 positive definite band matrix A using the Cholesky factorization
 A = U**H*U or A = L*L**H computed by CPBTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization
          A = U**H*U or A = L*L**H of the band matrix A, stored in the
          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpftrf (character TRANSR, character UPLO, integer N, complex, dimension( 0: * ) A, integer INFO)

CPFTRF

Purpose:

 CPFTRF computes the Cholesky factorization of a complex Hermitian
 positive definite matrix A.
 The factorization has the form
    A = U**H * U,  if UPLO = 'U', or
    A = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.
 This is the block version of the algorithm, calling Level 3 BLAS.

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of RFP A is stored;
          = 'L':  Lower triangle of RFP A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension ( N*(N+1)/2 );
          On entry, the Hermitian matrix A in RFP format. RFP format is
          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
          the Conjugate-transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the nt elements of
          upper packed A. If UPLO = 'L' the RFP A contains the elements
          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
          'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
          is odd. See the Note below for more details.
          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization RFP A = U**H*U or RFP A = L*L**H.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.
  Further Notes on RFP Format:
  ============================
  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
     AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpftri (character TRANSR, character UPLO, integer N, complex, dimension( 0: * ) A, integer INFO)

CPFTRI

Purpose:

 CPFTRI computes the inverse of a complex Hermitian positive definite
 matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
 computed by CPFTRF.

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension ( N*(N+1)/2 );
          On entry, the Hermitian matrix A in RFP format. RFP format is
          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
          the Conjugate-transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the nt elements of
          upper packed A. If UPLO = 'L' the RFP A contains the elements
          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
          'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
          is odd. See the Note below for more details.
          On exit, the Hermitian inverse of the original matrix, in the
          same storage format.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is
                zero, and the inverse could not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine cpftrs (character TRANSR, character UPLO, integer N, integer NRHS, complex, dimension( 0: * ) A, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CPFTRS

Purpose:

 CPFTRS solves a system of linear equations A*X = B with a Hermitian
 positive definite matrix A using the Cholesky factorization
 A = U**H*U or A = L*L**H computed by CPFTRF.

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of RFP A is stored;
          = 'L':  Lower triangle of RFP A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

A

          A is COMPLEX array, dimension ( N*(N+1)/2 );
          The triangular factor U or L from the Cholesky factorization
          of RFP A = U**H*U or RFP A = L*L**H, as computed by CPFTRF.
          See note below for more details about RFP A.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine cppcon (character UPLO, integer N, complex, dimension( * ) AP, real ANORM, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CPPCON

Purpose:

 CPPCON estimates the reciprocal of the condition number (in the
 1-norm) of a complex Hermitian positive definite packed matrix using
 the Cholesky factorization A = U**H*U or A = L*L**H computed by
 CPPTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization
          A = U**H*U or A = L*L**H, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

ANORM

          ANORM is REAL
          The 1-norm (or infinity-norm) of the Hermitian matrix A.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cppequ (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)

CPPEQU

Purpose:

 CPPEQU computes row and column scalings intended to equilibrate a
 Hermitian positive definite matrix A in packed storage and reduce
 its condition number (with respect to the two-norm).  S contains the
 scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
 B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
 This choice of S puts the condition number of B within a factor N of
 the smallest possible condition number over all possible diagonal
 scalings.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the Hermitian matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

S

          S is REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

SCOND

          SCOND is REAL
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

AMAX

          AMAX is REAL
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpprfs (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( * ) AFP, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CPPRFS

Purpose:

 CPPRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is Hermitian positive definite
 and packed, and provides error bounds and backward error estimates
 for the solution.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the Hermitian matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

AFP

          AFP is COMPLEX array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization
          A = U**H*U or A = L*L**H, as computed by SPPTRF/CPPTRF,
          packed columnwise in a linear array in the same format as A
          (see AP).

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by CPPTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

  ITMAX is the maximum number of steps of iterative refinement.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpptrf (character UPLO, integer N, complex, dimension( * ) AP, integer INFO)

CPPTRF

Purpose:

 CPPTRF computes the Cholesky factorization of a complex Hermitian
 positive definite matrix A stored in packed format.
 The factorization has the form
    A = U**H * U,  if UPLO = 'U', or
    A = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the Hermitian matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.
          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**H*U or A = L*L**H, in the same
          storage format as A.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':
  Two-dimensional storage of the Hermitian matrix A:
     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44
  Packed storage of the upper triangle of A:
  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

subroutine cpptri (character UPLO, integer N, complex, dimension( * ) AP, integer INFO)

CPPTRI

Purpose:

 CPPTRI computes the inverse of a complex Hermitian positive definite
 matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
 computed by CPPTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular factor is stored in AP;
          = 'L':  Lower triangular factor is stored in AP.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the triangular factor U or L from the Cholesky
          factorization A = U**H*U or A = L*L**H, packed columnwise as
          a linear array.  The j-th column of U or L is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
          On exit, the upper or lower triangle of the (Hermitian)
          inverse of A, overwriting the input factor U or L.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is
                zero, and the inverse could not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpptrs (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CPPTRS

Purpose:

 CPPTRS solves a system of linear equations A*X = B with a Hermitian
 positive definite matrix A in packed storage using the Cholesky
 factorization A = U**H*U or A = L*L**H computed by CPPTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization
          A = U**H*U or A = L*L**H, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpstf2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer INFO)

CPSTF2 computes the Cholesky factorization with complete pivoting of complex Hermitian positive semidefinite matrix.

Purpose:

 CPSTF2 computes the Cholesky factorization with complete
 pivoting of a complex Hermitian positive semidefinite matrix A.
 The factorization has the form
    P**T * A * P = U**H * U ,  if UPLO = 'U',
    P**T * A * P = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular, and
 P is stored as vector PIV.
 This algorithm does not attempt to check that A is positive
 semidefinite. This version of the algorithm calls level 2 BLAS.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization as above.

PIV

          PIV is INTEGER array, dimension (N)
          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.

RANK

          RANK is INTEGER
          The rank of A given by the number of steps the algorithm
          completed.

TOL

          TOL is REAL
          User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
          will be used. The algorithm terminates at the (K-1)st step
          if the pivot <= TOL.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

WORK

          WORK is REAL array, dimension (2*N)
          Work space.

INFO

          INFO is INTEGER
          < 0: If INFO = -K, the K-th argument had an illegal value,
          = 0: algorithm completed successfully, and
          > 0: the matrix A is either rank deficient with computed rank
               as returned in RANK, or is not positive semidefinite. See
               Section 7 of LAPACK Working Note #161 for further
               information.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cpstrf (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer INFO)

CPSTRF computes the Cholesky factorization with complete pivoting of complex Hermitian positive semidefinite matrix.

Purpose:

 CPSTRF computes the Cholesky factorization with complete
 pivoting of a complex Hermitian positive semidefinite matrix A.
 The factorization has the form
    P**T * A * P = U**H * U ,  if UPLO = 'U',
    P**T * A * P = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular, and
 P is stored as vector PIV.
 This algorithm does not attempt to check that A is positive
 semidefinite. This version of the algorithm calls level 3 BLAS.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization as above.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

PIV

          PIV is INTEGER array, dimension (N)
          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.

RANK

          RANK is INTEGER
          The rank of A given by the number of steps the algorithm
          completed.

TOL

          TOL is REAL
          User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
          will be used. The algorithm terminates at the (K-1)st step
          if the pivot <= TOL.

WORK

          WORK is REAL array, dimension (2*N)
          Work space.

INFO

          INFO is INTEGER
          < 0: If INFO = -K, the K-th argument had an illegal value,
          = 0: algorithm completed successfully, and
          > 0: the matrix A is either rank deficient with computed rank
               as returned in RANK, or is not positive semidefinite. See
               Section 7 of LAPACK Working Note #161 for further
               information.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cspcon (character UPLO, integer N, complex, dimension( * ) AP, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)

CSPCON

Purpose:

 CSPCON estimates the reciprocal of the condition number (in the
 1-norm) of a complex symmetric packed matrix A using the
 factorization A = U*D*U**T or A = L*D*L**T computed by CSPTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CSPTRF, stored as a
          packed triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CSPTRF.

ANORM

          ANORM is REAL
          The 1-norm of the original matrix A.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is COMPLEX array, dimension (2*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine csprfs (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( * ) AFP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CSPRFS

Purpose:

 CSPRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is symmetric indefinite
 and packed, and provides error bounds and backward error estimates
 for the solution.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

AFP

          AFP is COMPLEX array, dimension (N*(N+1)/2)
          The factored form of the matrix A.  AFP contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by CSPTRF, stored as a packed
          triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CSPTRF.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by CSPTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

  ITMAX is the maximum number of steps of iterative refinement.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine csptrf (character UPLO, integer N, complex, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)

CSPTRF

Purpose:

 CSPTRF computes the factorization of a complex symmetric matrix A
 stored in packed format using the Bunch-Kaufman diagonal pivoting
 method:
    A = U*D*U**T  or  A = L*D*L**T
 where U (or L) is a product of permutation and unit upper (lower)
 triangular matrices, and D is symmetric and block diagonal with
 1-by-1 and 2-by-2 diagonal blocks.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L, stored as a packed triangular
          matrix overwriting A (see below for further details).

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  5-96 - Based on modifications by J. Lewis, Boeing Computer Services
         Company
  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k
  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).
  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1
  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

subroutine csptri (character UPLO, integer N, complex, dimension( * ) AP, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer INFO)

CSPTRI

Purpose:

 CSPTRI computes the inverse of a complex symmetric indefinite matrix
 A in packed storage using the factorization A = U*D*U**T or
 A = L*D*L**T computed by CSPTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by CSPTRF,
          stored as a packed triangular matrix.
          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix, stored as a packed triangular matrix. The j-th column
          of inv(A) is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
          if UPLO = 'L',
             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CSPTRF.

WORK

          WORK is COMPLEX array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine csptrs (character UPLO, integer N, integer NRHS, complex, dimension( * ) AP, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CSPTRS

Purpose:

 CSPTRS solves a system of linear equations A*X = B with a complex
 symmetric matrix A stored in packed format using the factorization
 A = U*D*U**T or A = L*D*L**T computed by CSPTRF.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by CSPTRF, stored as a
          packed triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by CSPTRF.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cstedc (character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CSTEDC

Purpose:

 CSTEDC computes all eigenvalues and, optionally, eigenvectors of a
 symmetric tridiagonal matrix using the divide and conquer method.
 The eigenvectors of a full or band complex Hermitian matrix can also
 be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this
 matrix to tridiagonal form.
 This code makes very mild assumptions about floating point
 arithmetic. It will work on machines with a guard digit in
 add/subtract, or on those binary machines without guard digits
 which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.  See SLAED3 for details.

Parameters:

COMPZ

          COMPZ is CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.
          = 'V':  Compute eigenvectors of original Hermitian matrix
                  also.  On entry, Z contains the unitary matrix used
                  to reduce the original matrix to tridiagonal form.

N

          N is INTEGER
          The dimension of the symmetric tridiagonal matrix.  N >= 0.

D

          D is REAL array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is REAL array, dimension (N-1)
          On entry, the subdiagonal elements of the tridiagonal matrix.
          On exit, E has been destroyed.

Z

          Z is COMPLEX array, dimension (LDZ,N)
          On entry, if COMPZ = 'V', then Z contains the unitary
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original Hermitian matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If eigenvectors are desired, then LDZ >= max(1,N).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1.
          If COMPZ = 'V' and N > 1, LWORK must be at least N*N.
          Note that for COMPZ = 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LWORK need
          only be 1.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal sizes of the WORK, RWORK and
          IWORK arrays, returns these values as the first entries of
          the WORK, RWORK and IWORK arrays, and no error message
          related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (MAX(1,LRWORK))
          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

LRWORK

          LRWORK is INTEGER
          The dimension of the array RWORK.
          If COMPZ = 'N' or N <= 1, LRWORK must be at least 1.
          If COMPZ = 'V' and N > 1, LRWORK must be at least
                         1 + 3*N + 2*N*lg N + 4*N**2 ,
                         where lg( N ) = smallest integer k such
                         that 2**k >= N.
          If COMPZ = 'I' and N > 1, LRWORK must be at least
                         1 + 4*N + 2*N**2 .
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LRWORK
          need only be max(1,2*(N-1)).
          If LRWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK, RWORK
          and IWORK arrays, returns these values as the first entries
          of the WORK, RWORK and IWORK arrays, and no error message
          related to LWORK or LRWORK or LIWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If COMPZ = 'N' or N <= 1, LIWORK must be at least 1.
          If COMPZ = 'V' or N > 1,  LIWORK must be at least
                                    6 + 6*N + 5*N*lg N.
          If COMPZ = 'I' or N > 1,  LIWORK must be at least
                                    3 + 5*N .
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LIWORK
          need only be 1.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal sizes of the WORK, RWORK
          and IWORK arrays, returns these values as the first entries
          of the WORK, RWORK and IWORK arrays, and no error message
          related to LWORK or LRWORK or LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

subroutine cstegr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CSTEGR

Purpose:

 CSTEGR computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
 a well defined set of pairwise different real eigenvalues, the corresponding
 real eigenvectors are pairwise orthogonal.
 The spectrum may be computed either completely or partially by specifying
 either an interval (VL,VU] or a range of indices IL:IU for the desired
 eigenvalues.
 CSTEGR is a compatibility wrapper around the improved CSTEMR routine.
 See SSTEMR for further details.
 One important change is that the ABSTOL parameter no longer provides any
 benefit and hence is no longer used.
 Note : CSTEGR and CSTEMR work only on machines which follow
 IEEE-754 floating-point standard in their handling of infinities and
 NaNs.  Normal execution may create these exceptiona values and hence
 may abort due to a floating point exception in environments which
 do not conform to the IEEE-754 standard.

Parameters:

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          RANGE is CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

E

          E is REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

VL

          VL is REAL
          If RANGE='V', the lower bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is REAL
          If RANGE='V', the upper bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER
          If RANGE='I', the index of the
          smallest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER
          If RANGE='I', the index of the
          largest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

ABSTOL

          ABSTOL is REAL
          Unused.  Was the absolute error tolerance for the
          eigenvalues/eigenvectors in previous versions.

M

          M is INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

Z

          Z is COMPLEX array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.
          Supplying N columns is always safe.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

WORK

          WORK is REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in SLARRE,
                if INFO = 2X, internal error in CLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by SLARRE or
                CLARRV, respectively.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, LBNL/NERSC, USA

subroutine cstein (integer N, real, dimension( * ) D, real, dimension( * ) E, integer M, real, dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, complex, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

CSTEIN

Purpose:

 CSTEIN computes the eigenvectors of a real symmetric tridiagonal
 matrix T corresponding to specified eigenvalues, using inverse
 iteration.
 The maximum number of iterations allowed for each eigenvector is
 specified by an internal parameter MAXITS (currently set to 5).
 Although the eigenvectors are real, they are stored in a complex
 array, which may be passed to CUNMTR or CUPMTR for back
 transformation to the eigenvectors of a complex Hermitian matrix
 which was reduced to tridiagonal form.

Parameters:

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix
          T, stored in elements 1 to N-1.

M

          M is INTEGER
          The number of eigenvectors to be found.  0 <= M <= N.

W

          W is REAL array, dimension (N)
          The first M elements of W contain the eigenvalues for
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block.  ( The output array
          W from SSTEBZ with ORDER = 'B' is expected here. )

IBLOCK

          IBLOCK is INTEGER array, dimension (N)
          The submatrix indices associated with the corresponding
          eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
          the first submatrix from the top, =2 if W(i) belongs to
          the second submatrix, etc.  ( The output array IBLOCK
          from SSTEBZ is expected here. )

ISPLIT

          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.
          The first submatrix consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.
          ( The output array ISPLIT from SSTEBZ is expected here. )

Z

          Z is COMPLEX array, dimension (LDZ, M)
          The computed eigenvectors.  The eigenvector associated
          with the eigenvalue W(i) is stored in the i-th column of
          Z.  Any vector which fails to converge is set to its current
          iterate after MAXITS iterations.
          The imaginary parts of the eigenvectors are set to zero.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= max(1,N).

WORK

          WORK is REAL array, dimension (5*N)

IWORK

          IWORK is INTEGER array, dimension (N)

IFAIL

          IFAIL is INTEGER array, dimension (M)
          On normal exit, all elements of IFAIL are zero.
          If one or more eigenvectors fail to converge after
          MAXITS iterations, then their indices are stored in
          array IFAIL.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, then i eigenvectors failed to converge
               in MAXITS iterations.  Their indices are stored in
               array IFAIL.

Internal Parameters:

  MAXITS  INTEGER, default = 5
          The maximum number of iterations performed.
  EXTRA   INTEGER, default = 2
          The number of iterations performed after norm growth
          criterion is satisfied, should be at least 1.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cstemr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, integer M, real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, integer NZC, integer, dimension( * ) ISUPPZ, logical TRYRAC, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CSTEMR

Purpose:

 CSTEMR computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
 a well defined set of pairwise different real eigenvalues, the corresponding
 real eigenvectors are pairwise orthogonal.
 The spectrum may be computed either completely or partially by specifying
 either an interval (VL,VU] or a range of indices IL:IU for the desired
 eigenvalues.
 Depending on the number of desired eigenvalues, these are computed either
 by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
 computed by the use of various suitable L D L^T factorizations near clusters
 of close eigenvalues (referred to as RRRs, Relatively Robust
 Representations). An informal sketch of the algorithm follows.
 For each unreduced block (submatrix) of T,
    (a) Compute T - sigma I  = L D L^T, so that L and D
        define all the wanted eigenvalues to high relative accuracy.
        This means that small relative changes in the entries of D and L
        cause only small relative changes in the eigenvalues and
        eigenvectors. The standard (unfactored) representation of the
        tridiagonal matrix T does not have this property in general.
    (b) Compute the eigenvalues to suitable accuracy.
        If the eigenvectors are desired, the algorithm attains full
        accuracy of the computed eigenvalues only right before
        the corresponding vectors have to be computed, see steps c) and d).
    (c) For each cluster of close eigenvalues, select a new
        shift close to the cluster, find a new factorization, and refine
        the shifted eigenvalues to suitable accuracy.
    (d) For each eigenvalue with a large enough relative separation compute
        the corresponding eigenvector by forming a rank revealing twisted
        factorization. Go back to (c) for any clusters that remain.
 For more details, see:
 - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
 - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
   2004.  Also LAPACK Working Note 154.
 - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
   tridiagonal eigenvalue/eigenvector problem",
   Computer Science Division Technical Report No. UCB/CSD-97-971,
   UC Berkeley, May 1997.
 Further Details
 1.CSTEMR works only on machines which follow IEEE-754
 floating-point standard in their handling of infinities and NaNs.
 This permits the use of efficient inner loops avoiding a check for
 zero divisors.
 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
 real symmetric tridiagonal form.
 (Any complex Hermitean tridiagonal matrix has real values on its diagonal
 and potentially complex numbers on its off-diagonals. By applying a
 similarity transform with an appropriate diagonal matrix
 diag(1,e^{i hy_1},

... , e^{i hy_{n-1}}),
the complex Hermitean matrix can be transformed into a real symmetric matrix and complex arithmetic can be entirely avoided.) While the eigenvectors of the real symmetric tridiagonal matrix are real, the eigenvectors of original complex Hermitean matrix have complex entries in general. Since LAPACK drivers overwrite the matrix data with the eigenvectors, CSTEMR accepts complex workspace to facilitate interoperability with CUNMTR or CUPMTR.

Parameters:

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          RANGE is CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

E

          E is REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

VL

          VL is REAL
          If RANGE='V', the lower bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is REAL
          If RANGE='V', the upper bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER
          If RANGE='I', the index of the
          smallest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER
          If RANGE='I', the index of the
          largest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

M

          M is INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is REAL array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

Z

          Z is COMPLEX array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and can be computed with a workspace
          query by setting NZC = -1, see below.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

NZC

          NZC is INTEGER
          The number of eigenvectors to be held in the array Z.
          If RANGE = 'A', then NZC >= max(1,N).
          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
          If RANGE = 'I', then NZC >= IU-IL+1.
          If NZC = -1, then a workspace query is assumed; the
          routine calculates the number of columns of the array Z that
          are needed to hold the eigenvectors.
          This value is returned as the first entry of the Z array, and
          no error message related to NZC is issued by XERBLA.

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

TRYRAC

          TRYRAC is LOGICAL
          If TRYRAC.EQ..TRUE., indicates that the code should check whether
          the tridiagonal matrix defines its eigenvalues to high relative
          accuracy.  If so, the code uses relative-accuracy preserving
          algorithms that might be (a bit) slower depending on the matrix.
          If the matrix does not define its eigenvalues to high relative
          accuracy, the code can uses possibly faster algorithms.
          If TRYRAC.EQ..FALSE., the code is not required to guarantee
          relatively accurate eigenvalues and can use the fastest possible
          techniques.
          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
          does not define its eigenvalues to high relative accuracy.

WORK

          WORK is REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in SLARRE,
                if INFO = 2X, internal error in CLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by SLARRE or
                CLARRV, respectively.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA

subroutine csteqr (character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)

CSTEQR

Purpose:

 CSTEQR computes all eigenvalues and, optionally, eigenvectors of a
 symmetric tridiagonal matrix using the implicit QL or QR method.
 The eigenvectors of a full or band complex Hermitian matrix can also
 be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this
 matrix to tridiagonal form.

Parameters:

COMPZ

          COMPZ is CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvalues and eigenvectors of the original
                  Hermitian matrix.  On entry, Z must contain the
                  unitary matrix used to reduce the original matrix
                  to tridiagonal form.
          = 'I':  Compute eigenvalues and eigenvectors of the
                  tridiagonal matrix.  Z is initialized to the identity
                  matrix.

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is REAL array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

Z

          Z is COMPLEX array, dimension (LDZ, N)
          On entry, if  COMPZ = 'V', then Z contains the unitary
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original Hermitian matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          eigenvectors are desired, then  LDZ >= max(1,N).

WORK

          WORK is REAL array, dimension (max(1,2*N-2))
          If COMPZ = 'N', then WORK is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm has failed to find all the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero; on exit, D
                and E contain the elements of a symmetric tridiagonal
                matrix which is unitarily similar to the original
                matrix.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctbcon (character NORM, character UPLO, character DIAG, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTBCON

Purpose:

 CTBCON estimates the reciprocal of the condition number of a
 triangular band matrix A, in either the 1-norm or the infinity-norm.
 The norm of A is computed and an estimate is obtained for
 norm(inv(A)), then the reciprocal of the condition number is
 computed as
    RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctbrfs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTBRFS

Purpose:

 CTBRFS provides error bounds and backward error estimates for the
 solution to a system of linear equations with a triangular band
 coefficient matrix.
 The solution matrix X must be computed by CTBTRS or some other
 means before entering this routine.  CTBRFS does not do iterative
 refinement because doing so cannot improve the backward error.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          The solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctbtrs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CTBTRS

Purpose:

 CTBTRS solves a triangular system of the form
    A * X = B,  A**T * X = B,  or  A**H * X = B,
 where A is a triangular band matrix of order N, and B is an
 N-by-NRHS matrix.  A check is made to verify that A is nonsingular.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctfsm (character TRANSR, character SIDE, character UPLO, character TRANS, character DIAG, integer M, integer N, complex ALPHA, complex, dimension( 0: * ) A, complex, dimension( 0: ldb-1, 0: * ) B, integer LDB)

CTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

 Level 3 BLAS like routine for A in RFP Format.
 CTFSM solves the matrix equation
    op( A )*X = alpha*B  or  X*op( A ) = alpha*B
 where alpha is a scalar, X and B are m by n matrices, A is a unit, or
 non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
    op( A ) = A   or   op( A ) = A**H.
 A is in Rectangular Full Packed (RFP) Format.
 The matrix X is overwritten on B.

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal Form of RFP A is stored;
          = 'C':  The Conjugate-transpose Form of RFP A is stored.

SIDE

          SIDE is CHARACTER*1
           On entry, SIDE specifies whether op( A ) appears on the left
           or right of X as follows:
              SIDE = 'L' or 'l'   op( A )*X = alpha*B.
              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
           Unchanged on exit.

UPLO

          UPLO is CHARACTER*1
           On entry, UPLO specifies whether the RFP matrix A came from
           an upper or lower triangular matrix as follows:
           UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
           UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix
           Unchanged on exit.

TRANS

          TRANS is CHARACTER*1
           On entry, TRANS  specifies the form of op( A ) to be used
           in the matrix multiplication as follows:
              TRANS  = 'N' or 'n'   op( A ) = A.
              TRANS  = 'C' or 'c'   op( A ) = conjg( A' ).
           Unchanged on exit.

DIAG

          DIAG is CHARACTER*1
           On entry, DIAG specifies whether or not RFP A is unit
           triangular as follows:
              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
              DIAG = 'N' or 'n'   A is not assumed to be unit
                                  triangular.
           Unchanged on exit.

M

          M is INTEGER
           On entry, M specifies the number of rows of B. M must be at
           least zero.
           Unchanged on exit.

N

          N is INTEGER
           On entry, N specifies the number of columns of B.  N must be
           at least zero.
           Unchanged on exit.

ALPHA

          ALPHA is COMPLEX
           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
           zero then  A is not referenced and  B need not be set before
           entry.
           Unchanged on exit.

A

          A is COMPLEX array, dimension (N*(N+1)/2)
           NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
           RFP Format is described by TRANSR, UPLO and N as follows:
           If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
           K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
           TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A as
           defined when TRANSR = 'N'. The contents of RFP A are defined
           by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
           elements of upper packed A either in normal or
           conjugate-transpose Format. If UPLO = 'L' the RFP A contains
           the NT elements of lower packed A either in normal or
           conjugate-transpose Format. The LDA of RFP A is (N+1)/2 when
           TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is
           even and is N when is odd.
           See the Note below for more details. Unchanged on exit.

B

          B is COMPLEX array, dimension (LDB,N)
           Before entry,  the leading  m by n part of the array  B must
           contain  the  right-hand  side  matrix  B,  and  on exit  is
           overwritten by the solution matrix  X.

LDB

          LDB is INTEGER
           On entry, LDB specifies the first dimension of B as declared
           in  the  calling  (sub)  program.   LDB  must  be  at  least
           max( 1, m ).
           Unchanged on exit.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine ctftri (character TRANSR, character UPLO, character DIAG, integer N, complex, dimension( 0: * ) A, integer INFO)

CTFTRI

Purpose:

 CTFTRI computes the inverse of a triangular matrix A stored in RFP
 format.
 This is a Level 3 BLAS version of the algorithm.

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension ( N*(N+1)/2 );
          On entry, the triangular matrix A in RFP format. RFP format
          is described by TRANSR, UPLO, and N as follows: If TRANSR =
          'N' then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
          the Conjugate-transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the nt elements of
          upper packed A; If UPLO = 'L' the RFP A contains the nt
          elements of lower packed A. The LDA of RFP A is (N+1)/2 when
          TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is
          even and N is odd. See the Note below for more details.
          On exit, the (triangular) inverse of the original matrix, in
          the same storage format.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
               matrix is singular and its inverse can not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine ctfttp (character TRANSR, character UPLO, integer N, complex, dimension( 0: * ) ARF, complex, dimension( 0: * ) AP, integer INFO)

CTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).

Purpose:

 CTFTTP copies a triangular matrix A from rectangular full packed
 format (TF) to standard packed format (TP).

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  ARF is in Normal format;
          = 'C':  ARF is in Conjugate-transpose format;

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrix A. N >= 0.

ARF

          ARF is COMPLEX array, dimension ( N*(N+1)/2 ),
          On entry, the upper or lower triangular matrix A stored in
          RFP format. For a further discussion see Notes below.

AP

          AP is COMPLEX array, dimension ( N*(N+1)/2 ),
          On exit, the upper or lower triangular matrix A, packed
          columnwise in a linear array. The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine ctfttr (character TRANSR, character UPLO, integer N, complex, dimension( 0: * ) ARF, complex, dimension( 0: lda-1, 0: * ) A, integer LDA, integer INFO)

CTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).

Purpose:

 CTFTTR copies a triangular matrix A from rectangular full packed
 format (TF) to standard full format (TR).

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  ARF is in Normal format;
          = 'C':  ARF is in Conjugate-transpose format;

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

ARF

          ARF is COMPLEX array, dimension ( N*(N+1)/2 ),
          On entry, the upper or lower triangular matrix A stored in
          RFP format. For a further discussion see Notes below.

A

          A is COMPLEX array, dimension ( LDA, N )
          On exit, the triangular matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of the array A contains
          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of the array A contains
          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine ctgsen (integer IJOB, logical WANTQ, logical WANTZ, logical, dimension( * ) SELECT, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldz, * ) Z, integer LDZ, integer M, real PL, real PR, real, dimension( * ) DIF, complex, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CTGSEN

Purpose:

 CTGSEN reorders the generalized Schur decomposition of a complex
 matrix pair (A, B) (in terms of an unitary equivalence trans-
 formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues
 appears in the leading diagonal blocks of the pair (A,B). The leading
 columns of Q and Z form unitary bases of the corresponding left and
 right eigenspaces (deflating subspaces). (A, B) must be in
 generalized Schur canonical form, that is, A and B are both upper
 triangular.
 CTGSEN also computes the generalized eigenvalues
          w(j)= ALPHA(j) / BETA(j)
 of the reordered matrix pair (A, B).
 Optionally, the routine computes estimates of reciprocal condition
 numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
 (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
 between the matrix pairs (A11, B11) and (A22,B22) that correspond to
 the selected cluster and the eigenvalues outside the cluster, resp.,
 and norms of "projections" onto left and right eigenspaces w.r.t.
 the selected cluster in the (1,1)-block.

Parameters:

IJOB

          IJOB is INTEGER
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (PL and PR) or the deflating subspaces
          (Difu and Difl):
           =0: Only reorder w.r.t. SELECT. No extras.
           =1: Reciprocal of norms of "projections" onto left and right
               eigenspaces w.r.t. the selected cluster (PL and PR).
           =2: Upper bounds on Difu and Difl. F-norm-based estimate
               (DIF(1:2)).
           =3: Estimate of Difu and Difl. 1-norm-based estimate
               (DIF(1:2)).
               About 5 times as expensive as IJOB = 2.
           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
               version to get it all.
           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

WANTQ

          WANTQ is LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

WANTZ

          WANTZ is LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

SELECT

          SELECT is LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster. To
          select an eigenvalue w(j), SELECT(j) must be set to
          .TRUE..

N

          N is INTEGER
          The order of the matrices A and B. N >= 0.

A

          A is COMPLEX array, dimension(LDA,N)
          On entry, the upper triangular matrix A, in generalized
          Schur canonical form.
          On exit, A is overwritten by the reordered matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

B

          B is COMPLEX array, dimension(LDB,N)
          On entry, the upper triangular matrix B, in generalized
          Schur canonical form.
          On exit, B is overwritten by the reordered matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)
          The diagonal elements of A and B, respectively,
          when the pair (A,B) has been reduced to generalized Schur
          form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
          eigenvalues.

Q

          Q is COMPLEX array, dimension (LDQ,N)
          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
          On exit, Q has been postmultiplied by the left unitary
          transformation matrix which reorder (A, B); The leading M
          columns of Q form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTQ = .FALSE., Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= 1.
          If WANTQ = .TRUE., LDQ >= N.

Z

          Z is COMPLEX array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
          On exit, Z has been postmultiplied by the left unitary
          transformation matrix which reorder (A, B); The leading M
          columns of Z form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTZ = .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1.
          If WANTZ = .TRUE., LDZ >= N.

M

          M is INTEGER
          The dimension of the specified pair of left and right
          eigenspaces, (deflating subspaces) 0 <= M <= N.

PL

          PL is REAL

PR

          PR is REAL
          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
          reciprocal  of the norm of "projections" onto left and right
          eigenspace with respect to the selected cluster.
          0 < PL, PR <= 1.
          If M = 0 or M = N, PL = PR  = 1.
          If IJOB = 0, 2 or 3 PL, PR are not referenced.

DIF

          DIF is REAL array, dimension (2).
          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
          estimates of Difu and Difl, computed using reversed
          communication with CLACN2.
          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
          If IJOB = 0 or 1, DIF is not referenced.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >=  1
          If IJOB = 1, 2 or 4, LWORK >=  2*M*(N-M)
          If IJOB = 3 or 5, LWORK >=  4*M*(N-M)
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK. LIWORK >= 1.
          If IJOB = 1, 2 or 4, LIWORK >=  N+2;
          If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
            =0: Successful exit.
            <0: If INFO = -i, the i-th argument had an illegal value.
            =1: Reordering of (A, B) failed because the transformed
                matrix pair (A, B) would be too far from generalized
                Schur form; the problem is very ill-conditioned.
                (A, B) may have been partially reordered.
                If requested, 0 is returned in DIF(*), PL and PR.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

  CTGSEN first collects the selected eigenvalues by computing unitary
  U and W that move them to the top left corner of (A, B). In other
  words, the selected eigenvalues are the eigenvalues of (A11, B11) in
              U**H*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2
  where N = n1+n2 and U**H means the conjugate transpose of U. The first
  n1 columns of U and W span the specified pair of left and right
  eigenspaces (deflating subspaces) of (A, B).
  If (A, B) has been obtained from the generalized real Schur
  decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
  reordered generalized Schur form of (C, D) is given by
           (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,
  and the first n1 columns of Q*U and Z*W span the corresponding
  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
  Note that if the selected eigenvalue is sufficiently ill-conditioned,
  then its value may differ significantly from its value before
  reordering.
  The reciprocal condition numbers of the left and right eigenspaces
  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
  be returned in DIF(1:2), corresponding to Difu and Difl, resp.
  The Difu and Difl are defined as:
       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
  where sigma-min(Zu) is the smallest singular value of the
  (2*n1*n2)-by-(2*n1*n2) matrix
       Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
            [ kron(In2, B11)  -kron(B22**H, In1) ].
  Here, Inx is the identity matrix of size nx and A22**H is the
  conjuguate transpose of A22. kron(X, Y) is the Kronecker product between
  the matrices X and Y.
  When DIF(2) is small, small changes in (A, B) can cause large changes
  in the deflating subspace. An approximate (asymptotic) bound on the
  maximum angular error in the computed deflating subspaces is
       EPS * norm((A, B)) / DIF(2),
  where EPS is the machine precision.
  The reciprocal norm of the projectors on the left and right
  eigenspaces associated with (A11, B11) may be returned in PL and PR.
  They are computed as follows. First we compute L and R so that
  P*(A, B)*Q is block diagonal, where
       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2
  and (L, R) is the solution to the generalized Sylvester equation
       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12
  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is
       EPS * norm((A, B)) / PL.
  There are also global error bounds which valid for perturbations up
  to a certain restriction:  A lower bound (x) on the smallest
  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
  (i.e. (A + E, B + F), is
   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
  An approximate bound on x can be computed from DIF(1:2), PL and PR.
  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
  (L', R') and unperturbed (L, R) left and right deflating subspaces
  associated with the selected cluster in the (1,1)-blocks can be
  bounded as
   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
  See LAPACK User's Guide section 4.11 or the following references
  for more information.
  Note that if the default method for computing the Frobenius-norm-
  based estimate DIF is not wanted (see CLATDF), then the parameter
  IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF
  (IJOB = 2 will be used)). See CTGSYL for more details.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

subroutine ctgsja (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, real TOLA, real TOLB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( * ) WORK, integer NCYCLE, integer INFO)

CTGSJA

Purpose:

 CTGSJA computes the generalized singular value decomposition (GSVD)
 of two complex upper triangular (or trapezoidal) matrices A and B.
 On entry, it is assumed that matrices A and B have the following
 forms, which may be obtained by the preprocessing subroutine CGGSVP
 from a general M-by-N matrix A and P-by-N matrix B:
              N-K-L  K    L
    A =    K ( 0    A12  A13 ) if M-K-L >= 0;
           L ( 0     0   A23 )
       M-K-L ( 0     0    0  )
            N-K-L  K    L
    A =  K ( 0    A12  A13 ) if M-K-L < 0;
       M-K ( 0     0   A23 )
            N-K-L  K    L
    B =  L ( 0     0   B13 )
       P-L ( 0     0    0  )
 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
 upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
 otherwise A23 is (M-K)-by-L upper trapezoidal.
 On exit,
        U**H *A*Q = D1*( 0 R ),    V**H *B*Q = D2*( 0 R ),
 where U, V and Q are unitary matrices.
 R is a nonsingular upper triangular matrix, and D1
 and D2 are ``diagonal'' matrices, which are of the following
 structures:
 If M-K-L >= 0,
                     K  L
        D1 =     K ( I  0 )
                 L ( 0  C )
             M-K-L ( 0  0 )
                    K  L
        D2 = L   ( 0  S )
             P-L ( 0  0 )
                N-K-L  K    L
   ( 0 R ) = K (  0   R11  R12 ) K
             L (  0    0   R22 ) L
 where
   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
   S = diag( BETA(K+1),  ... , BETA(K+L) ),
   C**2 + S**2 = I.
   R is stored in A(1:K+L,N-K-L+1:N) on exit.
 If M-K-L < 0,
                K M-K K+L-M
     D1 =   K ( I  0    0   )
          M-K ( 0  C    0   )
                  K M-K K+L-M
     D2 =   M-K ( 0  S    0   )
          K+L-M ( 0  0    I   )
            P-L ( 0  0    0   )
                N-K-L  K   M-K  K+L-M
 ( 0 R ) =    K ( 0    R11  R12  R13  )
           M-K ( 0     0   R22  R23  )
         K+L-M ( 0     0    0   R33  )
 where
 C = diag( ALPHA(K+1), ... , ALPHA(M) ),
 S = diag( BETA(K+1),  ... , BETA(M) ),
 C**2 + S**2 = I.
 R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
     (  0  R22 R23 )
 in B(M-K+1:L,N+M-K-L+1:N) on exit.
 The computation of the unitary transformation matrices U, V or Q
 is optional.  These matrices may either be formed explicitly, or they
 may be postmultiplied into input matrices U1, V1, or Q1.

Parameters:

JOBU

          JOBU is CHARACTER*1
          = 'U':  U must contain a unitary matrix U1 on entry, and
                  the product U1*U is returned;
          = 'I':  U is initialized to the unit matrix, and the
                  unitary matrix U is returned;
          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1
          = 'V':  V must contain a unitary matrix V1 on entry, and
                  the product V1*V is returned;
          = 'I':  V is initialized to the unit matrix, and the
                  unitary matrix V is returned;
          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1
          = 'Q':  Q must contain a unitary matrix Q1 on entry, and
                  the product Q1*Q is returned;
          = 'I':  Q is initialized to the unit matrix, and the
                  unitary matrix Q is returned;
          = 'N':  Q is not computed.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.

K

          K is INTEGER

L

          L is INTEGER
          K and L specify the subblocks in the input matrices A and B:
          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
          of A and B, whose GSVD is going to be computed by CTGSJA.
          See Further Details.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
          matrix R or part of R.  See Purpose for details.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
          a part of R.  See Purpose for details.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

TOLA

          TOLA is REAL

TOLB

          TOLB is REAL
          TOLA and TOLB are the convergence criteria for the Jacobi-
          Kogbetliantz iteration procedure. Generally, they are the
          same as used in the preprocessing step, say
              TOLA = MAX(M,N)*norm(A)*MACHEPS,
              TOLB = MAX(P,N)*norm(B)*MACHEPS.

ALPHA

          ALPHA is REAL array, dimension (N)

BETA

          BETA is REAL array, dimension (N)
          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = diag(C),
            BETA(K+1:K+L)  = diag(S),
          or if M-K-L < 0,
            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
          Furthermore, if K+L < N,
            ALPHA(K+L+1:N) = 0
            BETA(K+L+1:N)  = 0.

U

          U is COMPLEX array, dimension (LDU,M)
          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
          the unitary matrix returned by CGGSVP).
          On exit,
          if JOBU = 'I', U contains the unitary matrix U;
          if JOBU = 'U', U contains the product U1*U.
          If JOBU = 'N', U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

V

          V is COMPLEX array, dimension (LDV,P)
          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
          the unitary matrix returned by CGGSVP).
          On exit,
          if JOBV = 'I', V contains the unitary matrix V;
          if JOBV = 'V', V contains the product V1*V.
          If JOBV = 'N', V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

Q

          Q is COMPLEX array, dimension (LDQ,N)
          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
          the unitary matrix returned by CGGSVP).
          On exit,
          if JOBQ = 'I', Q contains the unitary matrix Q;
          if JOBQ = 'Q', Q contains the product Q1*Q.
          If JOBQ = 'N', Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK

          WORK is COMPLEX array, dimension (2*N)

NCYCLE

          NCYCLE is INTEGER
          The number of cycles required for convergence.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the procedure does not converge after MAXIT cycles.

Internal Parameters:

  MAXIT   INTEGER
          MAXIT specifies the total loops that the iterative procedure
          may take. If after MAXIT cycles, the routine fails to
          converge, we return INFO = 1.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  CTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
  matrix B13 to the form:
           U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
  where U1, V1 and Q1 are unitary matrix.
  C1 and S1 are diagonal matrices satisfying
                C1**2 + S1**2 = I,
  and R1 is an L-by-L nonsingular upper triangular matrix.

subroutine ctgsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) S, real, dimension( * ) DIF, integer MM, integer M, complex, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

CTGSNA

Purpose:

 CTGSNA estimates reciprocal condition numbers for specified
 eigenvalues and/or eigenvectors of a matrix pair (A, B).
 (A, B) must be in generalized Schur canonical form, that is, A and
 B are both upper triangular.

Parameters:

JOB

          JOB is CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (DIF):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (DIF);
          = 'B': for both eigenvalues and eigenvectors (S and DIF).

HOWMNY

          HOWMNY is CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.

SELECT

          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the corresponding j-th eigenvalue and/or eigenvector,
          SELECT(j) must be set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.

N

          N is INTEGER
          The order of the square matrix pair (A, B). N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The upper triangular matrix A in the pair (A,B).

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB,N)
          The upper triangular matrix B in the pair (A, B).

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

VL

          VL is COMPLEX array, dimension (LDVL,M)
          IF JOB = 'E' or 'B', VL must contain left eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT.  The eigenvectors must be stored in consecutive
          columns of VL, as returned by CTGEVC.
          If JOB = 'V', VL is not referenced.

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL. LDVL >= 1; and
          If JOB = 'E' or 'B', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,M)
          IF JOB = 'E' or 'B', VR must contain right eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT.  The eigenvectors must be stored in consecutive
          columns of VR, as returned by CTGEVC.
          If JOB = 'V', VR is not referenced.

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR. LDVR >= 1;
          If JOB = 'E' or 'B', LDVR >= N.

S

          S is REAL array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the
          selected eigenvalues, stored in consecutive elements of the
          array.
          If JOB = 'V', S is not referenced.

DIF

          DIF is REAL array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array.
          If the eigenvalues cannot be reordered to compute DIF(j),
          DIF(j) is set to 0; this can only occur when the true value
          would be very small anyway.
          For each eigenvalue/vector specified by SELECT, DIF stores
          a Frobenius norm-based estimate of Difl.
          If JOB = 'E', DIF is not referenced.

MM

          MM is INTEGER
          The number of elements in the arrays S and DIF. MM >= M.

M

          M is INTEGER
          The number of elements of the arrays S and DIF used to store
          the specified condition numbers; for each selected eigenvalue
          one element is used. If HOWMNY = 'A', M is set to N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).

IWORK

          IWORK is INTEGER array, dimension (N+2)
          If JOB = 'E', IWORK is not referenced.

INFO

          INFO is INTEGER
          = 0: Successful exit
          < 0: If INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The reciprocal of the condition number of the i-th generalized
  eigenvalue w = (a, b) is defined as
          S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))
  where u and v are the right and left eigenvectors of (A, B)
  corresponding to w; |z| denotes the absolute value of the complex
  number, and norm(u) denotes the 2-norm of the vector u. The pair
  (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
  matrix pair (A, B). If both a and b equal zero, then (A,B) is
  singular and S(I) = -1 is returned.
  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is
          chord(w, lambda) <=   EPS * norm(A, B) / S(I),
  where EPS is the machine precision.
  The reciprocal of the condition number of the right eigenvector u
  and left eigenvector v corresponding to the generalized eigenvalue w
  is defined as follows. Suppose
                   (A, B) = ( a   *  ) ( b  *  )  1
                            ( 0  A22 ),( 0 B22 )  n-1
                              1  n-1     1 n-1
  Then the reciprocal condition number DIF(I) is
          Difl[(a, b), (A22, B22)]  = sigma-min( Zl )
  where sigma-min(Zl) denotes the smallest singular value of
         Zl = [ kron(a, In-1) -kron(1, A22) ]
              [ kron(b, In-1) -kron(1, B22) ].
  Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
  transpose of X. kron(X, Y) is the Kronecker product between the
  matrices X and Y.
  We approximate the smallest singular value of Zl with an upper
  bound. This is done by CLATDF.
  An approximate error bound for a computed eigenvector VL(i) or
  VR(i) is given by
                      EPS * norm(A, B) / DIF(i).
  See ref. [2-3] for more details and further references.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software, Report
      UMINF - 94.04, Department of Computing Science, Umea University,
      S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
      To appear in Numerical Algorithms, 1996.
  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,
      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75.
      To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

subroutine ctpcon (character NORM, character UPLO, character DIAG, integer N, complex, dimension( * ) AP, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTPCON

Purpose:

 CTPCON estimates the reciprocal of the condition number of a packed
 triangular matrix A, in either the 1-norm or the infinity-norm.
 The norm of A is computed and an estimate is obtained for
 norm(inv(A)), then the reciprocal of the condition number is
 computed as
    RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctpmqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer L, integer NB, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integer INFO)

CTPMQRT

Purpose:

 CTPMQRT applies a complex orthogonal matrix Q obtained from a
 "triangular-pentagonal" complex block reflector H to a general
 complex matrix C, which consists of two blocks A and B.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix B. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.

L

          L is INTEGER
          The order of the trapezoidal part of V.
          K >= L >= 0.  See Further Details.

NB

          NB is INTEGER
          The block size used for the storage of T.  K >= NB >= 1.
          This must be the same value of NB used to generate T
          in CTPQRT.

V

          V is COMPLEX array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CTPQRT in B.  See Further Details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If SIDE = 'L', LDV >= max(1,M);
          if SIDE = 'R', LDV >= max(1,N).

T

          T is COMPLEX array, dimension (LDT,K)
          The upper triangular factors of the block reflectors
          as returned by CTPQRT, stored as a NB-by-K matrix.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB.

A

          A is COMPLEX array, dimension
          (LDA,N) if SIDE = 'L' or
          (LDA,K) if SIDE = 'R'
          On entry, the K-by-N or M-by-K matrix A.
          On exit, A is overwritten by the corresponding block of
          Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDC >= max(1,K);
          If SIDE = 'R', LDC >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the M-by-N matrix B.
          On exit, B is overwritten by the corresponding block of
          Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.
          LDB >= max(1,M).

WORK

          WORK is COMPLEX array. The dimension of WORK is
           N*NB if SIDE = 'L', or  M*NB if SIDE = 'R'.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

Further Details:

  The columns of the pentagonal matrix V contain the elementary reflectors
  H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
  trapezoidal block V2:
        V = [V1]
            [V2].
  The size of the trapezoidal block V2 is determined by the parameter L,
  where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
  rows of a K-by-K upper triangular matrix.  If L=K, V2 is upper triangular;
  if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
  If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is M-by-K.
                      [B]
  If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is N-by-K.
  The complex orthogonal matrix Q is formed from V and T.
  If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
  If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C.
  If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
  If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

subroutine ctpqrt (integer M, integer N, integer L, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( * ) WORK, integer INFO)

CTPQRT

Purpose:

 CTPQRT computes a blocked QR factorization of a complex
 "triangular-pentagonal" matrix C, which is composed of a
 triangular block A and pentagonal block B, using the compact
 WY representation for Q.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix B.
          M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B, and the order of the
          triangular matrix A.
          N >= 0.

L

          L is INTEGER
          The number of rows of the upper trapezoidal part of B.
          MIN(M,N) >= L >= 0.  See Further Details.

NB

          NB is INTEGER
          The block size to be used in the blocked QR.  N >= NB >= 1.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the upper triangular N-by-N matrix A.
          On exit, the elements on and above the diagonal of the array
          contain the upper triangular matrix R.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the pentagonal M-by-N matrix B.  The first M-L rows
          are rectangular, and the last L rows are upper trapezoidal.
          On exit, B contains the pentagonal matrix V.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).

T

          T is COMPLEX array, dimension (LDT,N)
          The upper triangular block reflectors stored in compact form
          as a sequence of upper triangular blocks.  See Further Details.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB.

WORK

          WORK is COMPLEX array, dimension (NB*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The input matrix C is a (N+M)-by-N matrix
               C = [ A ]
                   [ B ]
  where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
  matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
  upper trapezoidal matrix B2:
               B = [ B1 ]  <- (M-L)-by-N rectangular
                   [ B2 ]  <-     L-by-N upper trapezoidal.
  The upper trapezoidal matrix B2 consists of the first L rows of a
  N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
  B is rectangular M-by-N; if M=L=N, B is upper triangular.
  The matrix W stores the elementary reflectors H(i) in the i-th column
  below the diagonal (of A) in the (N+M)-by-N input matrix C
               C = [ A ]  <- upper triangular N-by-N
                   [ B ]  <- M-by-N pentagonal
  so that W can be represented as
               W = [ I ]  <- identity, N-by-N
                   [ V ]  <- M-by-N, same form as B.
  Thus, all of information needed for W is contained on exit in B, which
  we call V above.  Note that V has the same form as B; that is,
               V = [ V1 ] <- (M-L)-by-N rectangular
                   [ V2 ] <-     L-by-N upper trapezoidal.
  The columns of V represent the vectors which define the H(i)'s.
  The number of blocks is B = ceiling(N/NB), where each
  block is of order NB except for the last block, which is of order
  IB = N - (B-1)*NB.  For each of the B blocks, a upper triangular block
  reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
  for the last block) T's are stored in the NB-by-N matrix T as
               T = [T1 T2 ... TB].

subroutine ctpqrt2 (integer M, integer N, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldt, * ) T, integer LDT, integer INFO)

CTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

 CTPQRT2 computes a QR factorization of a complex "triangular-pentagonal"
 matrix C, which is composed of a triangular block A and pentagonal block B,
 using the compact WY representation for Q.

Parameters:

M

          M is INTEGER
          The total number of rows of the matrix B.
          M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B, and the order of
          the triangular matrix A.
          N >= 0.

L

          L is INTEGER
          The number of rows of the upper trapezoidal part of B.
          MIN(M,N) >= L >= 0.  See Further Details.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the upper triangular N-by-N matrix A.
          On exit, the elements on and above the diagonal of the array
          contain the upper triangular matrix R.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the pentagonal M-by-N matrix B.  The first M-L rows
          are rectangular, and the last L rows are upper trapezoidal.
          On exit, B contains the pentagonal matrix V.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).

T

          T is COMPLEX array, dimension (LDT,N)
          The N-by-N upper triangular factor T of the block reflector.
          See Further Details.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= max(1,N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The input matrix C is a (N+M)-by-N matrix
               C = [ A ]
                   [ B ]
  where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
  matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
  upper trapezoidal matrix B2:
               B = [ B1 ]  <- (M-L)-by-N rectangular
                   [ B2 ]  <-     L-by-N upper trapezoidal.
  The upper trapezoidal matrix B2 consists of the first L rows of a
  N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
  B is rectangular M-by-N; if M=L=N, B is upper triangular.
  The matrix W stores the elementary reflectors H(i) in the i-th column
  below the diagonal (of A) in the (N+M)-by-N input matrix C
               C = [ A ]  <- upper triangular N-by-N
                   [ B ]  <- M-by-N pentagonal
  so that W can be represented as
               W = [ I ]  <- identity, N-by-N
                   [ V ]  <- M-by-N, same form as B.
  Thus, all of information needed for W is contained on exit in B, which
  we call V above.  Note that V has the same form as B; that is,
               V = [ V1 ] <- (M-L)-by-N rectangular
                   [ V2 ] <-     L-by-N upper trapezoidal.
  The columns of V represent the vectors which define the H(i)'s.
  The (M+N)-by-(M+N) block reflector H is then given by
               H = I - W * T * W**H
  where W**H is the conjugate transpose of W and T is the upper triangular
  factor of the block reflector.

subroutine ctprfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTPRFS

Purpose:

 CTPRFS provides error bounds and backward error estimates for the
 solution to a system of linear equations with a triangular packed
 coefficient matrix.
 The solution matrix X must be computed by CTPTRS or some other
 means before entering this routine.  CTPRFS does not do iterative
 refinement because doing so cannot improve the backward error.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          The solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctptri (character UPLO, character DIAG, integer N, complex, dimension( * ) AP, integer INFO)

CTPTRI

Purpose:

 CTPTRI computes the inverse of a complex upper or lower triangular
 matrix A stored in packed format.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangular matrix A, stored
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.
          On exit, the (triangular) inverse of the original matrix, in
          the same packed storage format.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, A(i,i) is exactly zero.  The triangular
                matrix is singular and its inverse can not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  A triangular matrix A can be transferred to packed storage using one
  of the following program segments:
  UPLO = 'U':                      UPLO = 'L':
        JC = 1                           JC = 1
        DO 2 J = 1, N                    DO 2 J = 1, N
           DO 1 I = 1, J                    DO 1 I = J, N
              AP(JC+I-1) = A(I,J)              AP(JC+I-J) = A(I,J)
      1    CONTINUE                    1    CONTINUE
           JC = JC + J                      JC = JC + N - J + 1
      2 CONTINUE                       2 CONTINUE

subroutine ctptrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex, dimension( * ) AP, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CTPTRS

Purpose:

 CTPTRS solves a triangular system of the form
    A * X = B,  A**T * X = B,  or  A**H * X = B,
 where A is a triangular matrix of order N stored in packed format,
 and B is an N-by-NRHS matrix.  A check is made to verify that A is
 nonsingular.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctpttf (character TRANSR, character UPLO, integer N, complex, dimension( 0: * ) AP, complex, dimension( 0: * ) ARF, integer INFO)

CTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).

Purpose:

 CTPTTF copies a triangular matrix A from standard packed format (TP)
 to rectangular full packed format (TF).

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  ARF in Normal format is wanted;
          = 'C':  ARF in Conjugate-transpose format is wanted.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension ( N*(N+1)/2 ),
          On entry, the upper or lower triangular matrix A, packed
          columnwise in a linear array. The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

ARF

          ARF is COMPLEX array, dimension ( N*(N+1)/2 ),
          On exit, the upper or lower triangular matrix A stored in
          RFP format. For a further discussion see Notes below.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine ctpttr (character UPLO, integer N, complex, dimension( * ) AP, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).

Purpose:

 CTPTTR copies a triangular matrix A from standard packed format (TP)
 to standard full format (TR).

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular.
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrix A. N >= 0.

AP

          AP is COMPLEX array, dimension ( N*(N+1)/2 ),
          On entry, the upper or lower triangular matrix A, packed
          columnwise in a linear array. The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

A

          A is COMPLEX array, dimension ( LDA, N )
          On exit, the triangular matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrcon (character NORM, character UPLO, character DIAG, integer N, complex, dimension( lda, * ) A, integer LDA, real RCOND, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTRCON

Purpose:

 CTRCON estimates the reciprocal of the condition number of a
 triangular matrix A, in either the 1-norm or the infinity-norm.
 The norm of A is computed and an estimate is obtained for
 norm(inv(A)), then the reciprocal of the condition number is
 computed as
    RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of
          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular
          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

RCOND

          RCOND is REAL
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTREVC

Purpose:

 CTREVC computes some or all of the right and/or left eigenvectors of
 a complex upper triangular matrix T.
 Matrices of this type are produced by the Schur factorization of
 a complex general matrix:  A = Q*T*Q**H, as computed by CHSEQR.
 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:
              T*x = w*x,     (y**H)*T = w*(y**H)
 where y**H denotes the conjugate transpose of the vector y.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal of T.
 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix.  If Q is the unitary factor that reduces a matrix A to
 Schur form T, then Q*X and Q*Y are the matrices of right and left
 eigenvectors of A.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.

HOWMNY

          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed using the matrices supplied in
                  VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT.

SELECT

          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed.
          The eigenvector corresponding to the j-th eigenvalue is
          computed if SELECT(j) = .TRUE..
          Not referenced if HOWMNY = 'A' or 'B'.

N

          N is INTEGER
          The order of the matrix T. N >= 0.

T

          T is COMPLEX array, dimension (LDT,N)
          The upper triangular matrix T.  T is modified, but restored
          on exit.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

VL

          VL is COMPLEX array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues.
          Not referenced if SIDE = 'R'.

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.  LDVL >= 1, and if
          SIDE = 'L' or 'B', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR).
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues.
          Not referenced if SIDE = 'L'.

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.  LDVR >= 1, and if
          SIDE = 'R' or 'B'; LDVR >= N.

MM

          MM is INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

M

          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
          is set to N.  Each selected eigenvector occupies one
          column.

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow.
  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|.

subroutine ctrevc3 (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer INFO)

CTREVC3

Purpose:

 CTREVC3 computes some or all of the right and/or left eigenvectors of
 a complex upper triangular matrix T.
 Matrices of this type are produced by the Schur factorization of
 a complex general matrix:  A = Q*T*Q**H, as computed by CHSEQR.
 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:
              T*x = w*x,     (y**H)*T = w*(y**H)
 where y**H denotes the conjugate transpose of the vector y.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal of T.
 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix. If Q is the unitary factor that reduces a matrix A to
 Schur form T, then Q*X and Q*Y are the matrices of right and left
 eigenvectors of A.
 This uses a Level 3 BLAS version of the back transformation.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.

HOWMNY

          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed using the matrices supplied in
                  VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT.

SELECT

          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed.
          The eigenvector corresponding to the j-th eigenvalue is
          computed if SELECT(j) = .TRUE..
          Not referenced if HOWMNY = 'A' or 'B'.

N

          N is INTEGER
          The order of the matrix T. N >= 0.

T

          T is COMPLEX array, dimension (LDT,N)
          The upper triangular matrix T.  T is modified, but restored
          on exit.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

VL

          VL is COMPLEX array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues.
          Not referenced if SIDE = 'R'.

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the unitary matrix Q of
          Schur vectors returned by CHSEQR).
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues.
          Not referenced if SIDE = 'L'.

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.

MM

          MM is INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

M

          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.
          If HOWMNY = 'A' or 'B', M is set to N.
          Each selected eigenvector occupies one column.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))

LWORK

          LWORK is INTEGER
          The dimension of array WORK. LWORK >= max(1,2*N).
          For optimum performance, LWORK >= N + 2*N*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (LRWORK)

LRWORK

          LRWORK is INTEGER
          The dimension of array RWORK. LRWORK >= max(1,N).
          If LRWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the RWORK array, returns
          this value as the first entry of the RWORK array, and no error
          message related to LRWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

Further Details:

  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow.
  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|.

subroutine ctrexc (character COMPQ, integer N, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldq, * ) Q, integer LDQ, integer IFST, integer ILST, integer INFO)

CTREXC

Purpose:

 CTREXC reorders the Schur factorization of a complex matrix
 A = Q*T*Q**H, so that the diagonal element of T with row index IFST
 is moved to row ILST.
 The Schur form T is reordered by a unitary similarity transformation
 Z**H*T*Z, and optionally the matrix Q of Schur vectors is updated by
 postmultplying it with Z.

Parameters:

COMPQ

          COMPQ is CHARACTER*1
          = 'V':  update the matrix Q of Schur vectors;
          = 'N':  do not update Q.

N

          N is INTEGER
          The order of the matrix T. N >= 0.
          If N == 0 arguments ILST and IFST may be any value.

T

          T is COMPLEX array, dimension (LDT,N)
          On entry, the upper triangular matrix T.
          On exit, the reordered upper triangular matrix.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

Q

          Q is COMPLEX array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          unitary transformation matrix Z which reorders T.
          If COMPQ = 'N', Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.  LDQ >= 1, and if
          COMPQ = 'V', LDQ >= max(1,N).

IFST

          IFST is INTEGER

ILST

          ILST is INTEGER
          Specify the reordering of the diagonal elements of T:
          The element with row index IFST is moved to row ILST by a
          sequence of transpositions between adjacent elements.
          1 <= IFST <= N; 1 <= ILST <= N.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrrfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)

CTRRFS

Purpose:

 CTRRFS provides error bounds and backward error estimates for the
 solution to a system of linear equations with a triangular
 coefficient matrix.
 The solution matrix X must be computed by CTRTRS or some other
 means before entering this routine.  CTRRFS does not do iterative
 refinement because doing so cannot improve the backward error.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of
          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular
          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)
          The solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is REAL array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is REAL array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrsen (character JOB, character COMPQ, logical, dimension( * ) SELECT, integer N, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( * ) W, integer M, real S, real SEP, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CTRSEN

Purpose:

 CTRSEN reorders the Schur factorization of a complex matrix
 A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
 the leading positions on the diagonal of the upper triangular matrix
 T, and the leading columns of Q form an orthonormal basis of the
 corresponding right invariant subspace.
 Optionally the routine computes the reciprocal condition numbers of
 the cluster of eigenvalues and/or the invariant subspace.

Parameters:

JOB

          JOB is CHARACTER*1
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (S) or the invariant subspace (SEP):
          = 'N': none;
          = 'E': for eigenvalues only (S);
          = 'V': for invariant subspace only (SEP);
          = 'B': for both eigenvalues and invariant subspace (S and
                 SEP).

COMPQ

          COMPQ is CHARACTER*1
          = 'V': update the matrix Q of Schur vectors;
          = 'N': do not update Q.

SELECT

          SELECT is LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster. To
          select the j-th eigenvalue, SELECT(j) must be set to .TRUE..

N

          N is INTEGER
          The order of the matrix T. N >= 0.

T

          T is COMPLEX array, dimension (LDT,N)
          On entry, the upper triangular matrix T.
          On exit, T is overwritten by the reordered matrix T, with the
          selected eigenvalues as the leading diagonal elements.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

Q

          Q is COMPLEX array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          unitary transformation matrix which reorders T; the leading M
          columns of Q form an orthonormal basis for the specified
          invariant subspace.
          If COMPQ = 'N', Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

W

          W is COMPLEX array, dimension (N)
          The reordered eigenvalues of T, in the same order as they
          appear on the diagonal of T.

M

          M is INTEGER
          The dimension of the specified invariant subspace.
          0 <= M <= N.

S

          S is REAL
          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
          condition number for the selected cluster of eigenvalues.
          S cannot underestimate the true reciprocal condition number
          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
          If JOB = 'N' or 'V', S is not referenced.

SEP

          SEP is REAL
          If JOB = 'V' or 'B', SEP is the estimated reciprocal
          condition number of the specified invariant subspace. If
          M = 0 or N, SEP = norm(T).
          If JOB = 'N' or 'E', SEP is not referenced.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If JOB = 'N', LWORK >= 1;
          if JOB = 'E', LWORK = max(1,M*(N-M));
          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  CTRSEN first collects the selected eigenvalues by computing a unitary
  transformation Z to move them to the top left corner of T. In other
  words, the selected eigenvalues are the eigenvalues of T11 in:
          Z**H * T * Z = ( T11 T12 ) n1
                         (  0  T22 ) n2
                            n1  n2
  where N = n1+n2. The first
  n1 columns of Z span the specified invariant subspace of T.
  If T has been obtained from the Schur factorization of a matrix
  A = Q*T*Q**H, then the reordered Schur factorization of A is given by
  A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, and the first n1 columns of Q*Z span the
  corresponding invariant subspace of A.
  The reciprocal condition number of the average of the eigenvalues of
  T11 may be returned in S. S lies between 0 (very badly conditioned)
  and 1 (very well conditioned). It is computed as follows. First we
  compute R so that
                         P = ( I  R ) n1
                             ( 0  0 ) n2
                               n1 n2
  is the projector on the invariant subspace associated with T11.
  R is the solution of the Sylvester equation:
                        T11*R - R*T22 = T12.
  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
  the two-norm of M. Then S is computed as the lower bound
                      (1 + F-norm(R)**2)**(-1/2)
  on the reciprocal of 2-norm(P), the true reciprocal condition number.
  S cannot underestimate 1 / 2-norm(P) by more than a factor of
  sqrt(N).
  An approximate error bound for the computed average of the
  eigenvalues of T11 is
                         EPS * norm(T) / S
  where EPS is the machine precision.
  The reciprocal condition number of the right invariant subspace
  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
  SEP is defined as the separation of T11 and T22:
                     sep( T11, T22 ) = sigma-min( C )
  where sigma-min(C) is the smallest singular value of the
  n1*n2-by-n1*n2 matrix
     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
  product. We estimate sigma-min(C) by the reciprocal of an estimate of
  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
  When SEP is small, small changes in T can cause large changes in
  the invariant subspace. An approximate bound on the maximum angular
  error in the computed right invariant subspace is
                      EPS * norm(T) / SEP

subroutine ctrsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) S, real, dimension( * ) SEP, integer MM, integer M, complex, dimension( ldwork, * ) WORK, integer LDWORK, real, dimension( * ) RWORK, integer INFO)

CTRSNA

Purpose:

 CTRSNA estimates reciprocal condition numbers for specified
 eigenvalues and/or right eigenvectors of a complex upper triangular
 matrix T (or of any matrix Q*T*Q**H with Q unitary).

Parameters:

JOB

          JOB is CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (SEP):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (SEP);
          = 'B': for both eigenvalues and eigenvectors (S and SEP).

HOWMNY

          HOWMNY is CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.

SELECT

          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the j-th eigenpair, SELECT(j) must be set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.

N

          N is INTEGER
          The order of the matrix T. N >= 0.

T

          T is COMPLEX array, dimension (LDT,N)
          The upper triangular matrix T.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

VL

          VL is COMPLEX array, dimension (LDVL,M)
          If JOB = 'E' or 'B', VL must contain left eigenvectors of T
          (or of any Q*T*Q**H with Q unitary), corresponding to the
          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
          must be stored in consecutive columns of VL, as returned by
          CHSEIN or CTREVC.
          If JOB = 'V', VL is not referenced.

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,M)
          If JOB = 'E' or 'B', VR must contain right eigenvectors of T
          (or of any Q*T*Q**H with Q unitary), corresponding to the
          eigenpairs specified by HOWMNY and SELECT. The eigenvectors
          must be stored in consecutive columns of VR, as returned by
          CHSEIN or CTREVC.
          If JOB = 'V', VR is not referenced.

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.

S

          S is REAL array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the
          selected eigenvalues, stored in consecutive elements of the
          array. Thus S(j), SEP(j), and the j-th columns of VL and VR
          all correspond to the same eigenpair (but not in general the
          j-th eigenpair, unless all eigenpairs are selected).
          If JOB = 'V', S is not referenced.

SEP

          SEP is REAL array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array.
          If JOB = 'E', SEP is not referenced.

MM

          MM is INTEGER
          The number of elements in the arrays S (if JOB = 'E' or 'B')
           and/or SEP (if JOB = 'V' or 'B'). MM >= M.

M

          M is INTEGER
          The number of elements of the arrays S and/or SEP actually
          used to store the estimated condition numbers.
          If HOWMNY = 'A', M is set to N.

WORK

          WORK is COMPLEX array, dimension (LDWORK,N+6)
          If JOB = 'E', WORK is not referenced.

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK.
          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

RWORK

          RWORK is REAL array, dimension (N)
          If JOB = 'E', RWORK is not referenced.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The reciprocal of the condition number of an eigenvalue lambda is
  defined as
          S(lambda) = |v**H*u| / (norm(u)*norm(v))
  where u and v are the right and left eigenvectors of T corresponding
  to lambda; v**H denotes the conjugate transpose of v, and norm(u)
  denotes the Euclidean norm. These reciprocal condition numbers always
  lie between zero (very badly conditioned) and one (very well
  conditioned). If n = 1, S(lambda) is defined to be 1.
  An approximate error bound for a computed eigenvalue W(i) is given by
                      EPS * norm(T) / S(i)
  where EPS is the machine precision.
  The reciprocal of the condition number of the right eigenvector u
  corresponding to lambda is defined as follows. Suppose
              T = ( lambda  c  )
                  (   0    T22 )
  Then the reciprocal condition number is
          SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
  where sigma-min denotes the smallest singular value. We approximate
  the smallest singular value by the reciprocal of an estimate of the
  one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
  defined to be abs(T(1,1)).
  An approximate error bound for a computed right eigenvector VR(i)
  is given by
                      EPS * norm(T) / SEP(i)

subroutine ctrti2 (character UPLO, character DIAG, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).

Purpose:

 CTRTI2 computes the inverse of a complex upper or lower triangular
 matrix.
 This is the Level 2 BLAS version of the algorithm.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading n by n upper triangular part of the array A contains
          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of the array A contains
          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.  If DIAG = 'U', the
          diagonal elements of A are also not referenced and are
          assumed to be 1.
          On exit, the (triangular) inverse of the original matrix, in
          the same storage format.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrtri (character UPLO, character DIAG, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CTRTRI

Purpose:

 CTRTRI computes the inverse of a complex upper or lower triangular
 matrix A.
 This is the Level 3 BLAS version of the algorithm.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of the array A contains
          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of the array A contains
          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.  If DIAG = 'U', the
          diagonal elements of A are also not referenced and are
          assumed to be 1.
          On exit, the (triangular) inverse of the original matrix, in
          the same storage format.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
               matrix is singular and its inverse can not be computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrtrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CTRTRS

Purpose:

 CTRTRS solves a triangular system of the form
    A * X = B,  A**T * X = B,  or  A**H * X = B,
 where A is a triangular matrix of order N, and B is an N-by-NRHS
 matrix.  A check is made to verify that A is nonsingular.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of
          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
          triangular part of the array A contains the lower triangular
          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the i-th diagonal element of A is zero,
               indicating that the matrix is singular and the solutions
               X have not been computed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctrttf (character TRANSR, character UPLO, integer N, complex, dimension( 0: lda-1, 0: * ) A, integer LDA, complex, dimension( 0: * ) ARF, integer INFO)

CTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).

Purpose:

 CTRTTF copies a triangular matrix A from standard full format (TR)
 to rectangular full packed format (TF) .

Parameters:

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  ARF in Normal mode is wanted;
          = 'C':  ARF in Conjugate Transpose mode is wanted;

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension ( LDA, N )
          On entry, the triangular matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of the array A contains
          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of the array A contains
          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.

LDA

          LDA is INTEGER
          The leading dimension of the matrix A.  LDA >= max(1,N).

ARF

          ARF is COMPLEX array, dimension ( N*(N+1)/2 ),
          On exit, the upper or lower triangular matrix A stored in
          RFP format. For a further discussion see Notes below.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.
         RFP A                   RFP A
                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55
        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.
         RFP A                   RFP A
                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44
        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42
  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

subroutine ctrttp (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) AP, integer INFO)

CTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).

Purpose:

 CTRTTP copies a triangular matrix A from full format (TR) to standard
 packed format (TP).

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrices AP and A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the triangular matrix A.  If UPLO = 'U', the leading
          N-by-N upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

AP

          AP is COMPLEX array, dimension ( N*(N+1)/2 ),
          On exit, the upper or lower triangular matrix A, packed
          columnwise in a linear array. The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctzrzf (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CTZRZF

Purpose:

 CTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
 to upper triangular form by means of unitary transformations.
 The upper trapezoidal matrix A is factored as
    A = ( R  0 ) * Z,
 where Z is an N-by-N unitary matrix and R is an M-by-M upper
 triangular matrix.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= M.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements M+1 to
          N of the first M rows of A, with the array TAU, represent the
          unitary matrix Z as a product of M elementary reflectors.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (M)
          The scalar factors of the elementary reflectors.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.  LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

  The N-by-N matrix Z can be computed by
     Z =  Z(1)*Z(2)* ... *Z(M)
  where each N-by-N Z(k) is given by
     Z(k) = I - tau(k)*v(k)*v(k)**H
  with v(k) is the kth row vector of the M-by-N matrix
     V = ( I   A(:,M+1:N) )
  I is the M-by-M identity matrix, A(:,M+1:N)
  is the output stored in A on exit from DTZRZF,
  and tau(k) is the kth element of the array TAU.

subroutine cunbdb (character TRANS, character SIGNS, integer M, integer P, integer Q, complex, dimension( ldx11, * ) X11, integer LDX11, complex, dimension( ldx12, * ) X12, integer LDX12, complex, dimension( ldx21, * ) X21, integer LDX21, complex, dimension( ldx22, * ) X22, integer LDX22, real, dimension( * ) THETA, real, dimension( * ) PHI, complex, dimension( * ) TAUP1, complex, dimension( * ) TAUP2, complex, dimension( * ) TAUQ1, complex, dimension( * ) TAUQ2, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNBDB

Purpose:

 CUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
 partitioned unitary matrix X:
                                 [ B11 | B12 0  0 ]
     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**H
 X = [-----------] = [---------] [----------------] [---------]   .
     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
                                 [  0  |  0  0  I ]
 X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
 not the case, then X must be transposed and/or permuted. This can be
 done in constant time using the TRANS and SIGNS options. See CUNCSD
 for details.)
 The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
 (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
 represented implicitly by Householder vectors.
 B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
 implicitly by angles THETA, PHI.

Parameters:

TRANS

          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.

SIGNS

          SIGNS is CHARACTER
          = 'O':      The lower-left block is made nonpositive (the
                      "other" convention);
          otherwise:  The upper-right block is made nonpositive (the
                      "default" convention).

M

          M is INTEGER
          The number of rows and columns in X.

P

          P is INTEGER
          The number of rows in X11 and X12. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <=
          MIN(P,M-P,M-Q).

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
          On entry, the top-left block of the unitary matrix to be
          reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the columns of tril(X11) specify reflectors for P1,
             the rows of triu(X11,1) specify reflectors for Q1;
          else TRANS = 'T', and
             the rows of triu(X11) specify reflectors for P1,
             the columns of tril(X11,-1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
          P; else LDX11 >= Q.

X12

          X12 is COMPLEX array, dimension (LDX12,M-Q)
          On entry, the top-right block of the unitary matrix to
          be reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the rows of triu(X12) specify the first P reflectors for
             Q2;
          else TRANS = 'T', and
             the columns of tril(X12) specify the first P reflectors
             for Q2.

LDX12

          LDX12 is INTEGER
          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
          P; else LDX11 >= M-Q.

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
          On entry, the bottom-left block of the unitary matrix to
          be reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the columns of tril(X21) specify reflectors for P2;
          else TRANS = 'T', and
             the rows of triu(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
          M-P; else LDX21 >= Q.

X22

          X22 is COMPLEX array, dimension (LDX22,M-Q)
          On entry, the bottom-right block of the unitary matrix to
          be reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
             M-P-Q reflectors for Q2,
          else TRANS = 'T', and
             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
             M-P-Q reflectors for P2.

LDX22

          LDX22 is INTEGER
          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
          M-P; else LDX22 >= M-Q.

THETA

          THETA is REAL array, dimension (Q)
          The entries of the bidiagonal blocks B11, B12, B21, B22 can
          be computed from the angles THETA and PHI. See Further
          Details.

PHI

          PHI is REAL array, dimension (Q-1)
          The entries of the bidiagonal blocks B11, B12, B21, B22 can
          be computed from the angles THETA and PHI. See Further
          Details.

TAUP1

          TAUP1 is COMPLEX array, dimension (P)
          The scalar factors of the elementary reflectors that define
          P1.

TAUP2

          TAUP2 is COMPLEX array, dimension (M-P)
          The scalar factors of the elementary reflectors that define
          P2.

TAUQ1

          TAUQ1 is COMPLEX array, dimension (Q)
          The scalar factors of the elementary reflectors that define
          Q1.

TAUQ2

          TAUQ2 is COMPLEX array, dimension (M-Q)
          The scalar factors of the elementary reflectors that define
          Q2.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= M-Q.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The bidiagonal blocks B11, B12, B21, and B22 are represented
  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
  lower bidiagonal. Every entry in each bidiagonal band is a product
  of a sine or cosine of a THETA with a sine or cosine of a PHI. See
  [1] or CUNCSD for details.
  P1, P2, Q1, and Q2 are represented as products of elementary
  reflectors. See CUNCSD for details on generating P1, P2, Q1, and Q2
  using CUNGQR and CUNGLQ.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

subroutine cunbdb1 (integer M, integer P, integer Q, complex, dimension(ldx11,*) X11, integer LDX11, complex, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, complex, dimension(*) TAUP1, complex, dimension(*) TAUP2, complex, dimension(*) TAUQ1, complex, dimension(*) WORK, integer LWORK, integer INFO)

CUNBDB1

Purpose:

 CUNBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
 M-P, or M-Q. Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in
 which Q is not the minimum dimension.
 The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
 angles THETA, PHI.

Parameters:

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= M.

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <=
           MIN(P,M-P,M-Q).

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is REAL array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is REAL array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is COMPLEX array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is COMPLEX array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is COMPLEX array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

July 2012

Further Details:

The upper-bidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry in each bidiagonal band is a product of a sine or cosine of a THETA with a sine or cosine of a PHI. See [1] or CUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR and CUNGLQ.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

subroutine cunbdb2 (integer M, integer P, integer Q, complex, dimension(ldx11,*) X11, integer LDX11, complex, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, complex, dimension(*) TAUP1, complex, dimension(*) TAUP2, complex, dimension(*) TAUQ1, complex, dimension(*) WORK, integer LWORK, integer INFO)

CUNBDB2

Purpose:

 CUNBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
 Q, or M-Q. Routines CUNBDB1, CUNBDB3, and CUNBDB4 handle cases in
 which P is not the minimum dimension.
 The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
 angles THETA, PHI.

Parameters:

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is REAL array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is REAL array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is COMPLEX array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is COMPLEX array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is COMPLEX array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

July 2012

Further Details:

  The upper-bidiagonal blocks B11, B21 are represented implicitly by
  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
  in each bidiagonal band is a product of a sine or cosine of a THETA
  with a sine or cosine of a PHI. See [1] or CUNCSD for details.
  P1, P2, and Q1 are represented as products of elementary reflectors.
  See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR
  and CUNGLQ.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

subroutine cunbdb3 (integer M, integer P, integer Q, complex, dimension(ldx11,*) X11, integer LDX11, complex, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, complex, dimension(*) TAUP1, complex, dimension(*) TAUP2, complex, dimension(*) TAUQ1, complex, dimension(*) WORK, integer LWORK, integer INFO)

CUNBDB3

Purpose:

 CUNBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
 Q, or M-Q. Routines CUNBDB1, CUNBDB2, and CUNBDB4 handle cases in
 which M-P is not the minimum dimension.
 The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
 implicitly by angles THETA, PHI.

Parameters:

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is REAL array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is REAL array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is COMPLEX array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is COMPLEX array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is COMPLEX array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

July 2012

Further Details:

The upper-bidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry in each bidiagonal band is a product of a sine or cosine of a THETA with a sine or cosine of a PHI. See [1] or CUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR and CUNGLQ.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

subroutine cunbdb4 (integer M, integer P, integer Q, complex, dimension(ldx11,*) X11, integer LDX11, complex, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, complex, dimension(*) TAUP1, complex, dimension(*) TAUP2, complex, dimension(*) TAUQ1, complex, dimension(*) PHANTOM, complex, dimension(*) WORK, integer LWORK, integer INFO)

CUNBDB4

Purpose:

 CUNBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
 M-P, or Q. Routines CUNBDB1, CUNBDB2, and CUNBDB3 handle cases in
 which M-Q is not the minimum dimension.
 The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
 implicitly by angles THETA, PHI.

Parameters:

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= M.

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M and
           M-Q <= min(P,M-P,Q).

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is REAL array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is REAL array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is COMPLEX array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is COMPLEX array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is COMPLEX array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

PHANTOM

          PHANTOM is COMPLEX array, dimension (M)
           The routine computes an M-by-1 column vector Y that is
           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
           Y(P+1:M), respectively.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

July 2012

Further Details:

The upper-bidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry in each bidiagonal band is a product of a sine or cosine of a THETA with a sine or cosine of a PHI. See [1] or CUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR and CUNGLQ.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

subroutine cunbdb5 (integer M1, integer M2, integer N, complex, dimension(*) X1, integer INCX1, complex, dimension(*) X2, integer INCX2, complex, dimension(ldq1,*) Q1, integer LDQ1, complex, dimension(ldq2,*) Q2, integer LDQ2, complex, dimension(*) WORK, integer LWORK, integer INFO)

CUNBDB5

Purpose:

 CUNBDB5 orthogonalizes the column vector
      X = [ X1 ]
          [ X2 ]
 with respect to the columns of
      Q = [ Q1 ] .
          [ Q2 ]
 The columns of Q must be orthonormal.
 If the projection is zero according to Kahan's "twice is enough"
 criterion, then some other vector from the orthogonal complement
 is returned. This vector is chosen in an arbitrary but deterministic
 way.

Parameters:

M1

          M1 is INTEGER
           The dimension of X1 and the number of rows in Q1. 0 <= M1.

M2

          M2 is INTEGER
           The dimension of X2 and the number of rows in Q2. 0 <= M2.

N

          N is INTEGER
           The number of columns in Q1 and Q2. 0 <= N.

X1

          X1 is COMPLEX array, dimension (M1)
           On entry, the top part of the vector to be orthogonalized.
           On exit, the top part of the projected vector.

INCX1

          INCX1 is INTEGER
           Increment for entries of X1.

X2

          X2 is COMPLEX array, dimension (M2)
           On entry, the bottom part of the vector to be
           orthogonalized. On exit, the bottom part of the projected
           vector.

INCX2

          INCX2 is INTEGER
           Increment for entries of X2.

Q1

          Q1 is COMPLEX array, dimension (LDQ1, N)
           The top part of the orthonormal basis matrix.

LDQ1

          LDQ1 is INTEGER
           The leading dimension of Q1. LDQ1 >= M1.

Q2

          Q2 is COMPLEX array, dimension (LDQ2, N)
           The bottom part of the orthonormal basis matrix.

LDQ2

          LDQ2 is INTEGER
           The leading dimension of Q2. LDQ2 >= M2.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= N.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

July 2012

subroutine cunbdb6 (integer M1, integer M2, integer N, complex, dimension(*) X1, integer INCX1, complex, dimension(*) X2, integer INCX2, complex, dimension(ldq1,*) Q1, integer LDQ1, complex, dimension(ldq2,*) Q2, integer LDQ2, complex, dimension(*) WORK, integer LWORK, integer INFO)

CUNBDB6

Purpose:

 CUNBDB6 orthogonalizes the column vector
      X = [ X1 ]
          [ X2 ]
 with respect to the columns of
      Q = [ Q1 ] .
          [ Q2 ]
 The columns of Q must be orthonormal.
 If the projection is zero according to Kahan's "twice is enough"
 criterion, then the zero vector is returned.

Parameters:

M1

          M1 is INTEGER
           The dimension of X1 and the number of rows in Q1. 0 <= M1.

M2

          M2 is INTEGER
           The dimension of X2 and the number of rows in Q2. 0 <= M2.

N

          N is INTEGER
           The number of columns in Q1 and Q2. 0 <= N.

X1

          X1 is COMPLEX array, dimension (M1)
           On entry, the top part of the vector to be orthogonalized.
           On exit, the top part of the projected vector.

INCX1

          INCX1 is INTEGER
           Increment for entries of X1.

X2

          X2 is COMPLEX array, dimension (M2)
           On entry, the bottom part of the vector to be
           orthogonalized. On exit, the bottom part of the projected
           vector.

INCX2

          INCX2 is INTEGER
           Increment for entries of X2.

Q1

          Q1 is COMPLEX array, dimension (LDQ1, N)
           The top part of the orthonormal basis matrix.

LDQ1

          LDQ1 is INTEGER
           The leading dimension of Q1. LDQ1 >= M1.

Q2

          Q2 is COMPLEX array, dimension (LDQ2, N)
           The bottom part of the orthonormal basis matrix.

LDQ2

          LDQ2 is INTEGER
           The leading dimension of Q2. LDQ2 >= M2.

WORK

          WORK is COMPLEX array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= N.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

July 2012

recursive subroutine cuncsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, character SIGNS, integer M, integer P, integer Q, complex, dimension( ldx11, * ) X11, integer LDX11, complex, dimension( ldx12, * ) X12, integer LDX12, complex, dimension( ldx21, * ) X21, integer LDX21, complex, dimension( ldx22, * ) X22, integer LDX22, real, dimension( * ) THETA, complex, dimension( ldu1, * ) U1, integer LDU1, complex, dimension( ldu2, * ) U2, integer LDU2, complex, dimension( ldv1t, * ) V1T, integer LDV1T, complex, dimension( ldv2t, * ) V2T, integer LDV2T, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer INFO)

CUNCSD

Purpose:

 CUNCSD computes the CS decomposition of an M-by-M partitioned
 unitary matrix X:
                                 [  I  0  0 |  0  0  0 ]
                                 [  0  C  0 |  0 -S  0 ]
     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**H
 X = [-----------] = [---------] [---------------------] [---------]   .
     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
                                 [  0  S  0 |  0  C  0 ]
                                 [  0  0  I |  0  0  0 ]
 X11 is P-by-Q. The unitary matrices U1, U2, V1, and V2 are P-by-P,
 (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
 R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
 which R = MIN(P,M-P,Q,M-Q).

Parameters:

JOBU1

          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed.

JOBU2

          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed.

JOBV1T

          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed.

JOBV2T

          JOBV2T is CHARACTER
          = 'Y':      V2T is computed;
          otherwise:  V2T is not computed.

TRANS

          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.

SIGNS

          SIGNS is CHARACTER
          = 'O':      The lower-left block is made nonpositive (the
                      "other" convention);
          otherwise:  The upper-right block is made nonpositive (the
                      "default" convention).

M

          M is INTEGER
          The number of rows and columns in X.

P

          P is INTEGER
          The number of rows in X11 and X12. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
          On entry, part of the unitary matrix whose CSD is desired.

LDX11

          LDX11 is INTEGER
          The leading dimension of X11. LDX11 >= MAX(1,P).

X12

          X12 is COMPLEX array, dimension (LDX12,M-Q)
          On entry, part of the unitary matrix whose CSD is desired.

LDX12

          LDX12 is INTEGER
          The leading dimension of X12. LDX12 >= MAX(1,P).

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
          On entry, part of the unitary matrix whose CSD is desired.

LDX21

          LDX21 is INTEGER
          The leading dimension of X11. LDX21 >= MAX(1,M-P).

X22

          X22 is COMPLEX array, dimension (LDX22,M-Q)
          On entry, part of the unitary matrix whose CSD is desired.

LDX22

          LDX22 is INTEGER
          The leading dimension of X11. LDX22 >= MAX(1,M-P).

THETA

          THETA is REAL array, dimension (R), in which R =
          MIN(P,M-P,Q,M-Q).
          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1

          U1 is COMPLEX array, dimension (LDU1,P)
          If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1.

LDU1

          LDU1 is INTEGER
          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P).

U2

          U2 is COMPLEX array, dimension (LDU2,M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
          matrix U2.

LDU2

          LDU2 is INTEGER
          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P).

V1T

          V1T is COMPLEX array, dimension (LDV1T,Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
          matrix V1**H.

LDV1T

          LDV1T is INTEGER
          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q).

V2T

          V2T is COMPLEX array, dimension (LDV2T,M-Q)
          If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary
          matrix V2**H.

LDV2T

          LDV2T is INTEGER
          The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
          MAX(1,M-Q).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the work array, and no error
          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension MAX(1,LRWORK)
          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
          If INFO > 0 on exit, RWORK(2:R) contains the values PHI(1),
          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
          define the matrix in intermediate bidiagonal-block form
          remaining after nonconvergence. INFO specifies the number
          of nonzero PHI's.

LRWORK

          LRWORK is INTEGER
          The dimension of the array RWORK.
          If LRWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the RWORK array, returns
          this value as the first entry of the work array, and no error
          message related to LRWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  CBBCSD did not converge. See the description of RWORK
                above for details.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine cuncsd2by1 (character JOBU1, character JOBU2, character JOBV1T, integer M, integer P, integer Q, complex, dimension(ldx11,*) X11, integer LDX11, complex, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, complex, dimension(ldu1,*) U1, integer LDU1, complex, dimension(ldu2,*) U2, integer LDU2, complex, dimension(ldv1t,*) V1T, integer LDV1T, complex, dimension(*) WORK, integer LWORK, real, dimension(*) RWORK, integer LRWORK, integer, dimension(*) IWORK, integer INFO)

CUNCSD2BY1

Purpose:

 CUNCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
 orthonormal columns that has been partitioned into a 2-by-1 block
 structure:
                                [  I1 0  0 ]
                                [  0  C  0 ]
          [ X11 ]   [ U1 |    ] [  0  0  0 ]
      X = [-----] = [---------] [----------] V1**T .
          [ X21 ]   [    | U2 ] [  0  0  0 ]
                                [  0  S  0 ]
                                [  0  0  I2]
 X11 is P-by-Q. The unitary matrices U1, U2, and V1 are P-by-P,
 (M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R
 nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which
 R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a
 K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

Parameters:

JOBU1

          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed.

JOBU2

          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed.

JOBV1T

          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed.

M

          M is INTEGER
          The number of rows in X.

P

          P is INTEGER
          The number of rows in X11. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is COMPLEX array, dimension (LDX11,Q)
          On entry, part of the unitary matrix whose CSD is desired.

LDX11

          LDX11 is INTEGER
          The leading dimension of X11. LDX11 >= MAX(1,P).

X21

          X21 is COMPLEX array, dimension (LDX21,Q)
          On entry, part of the unitary matrix whose CSD is desired.

LDX21

          LDX21 is INTEGER
          The leading dimension of X21. LDX21 >= MAX(1,M-P).

THETA

          THETA is REAL array, dimension (R), in which R =
          MIN(P,M-P,Q,M-Q).
          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1

          U1 is COMPLEX array, dimension (P)
          If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1.

LDU1

          LDU1 is INTEGER
          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P).

U2

          U2 is COMPLEX array, dimension (M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
          matrix U2.

LDU2

          LDU2 is INTEGER
          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P).

V1T

          V1T is COMPLEX array, dimension (Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
          matrix V1**T.

LDV1T

          LDV1T is INTEGER
          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the work array, and no error
          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (MAX(1,LRWORK))
          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
          If INFO > 0 on exit, RWORK(2:R) contains the values PHI(1),
          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
          define the matrix in intermediate bidiagonal-block form
          remaining after nonconvergence. INFO specifies the number
          of nonzero PHI's.

LRWORK

          LRWORK is INTEGER
          The dimension of the array RWORK.
          If LRWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the RWORK array, returns
          this value as the first entry of the work array, and no error
          message related to LRWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  CBBCSD did not converge. See the description of WORK
                above for details.

References:

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine cung2l (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

CUNG2L generates all or part of the unitary matrix Q from a QL factorization determined by cgeqlf (unblocked algorithm).

Purpose:

 CUNG2L generates an m by n complex matrix Q with orthonormal columns,
 which is defined as the last n columns of a product of k elementary
 reflectors of order m
       Q  =  H(k) . . . H(2) H(1)
 as returned by CGEQLF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by CGEQLF in the last k columns of its array
          argument A.
          On exit, the m-by-n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQLF.

WORK

          WORK is COMPLEX array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cung2r (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

CUNG2R

Purpose:

 CUNG2R generates an m by n complex matrix Q with orthonormal columns,
 which is defined as the first n columns of a product of k elementary
 reflectors of order m
       Q  =  H(1) H(2) . . . H(k)
 as returned by CGEQRF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by CGEQRF in the first k columns of its array
          argument A.
          On exit, the m by n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQRF.

WORK

          WORK is COMPLEX array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunghr (integer N, integer ILO, integer IHI, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNGHR

Purpose:

 CUNGHR generates a complex unitary matrix Q which is defined as the
 product of IHI-ILO elementary reflectors of order N, as returned by
 CGEHRD:
 Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters:

N

          N is INTEGER
          The order of the matrix Q. N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI must have the same values as in the previous call
          of CGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by CGEHRD.
          On exit, the N-by-N unitary matrix Q.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

TAU

          TAU is COMPLEX array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEHRD.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= IHI-ILO.
          For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cungl2 (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

CUNGL2 generates all or part of the unitary matrix Q from an LQ factorization determined by cgelqf (unblocked algorithm).

Purpose:

 CUNGL2 generates an m-by-n complex matrix Q with orthonormal rows,
 which is defined as the first m rows of a product of k elementary
 reflectors of order n
       Q  =  H(k)**H . . . H(2)**H H(1)**H
 as returned by CGELQF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines
          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by CGELQF in the first k rows of its array argument A.
          On exit, the m by n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGELQF.

WORK

          WORK is COMPLEX array, dimension (M)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunglq (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNGLQ

Purpose:

 CUNGLQ generates an M-by-N complex matrix Q with orthonormal rows,
 which is defined as the first M rows of a product of K elementary
 reflectors of order N
       Q  =  H(k)**H . . . H(2)**H H(1)**H
 as returned by CGELQF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines
          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by CGELQF in the first k rows of its array argument A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGELQF.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cungql (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNGQL

Purpose:

 CUNGQL generates an M-by-N complex matrix Q with orthonormal columns,
 which is defined as the last N columns of a product of K elementary
 reflectors of order M
       Q  =  H(k) . . . H(2) H(1)
 as returned by CGEQLF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by CGEQLF in the last k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQLF.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cungqr (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNGQR

Purpose:

 CUNGQR generates an M-by-N complex matrix Q with orthonormal columns,
 which is defined as the first N columns of a product of K elementary
 reflectors of order M
       Q  =  H(1) H(2) . . . H(k)
 as returned by CGEQRF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by CGEQRF in the first k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQRF.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cungr2 (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer INFO)

CUNGR2 generates all or part of the unitary matrix Q from an RQ factorization determined by cgerqf (unblocked algorithm).

Purpose:

 CUNGR2 generates an m by n complex matrix Q with orthonormal rows,
 which is defined as the last m rows of a product of k elementary
 reflectors of order n
       Q  =  H(1)**H H(2)**H . . . H(k)**H
 as returned by CGERQF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by CGERQF in the last k rows of its array argument
          A.
          On exit, the m-by-n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGERQF.

WORK

          WORK is COMPLEX array, dimension (M)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cungrq (integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNGRQ

Purpose:

 CUNGRQ generates an M-by-N complex matrix Q with orthonormal rows,
 which is defined as the last M rows of a product of K elementary
 reflectors of order N
       Q  =  H(1)**H H(2)**H . . . H(k)**H
 as returned by CGERQF.

Parameters:

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by CGERQF in the last k rows of its array argument
          A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGERQF.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cungtr (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNGTR

Purpose:

 CUNGTR generates a complex unitary matrix Q which is defined as the
 product of n-1 elementary reflectors of order N, as returned by
 CHETRD:
 if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
 if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from CHETRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from CHETRD.

N

          N is INTEGER
          The order of the matrix Q. N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by CHETRD.
          On exit, the N-by-N unitary matrix Q.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= N.

TAU

          TAU is COMPLEX array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CHETRD.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= N-1.
          For optimum performance LWORK >= (N-1)*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunm22 (character SIDE, character TRANS, integer M, integer N, integer N1, integer N2, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNM22 multiplies a general matrix by a banded unitary matrix.

Purpose

  CUNM22 overwrites the general complex M-by-N matrix C with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'C':      Q**H * C       C * Q**H
  where Q is a complex unitary matrix of order NQ, with NQ = M if
  SIDE = 'L' and NQ = N if SIDE = 'R'.
  The unitary matrix Q processes a 2-by-2 block structure
         [  Q11  Q12  ]
     Q = [            ]
         [  Q21  Q22  ],
  where Q12 is an N1-by-N1 lower triangular matrix and Q21 is an
  N2-by-N2 upper triangular matrix.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  apply Q (No transpose);
          = 'C':  apply Q**H (Conjugate transpose).

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

N1
N2

          N1 is INTEGER
          N2 is INTEGER
          The dimension of Q12 and Q21, respectively. N1, N2 >= 0.
          The following requirement must be satisfied:
          N1 + N2 = M if SIDE = 'L' and N1 + N2 = N if SIDE = 'R'.

Q

          Q is COMPLEX array, dimension
                              (LDQ,M) if SIDE = 'L'
                              (LDQ,N) if SIDE = 'R'

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if SIDE = 'L'; LDQ >= max(1,N) if SIDE = 'R'.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= M*N.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine cunm2l (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CUNM2L multiplies a general matrix by the unitary matrix from a QL factorization determined by cgeqlf (unblocked algorithm).

Purpose:

 CUNM2L overwrites the general complex m-by-n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**H* C  if SIDE = 'L' and TRANS = 'C', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**H if SIDE = 'R' and TRANS = 'C',
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(k) . . . H(2) H(1)
 as returned by CGEQLF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left
          = 'R': apply Q or Q**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'C': apply Q**H (Conjugate transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGEQLF in the last k columns of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQLF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunm2r (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf (unblocked algorithm).

Purpose:

 CUNM2R overwrites the general complex m-by-n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**H* C  if SIDE = 'L' and TRANS = 'C', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**H if SIDE = 'R' and TRANS = 'C',
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by CGEQRF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left
          = 'R': apply Q or Q**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'C': apply Q**H (Conjugate transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGEQRF in the first k columns of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQRF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmbr (character VECT, character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMBR

Purpose:

 If VECT = 'Q', CUNMBR overwrites the general complex M-by-N matrix C
 with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 If VECT = 'P', CUNMBR overwrites the general complex M-by-N matrix C
 with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      P * C          C * P
 TRANS = 'C':      P**H * C       C * P**H
 Here Q and P**H are the unitary matrices determined by CGEBRD when
 reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. Q
 and P**H are defined as products of elementary reflectors H(i) and
 G(i) respectively.
 Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
 order of the unitary matrix Q or P**H that is applied.
 If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
 if nq >= k, Q = H(1) H(2) . . . H(k);
 if nq < k, Q = H(1) H(2) . . . H(nq-1).
 If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
 if k < nq, P = G(1) G(2) . . . G(k);
 if k >= nq, P = G(1) G(2) . . . G(nq-1).

Parameters:

VECT

          VECT is CHARACTER*1
          = 'Q': apply Q or Q**H;
          = 'P': apply P or P**H.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q, Q**H, P or P**H from the Left;
          = 'R': apply Q, Q**H, P or P**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q or P;
          = 'C':  Conjugate transpose, apply Q**H or P**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          If VECT = 'Q', the number of columns in the original
          matrix reduced by CGEBRD.
          If VECT = 'P', the number of rows in the original
          matrix reduced by CGEBRD.
          K >= 0.

A

          A is COMPLEX array, dimension
                                (LDA,min(nq,K)) if VECT = 'Q'
                                (LDA,nq)        if VECT = 'P'
          The vectors which define the elementary reflectors H(i) and
          G(i), whose products determine the matrices Q and P, as
          returned by CGEBRD.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If VECT = 'Q', LDA >= max(1,nq);
          if VECT = 'P', LDA >= max(1,min(nq,K)).

TAU

          TAU is COMPLEX array, dimension (min(nq,K))
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i) which determines Q or P, as returned
          by CGEBRD in the array argument TAUQ or TAUP.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q
          or P*C or P**H*C or C*P or C*P**H.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M);
          if N = 0 or M = 0, LWORK >= 1.
          For optimum performance LWORK >= max(1,N*NB) if SIDE = 'L',
          and LWORK >= max(1,M*NB) if SIDE = 'R', where NB is the
          optimal blocksize. (NB = 0 if M = 0 or N = 0.)
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmhr (character SIDE, character TRANS, integer M, integer N, integer ILO, integer IHI, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMHR

Purpose:

 CUNMHR overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix of order nq, with nq = m if
 SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
 IHI-ILO elementary reflectors, as returned by CGEHRD:
 Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'C': apply Q**H (Conjugate transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI must have the same values as in the previous call
          of CGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
          ILO = 1 and IHI = 0, if M = 0;
          if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
          ILO = 1 and IHI = 0, if N = 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by CGEHRD.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

TAU

          TAU is COMPLEX array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEHRD.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunml2 (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CUNML2 multiplies a general matrix by the unitary matrix from a LQ factorization determined by cgelqf (unblocked algorithm).

Purpose:

 CUNML2 overwrites the general complex m-by-n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**H* C  if SIDE = 'L' and TRANS = 'C', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**H if SIDE = 'R' and TRANS = 'C',
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(k)**H . . . H(2)**H H(1)**H
 as returned by CGELQF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left
          = 'R': apply Q or Q**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'C': apply Q**H (Conjugate transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGELQF in the first k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGELQF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmlq (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMLQ

Purpose:

 CUNMLQ overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(k)**H . . . H(2)**H H(1)**H
 as returned by CGELQF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Conjugate transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGELQF in the first k rows of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGELQF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmql (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMQL

Purpose:

 CUNMQL overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(k) . . . H(2) H(1)
 as returned by CGEQLF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGEQLF in the last k columns of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQLF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmqr (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMQR

Purpose:

 CUNMQR overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by CGEQRF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Conjugate transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGEQRF in the first k columns of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGEQRF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmr2 (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CUNMR2 multiplies a general matrix by the unitary matrix from a RQ factorization determined by cgerqf (unblocked algorithm).

Purpose:

 CUNMR2 overwrites the general complex m-by-n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**H* C  if SIDE = 'L' and TRANS = 'C', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**H if SIDE = 'R' and TRANS = 'C',
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(1)**H H(2)**H . . . H(k)**H
 as returned by CGERQF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left
          = 'R': apply Q or Q**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'C': apply Q**H (Conjugate transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGERQF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGERQF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmr3 (character SIDE, character TRANS, integer M, integer N, integer K, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CUNMR3 multiplies a general matrix by the unitary matrix from a RZ factorization determined by ctzrzf (unblocked algorithm).

Purpose:

 CUNMR3 overwrites the general complex m by n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**H* C  if SIDE = 'L' and TRANS = 'C', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**H if SIDE = 'R' and TRANS = 'C',
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by CTZRZF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left
          = 'R': apply Q or Q**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'C': apply Q**H (Conjugate transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

L

          L is INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CTZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CTZRZF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

 

subroutine cunmrq (character SIDE, character TRANS, integer M, integer N, integer K, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMRQ

Purpose:

 CUNMRQ overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(1)**H H(2)**H . . . H(k)**H
 as returned by CGERQF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGERQF in the last k rows of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CGERQF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cunmrz (character SIDE, character TRANS, integer M, integer N, integer K, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMRZ

Purpose:

 CUNMRZ overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by CTZRZF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Conjugate transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

L

          L is INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CTZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CTZRZF.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

 

subroutine cunmtr (character SIDE, character UPLO, character TRANS, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CUNMTR

Purpose:

 CUNMTR overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix of order nq, with nq = m if
 SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
 nq-1 elementary reflectors, as returned by CHETRD:
 if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
 if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from CHETRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from CHETRD.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Conjugate transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

A

          A is COMPLEX array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by CHETRD.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

TAU

          TAU is COMPLEX array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CHETRD.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >=M*NB if SIDE = 'R', where NB is the optimal
          blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cupgtr (character UPLO, integer N, complex, dimension( * ) AP, complex, dimension( * ) TAU, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( * ) WORK, integer INFO)

CUPGTR

Purpose:

 CUPGTR generates a complex unitary matrix Q which is defined as the
 product of n-1 elementary reflectors H(i) of order n, as returned by
 CHPTRD using packed storage:
 if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
 if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to CHPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to CHPTRD.

N

          N is INTEGER
          The order of the matrix Q. N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The vectors which define the elementary reflectors, as
          returned by CHPTRD.

TAU

          TAU is COMPLEX array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CHPTRD.

Q

          Q is COMPLEX array, dimension (LDQ,N)
          The N-by-N unitary matrix Q.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N).

WORK

          WORK is COMPLEX array, dimension (N-1)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine cupmtr (character SIDE, character UPLO, character TRANS, integer M, integer N, complex, dimension( * ) AP, complex, dimension( * ) TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CUPMTR

Purpose:

 CUPMTR overwrites the general complex M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'C':      Q**H * C       C * Q**H
 where Q is a complex unitary matrix of order nq, with nq = m if
 SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
 nq-1 elementary reflectors, as returned by CHPTRD using packed
 storage:
 if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
 if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to CHPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to CHPTRD.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Conjugate transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

AP

          AP is COMPLEX array, dimension
                               (M*(M+1)/2) if SIDE = 'L'
                               (N*(N+1)/2) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by CHPTRD.  AP is modified by the routine but
          restored on exit.

TAU

          TAU is COMPLEX array, dimension (M-1) if SIDE = 'L'
                                     or (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by CHPTRD.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                                   (N) if SIDE = 'L'
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dorm22 (character SIDE, character TRANS, integer M, integer N, integer N1, integer N2, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DORM22 multiplies a general matrix by a banded orthogonal matrix.

Purpose

  DORM22 overwrites the general real M-by-N matrix C with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T
  where Q is a real orthogonal matrix of order NQ, with NQ = M if
  SIDE = 'L' and NQ = N if SIDE = 'R'.
  The orthogonal matrix Q processes a 2-by-2 block structure
         [  Q11  Q12  ]
     Q = [            ]
         [  Q21  Q22  ],
  where Q12 is an N1-by-N1 lower triangular matrix and Q21 is an
  N2-by-N2 upper triangular matrix.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  apply Q (No transpose);
          = 'C':  apply Q**T (Conjugate transpose).

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

N1
N2

          N1 is INTEGER
          N2 is INTEGER
          The dimension of Q12 and Q21, respectively. N1, N2 >= 0.
          The following requirement must be satisfied:
          N1 + N2 = M if SIDE = 'L' and N1 + N2 = N if SIDE = 'R'.

Q

          Q is DOUBLE PRECISION array, dimension
                                       (LDQ,M) if SIDE = 'L'
                                       (LDQ,N) if SIDE = 'R'

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if SIDE = 'L'; LDQ >= max(1,N) if SIDE = 'R'.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= M*N.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine sorm22 (character SIDE, character TRANS, integer M, integer N, integer N1, integer N2, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)

SORM22 multiplies a general matrix by a banded orthogonal matrix.

Purpose

  SORM22 overwrites the general real M-by-N matrix C with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q**T * C       C * Q**T
  where Q is a real orthogonal matrix of order NQ, with NQ = M if
  SIDE = 'L' and NQ = N if SIDE = 'R'.
  The orthogonal matrix Q processes a 2-by-2 block structure
         [  Q11  Q12  ]
     Q = [            ]
         [  Q21  Q22  ],
  where Q12 is an N1-by-N1 lower triangular matrix and Q21 is an
  N2-by-N2 upper triangular matrix.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  apply Q (No transpose);
          = 'C':  apply Q**T (Conjugate transpose).

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

N1
N2

          N1 is INTEGER
          N2 is INTEGER
          The dimension of Q12 and Q21, respectively. N1, N2 >= 0.
          The following requirement must be satisfied:
          N1 + N2 = M if SIDE = 'L' and N1 + N2 = N if SIDE = 'R'.

Q

          Q is REAL array, dimension
                              (LDQ,M) if SIDE = 'L'
                              (LDQ,N) if SIDE = 'R'

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if SIDE = 'L'; LDQ >= max(1,N) if SIDE = 'R'.

C

          C is REAL array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= M*N.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine zunm22 (character SIDE, character TRANS, integer M, integer N, integer N1, integer N2, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)

ZUNM22 multiplies a general matrix by a banded unitary matrix.

Purpose

  ZUNM22 overwrites the general complex M-by-N matrix C with
                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'C':      Q**H * C       C * Q**H
  where Q is a complex unitary matrix of order NQ, with NQ = M if
  SIDE = 'L' and NQ = N if SIDE = 'R'.
  The unitary matrix Q processes a 2-by-2 block structure
         [  Q11  Q12  ]
     Q = [            ]
         [  Q21  Q22  ],
  where Q12 is an N1-by-N1 lower triangular matrix and Q21 is an
  N2-by-N2 upper triangular matrix.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  apply Q (No transpose);
          = 'C':  apply Q**H (Conjugate transpose).

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

N1
N2

          N1 is INTEGER
          N2 is INTEGER
          The dimension of Q12 and Q21, respectively. N1, N2 >= 0.
          The following requirement must be satisfied:
          N1 + N2 = M if SIDE = 'L' and N1 + N2 = N if SIDE = 'R'.

Q

          Q is COMPLEX*16 array, dimension
                              (LDQ,M) if SIDE = 'L'
                              (LDQ,N) if SIDE = 'R'

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if SIDE = 'L'; LDQ >= max(1,N) if SIDE = 'R'.

C

          C is COMPLEX*16 array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= M*N.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

Author

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