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doubleOTHERauxiliary(3) LAPACK doubleOTHERauxiliary(3)

NAME

doubleOTHERauxiliary

SYNOPSIS

Functions


subroutine dlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. subroutine dlacn2 (N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine dlacon (N, V, X, ISGN, EST, KASE)
DLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine dladiv (A, B, C, D, P, Q)
DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow. subroutine dladiv1 (A, B, C, D, P, Q)
double precision function dladiv2 (A, B, C, D, R, T)
subroutine dlaein (RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO)
DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. subroutine dlaexc (WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, INFO)
DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation. subroutine dlag2 (A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. subroutine dlag2s (M, N, A, LDA, SA, LDSA, INFO)
DLAG2S converts a double precision matrix to a single precision matrix. subroutine dlags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel. subroutine dlagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
DLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1. subroutine dlagv2 (A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR, SNR)
DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. subroutine dlahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)
DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. subroutine dlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. subroutine dlaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
DLAIC1 applies one step of incremental condition estimation. subroutine dlaln2 (LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO)
DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form. double precision function dlangt (NORM, N, DL, D, DU)
DLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. double precision function dlanhs (NORM, N, A, LDA, WORK)
DLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix. double precision function dlansb (NORM, UPLO, N, K, AB, LDAB, WORK)
DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix. double precision function dlansp (NORM, UPLO, N, AP, WORK)
DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form. double precision function dlantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. double precision function dlantp (NORM, UPLO, DIAG, N, AP, WORK)
DLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form. double precision function dlantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
DLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix. subroutine dlanv2 (A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN)
DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form. subroutine dlapll (N, X, INCX, Y, INCY, SSMIN)
DLAPLL measures the linear dependence of two vectors. subroutine dlapmr (FORWRD, M, N, X, LDX, K)
DLAPMR rearranges rows of a matrix as specified by a permutation vector. subroutine dlapmt (FORWRD, M, N, X, LDX, K)
DLAPMT performs a forward or backward permutation of the columns of a matrix. subroutine dlaqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
DLAQP2 computes a QR factorization with column pivoting of the matrix block. subroutine dlaqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. subroutine dlaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine dlaqr1 (N, H, LDH, SR1, SI1, SR2, SI2, V)
DLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts. subroutine dlaqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine dlaqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine dlaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine dlaqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
DLAQR5 performs a single small-bulge multi-shift QR sweep. subroutine dlaqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
DLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ. subroutine dlaqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ. subroutine dlaqtr (LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, INFO)
DLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic. subroutine dlar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. subroutine dlar2v (N, X, Y, Z, INCX, C, S, INCC)
DLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices. subroutine dlarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix. subroutine dlarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
DLARFB applies a block reflector or its transpose to a general rectangular matrix. subroutine dlarfg (N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix). subroutine dlarfgp (N, ALPHA, X, INCX, TAU)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta. subroutine dlarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
DLARFT forms the triangular factor T of a block reflector H = I - vtvH subroutine dlarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
DLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10. subroutine dlargv (N, X, INCX, Y, INCY, C, INCC)
DLARGV generates a vector of plane rotations with real cosines and real sines. subroutine dlarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)
DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. subroutine dlartv (N, X, INCX, Y, INCY, C, S, INCC)
DLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors. subroutine dlaswp (N, A, LDA, K1, K2, IPIV, INCX)
DLASWP performs a series of row interchanges on a general rectangular matrix. subroutine dlat2s (UPLO, N, A, LDA, SA, LDSA, INFO)
DLAT2S converts a double-precision triangular matrix to a single-precision triangular matrix. subroutine dlatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
DLATBS solves a triangular banded system of equations. subroutine dlatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine dlatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
DLATPS solves a triangular system of equations with the matrix held in packed storage. subroutine dlatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. subroutine dlatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
DLATRS solves a triangular system of equations with the scale factor set to prevent overflow. subroutine dlauu2 (UPLO, N, A, LDA, INFO)
DLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm). subroutine dlauum (UPLO, N, A, LDA, INFO)
DLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm). subroutine drscl (N, SA, SX, INCX)
DRSCL multiplies a vector by the reciprocal of a real scalar. subroutine dtprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
DTPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks. subroutine slatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Detailed Description

This is the group of double other auxiliary routines

Function Documentation

subroutine dlabrd (integer M, integer N, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( ldy, * ) Y, integer LDY)

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

 DLABRD reduces the first NB rows and columns of a real general
 m by n matrix A to upper or lower bidiagonal form by an orthogonal
 transformation Q**T * A * P, and returns the matrices X and Y which
 are needed to apply the transformation to the unreduced part of A.
 If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
 bidiagonal form.
 This is an auxiliary routine called by DGEBRD

Parameters:

M

          M is INTEGER
          The number of rows in the matrix A.

N

          N is INTEGER
          The number of columns in the matrix A.

NB

          NB is INTEGER
          The number of leading rows and columns of A to be reduced.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the m by n general matrix to be reduced.
          On exit, the first NB rows and columns of the matrix are
          overwritten; the rest of the array is unchanged.
          If m >= n, elements on and below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the
            array TAUP, represent the orthogonal matrix P as a product
            of elementary reflectors.
          If m < n, elements below the diagonal in the first NB
            columns, with the array TAUQ, represent the orthogonal
            matrix Q as a product of elementary reflectors, and
            elements on and above the diagonal in the first NB rows,
            with the array TAUP, represent the orthogonal matrix P as
            a product of elementary reflectors.
          See Further Details.

LDA

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

D

          D is DOUBLE PRECISION array, dimension (NB)
          The diagonal elements of the first NB rows and columns of
          the reduced matrix.  D(i) = A(i,i).

E

          E is DOUBLE PRECISION array, dimension (NB)
          The off-diagonal elements of the first NB rows and columns of
          the reduced matrix.

TAUQ

          TAUQ is DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.

TAUP

          TAUP is DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.

X

          X is DOUBLE PRECISION array, dimension (LDX,NB)
          The m-by-nb matrix X required to update the unreduced part
          of A.

LDX

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

Y

          Y is DOUBLE PRECISION array, dimension (LDY,NB)
          The n-by-nb matrix Y required to update the unreduced part
          of A.

LDY

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Further Details:

  The matrices Q and P are represented as products of elementary
  reflectors:
     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
  Each H(i) and G(i) has the form:
     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T
  where tauq and taup are real scalars, and v and u are real vectors.
  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
  The elements of the vectors v and u together form the m-by-nb matrix
  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
  the transformation to the unreduced part of the matrix, using a block
  update of the form:  A := A - V*Y**T - X*U**T.
  The contents of A on exit are illustrated by the following examples
  with nb = 2:
  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )
  where a denotes an element of the original matrix which is unchanged,
  vi denotes an element of the vector defining H(i), and ui an element
  of the vector defining G(i).

subroutine dlacn2 (integer N, double precision, dimension( * ) V, double precision, dimension( * ) X, integer, dimension( * ) ISGN, double precision EST, integer KASE, integer, dimension( 3 ) ISAVE)

DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

Purpose:

 DLACN2 estimates the 1-norm of a square, real matrix A.
 Reverse communication is used for evaluating matrix-vector products.

Parameters:

N

          N is INTEGER
         The order of the matrix.  N >= 1.

V

          V is DOUBLE PRECISION array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

X

          X is DOUBLE PRECISION array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A**T * X,  if KASE=2,
         and DLACN2 must be re-called with all the other parameters
         unchanged.

ISGN

          ISGN is INTEGER array, dimension (N)

EST

          EST is DOUBLE PRECISION
         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
         unchanged from the previous call to DLACN2.
         On exit, EST is an estimate (a lower bound) for norm(A).

KASE

          KASE is INTEGER
         On the initial call to DLACN2, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A**T * X.
         On the final return from DLACN2, KASE will again be 0.

ISAVE

          ISAVE is INTEGER array, dimension (3)
         ISAVE is used to save variables between calls to DLACN2

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  Originally named SONEST, dated March 16, 1988.
  This is a thread safe version of DLACON, which uses the array ISAVE
  in place of a SAVE statement, as follows:
     DLACON     DLACN2
      JUMP     ISAVE(1)
      J        ISAVE(2)
      ITER     ISAVE(3)

Contributors:

Nick Higham, University of Manchester

References:

N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

subroutine dlacon (integer N, double precision, dimension( * ) V, double precision, dimension( * ) X, integer, dimension( * ) ISGN, double precision EST, integer KASE)

DLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

Purpose:

 DLACON estimates the 1-norm of a square, real matrix A.
 Reverse communication is used for evaluating matrix-vector products.

Parameters:

N

          N is INTEGER
         The order of the matrix.  N >= 1.

V

          V is DOUBLE PRECISION array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

X

          X is DOUBLE PRECISION array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A**T * X,  if KASE=2,
         and DLACON must be re-called with all the other parameters
         unchanged.

ISGN

          ISGN is INTEGER array, dimension (N)

EST

          EST is DOUBLE PRECISION
         On entry with KASE = 1 or 2 and JUMP = 3, EST should be
         unchanged from the previous call to DLACON.
         On exit, EST is an estimate (a lower bound) for norm(A).

KASE

          KASE is INTEGER
         On the initial call to DLACON, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A**T * X.
         On the final return from DLACON, KASE will again be 0.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Nick Higham, University of Manchester. Originally named SONEST, dated March 16, 1988.

References:

N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

subroutine dladiv (double precision A, double precision B, double precision C, double precision D, double precision P, double precision Q)

DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.

Purpose:

 DLADIV performs complex division in  real arithmetic
                       a + i*b
            p + i*q = ---------
                       c + i*d
 The algorithm is due to Michael Baudin and Robert L. Smith
 and can be found in the paper
 "A Robust Complex Division in Scilab"

Parameters:

A

          A is DOUBLE PRECISION

B

          B is DOUBLE PRECISION

C

          C is DOUBLE PRECISION

D

          D is DOUBLE PRECISION
          The scalars a, b, c, and d in the above expression.

P

          P is DOUBLE PRECISION

Q

          Q is DOUBLE PRECISION
          The scalars p and q in the above expression.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2013

subroutine dlaein (logical RIGHTV, logical NOINIT, integer N, double precision, dimension( ldh, * ) H, integer LDH, double precision WR, double precision WI, double precision, dimension( * ) VR, double precision, dimension( * ) VI, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, double precision EPS3, double precision SMLNUM, double precision BIGNUM, integer INFO)

DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

Purpose:

 DLAEIN uses inverse iteration to find a right or left eigenvector
 corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg
 matrix H.

Parameters:

RIGHTV

          RIGHTV is LOGICAL
          = .TRUE. : compute right eigenvector;
          = .FALSE.: compute left eigenvector.

NOINIT

          NOINIT is LOGICAL
          = .TRUE. : no initial vector supplied in (VR,VI).
          = .FALSE.: initial vector supplied in (VR,VI).

N

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

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
          The upper Hessenberg matrix H.

LDH

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

WR

          WR is DOUBLE PRECISION

WI

          WI is DOUBLE PRECISION
          The real and imaginary parts of the eigenvalue of H whose
          corresponding right or left eigenvector is to be computed.

VR

          VR is DOUBLE PRECISION array, dimension (N)

VI

          VI is DOUBLE PRECISION array, dimension (N)
          On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
          a real starting vector for inverse iteration using the real
          eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
          must contain the real and imaginary parts of a complex
          starting vector for inverse iteration using the complex
          eigenvalue (WR,WI); otherwise VR and VI need not be set.
          On exit, if WI = 0.0 (real eigenvalue), VR contains the
          computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
          VR and VI contain the real and imaginary parts of the
          computed complex eigenvector. The eigenvector is normalized
          so that the component of largest magnitude has magnitude 1;
          here the magnitude of a complex number (x,y) is taken to be
          |x| + |y|.
          VI is not referenced if WI = 0.0.

B

          B is DOUBLE PRECISION array, dimension (LDB,N)

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= N+1.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

EPS3

          EPS3 is DOUBLE PRECISION
          A small machine-dependent value which is used to perturb
          close eigenvalues, and to replace zero pivots.

SMLNUM

          SMLNUM is DOUBLE PRECISION
          A machine-dependent value close to the underflow threshold.

BIGNUM

          BIGNUM is DOUBLE PRECISION
          A machine-dependent value close to the overflow threshold.

INFO

          INFO is INTEGER
          = 0:  successful exit
          = 1:  inverse iteration did not converge; VR is set to the
                last iterate, and so is VI if WI.ne.0.0.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaexc (logical WANTQ, integer N, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( ldq, * ) Q, integer LDQ, integer J1, integer N1, integer N2, double precision, dimension( * ) WORK, integer INFO)

DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.

Purpose:

 DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
 an upper quasi-triangular matrix T by an orthogonal similarity
 transformation.
 T must be in Schur canonical form, that is, block upper triangular
 with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
 has its diagonal elemnts equal and its off-diagonal elements of
 opposite sign.

Parameters:

WANTQ

          WANTQ is LOGICAL
          = .TRUE. : accumulate the transformation in the matrix Q;
          = .FALSE.: do not accumulate the transformation.

N

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

T

          T is DOUBLE PRECISION array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          canonical form.
          On exit, the updated matrix T, again in Schur canonical form.

LDT

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

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
          On exit, if WANTQ is .TRUE., the updated matrix Q.
          If WANTQ is .FALSE., Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.

J1

          J1 is INTEGER
          The index of the first row of the first block T11.

N1

          N1 is INTEGER
          The order of the first block T11. N1 = 0, 1 or 2.

N2

          N2 is INTEGER
          The order of the second block T22. N2 = 0, 1 or 2.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          = 1: the transformed matrix T would be too far from Schur
               form; the blocks are not swapped and T and Q are
               unchanged.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlag2 (double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision SAFMIN, double precision SCALE1, double precision SCALE2, double precision WR1, double precision WR2, double precision WI)

DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Purpose:

 DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
 problem  A - w B, with scaling as necessary to avoid over-/underflow.
 The scaling factor "s" results in a modified eigenvalue equation
     s A - w B
 where  s  is a non-negative scaling factor chosen so that  w,  w B,
 and  s A  do not overflow and, if possible, do not underflow, either.

Parameters:

A

          A is DOUBLE PRECISION array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
          is less than 1/SAFMIN.  Entries less than
          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= 2.

B

          B is DOUBLE PRECISION array, dimension (LDB, 2)
          On entry, the 2 x 2 upper triangular matrix B.  It is
          assumed that the one-norm of B is less than 1/SAFMIN.  The
          diagonals should be at least sqrt(SAFMIN) times the largest
          element of B (in absolute value); if a diagonal is smaller
          than that, then  +/- sqrt(SAFMIN) will be used instead of
          that diagonal.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= 2.

SAFMIN

          SAFMIN is DOUBLE PRECISION
          The smallest positive number s.t. 1/SAFMIN does not
          overflow.  (This should always be DLAMCH('S') -- it is an
          argument in order to avoid having to call DLAMCH frequently.)

SCALE1

          SCALE1 is DOUBLE PRECISION
          A scaling factor used to avoid over-/underflow in the
          eigenvalue equation which defines the first eigenvalue.  If
          the eigenvalues are complex, then the eigenvalues are
          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
          exponent range of the machine), SCALE1=SCALE2, and SCALE1
          will always be positive.  If the eigenvalues are real, then
          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
          overflow or underflow, and in fact, SCALE1 may be zero or
          less than the underflow threshold if the exact eigenvalue
          is sufficiently large.

SCALE2

          SCALE2 is DOUBLE PRECISION
          A scaling factor used to avoid over-/underflow in the
          eigenvalue equation which defines the second eigenvalue.  If
          the eigenvalues are complex, then SCALE2=SCALE1.  If the
          eigenvalues are real, then the second (real) eigenvalue is
          WR2 / SCALE2 , but this may overflow or underflow, and in
          fact, SCALE2 may be zero or less than the underflow
          threshold if the exact eigenvalue is sufficiently large.

WR1

          WR1 is DOUBLE PRECISION
          If the eigenvalue is real, then WR1 is SCALE1 times the
          eigenvalue closest to the (2,2) element of A B**(-1).  If the
          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
          part of the eigenvalues.

WR2

          WR2 is DOUBLE PRECISION
          If the eigenvalue is real, then WR2 is SCALE2 times the
          other eigenvalue.  If the eigenvalue is complex, then
          WR1=WR2 is SCALE1 times the real part of the eigenvalues.

WI

          WI is DOUBLE PRECISION
          If the eigenvalue is real, then WI is zero.  If the
          eigenvalue is complex, then WI is SCALE1 times the imaginary
          part of the eigenvalues.  WI will always be non-negative.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

subroutine dlag2s (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, real, dimension( ldsa, * ) SA, integer LDSA, integer INFO)

DLAG2S converts a double precision matrix to a single precision matrix.

Purpose:

 DLAG2S converts a DOUBLE PRECISION matrix, SA, to a SINGLE
 PRECISION matrix, A.
 RMAX is the overflow for the SINGLE PRECISION arithmetic
 DLAG2S checks that all the entries of A are between -RMAX and
 RMAX. If not the conversion is aborted and a flag is raised.
 This is an auxiliary routine so there is no argument checking.

Parameters:

M

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

N

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

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N coefficient matrix A.

LDA

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

SA

          SA is REAL array, dimension (LDSA,N)
          On exit, if INFO=0, the M-by-N coefficient matrix SA; if
          INFO>0, the content of SA is unspecified.

LDSA

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

INFO

          INFO is INTEGER
          = 0:  successful exit.
          = 1:  an entry of the matrix A is greater than the SINGLE
                PRECISION overflow threshold, in this case, the content
                of SA in exit is unspecified.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlags2 (logical UPPER, double precision A1, double precision A2, double precision A3, double precision B1, double precision B2, double precision B3, double precision CSU, double precision SNU, double precision CSV, double precision SNV, double precision CSQ, double precision SNQ)

DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.

Purpose:

 DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
 that if ( UPPER ) then
           U**T *A*Q = U**T *( A1 A2 )*Q = ( x  0  )
                             ( 0  A3 )     ( x  x  )
 and
           V**T*B*Q = V**T *( B1 B2 )*Q = ( x  0  )
                            ( 0  B3 )     ( x  x  )
 or if ( .NOT.UPPER ) then
           U**T *A*Q = U**T *( A1 0  )*Q = ( x  x  )
                             ( A2 A3 )     ( 0  x  )
 and
           V**T*B*Q = V**T*( B1 0  )*Q = ( x  x  )
                           ( B2 B3 )     ( 0  x  )
 The rows of the transformed A and B are parallel, where
   U = (  CSU  SNU ), V = (  CSV SNV ), Q = (  CSQ   SNQ )
       ( -SNU  CSU )      ( -SNV CSV )      ( -SNQ   CSQ )
 Z**T denotes the transpose of Z.

Parameters:

UPPER

          UPPER is LOGICAL
          = .TRUE.: the input matrices A and B are upper triangular.
          = .FALSE.: the input matrices A and B are lower triangular.

A1

          A1 is DOUBLE PRECISION

A2

          A2 is DOUBLE PRECISION

A3

          A3 is DOUBLE PRECISION
          On entry, A1, A2 and A3 are elements of the input 2-by-2
          upper (lower) triangular matrix A.

B1

          B1 is DOUBLE PRECISION

B2

          B2 is DOUBLE PRECISION

B3

          B3 is DOUBLE PRECISION
          On entry, B1, B2 and B3 are elements of the input 2-by-2
          upper (lower) triangular matrix B.

CSU

          CSU is DOUBLE PRECISION

SNU

          SNU is DOUBLE PRECISION
          The desired orthogonal matrix U.

CSV

          CSV is DOUBLE PRECISION

SNV

          SNV is DOUBLE PRECISION
          The desired orthogonal matrix V.

CSQ

          CSQ is DOUBLE PRECISION

SNQ

          SNQ is DOUBLE PRECISION
          The desired orthogonal matrix Q.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlagtm (character TRANS, integer N, integer NRHS, double precision ALPHA, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( ldx, * ) X, integer LDX, double precision BETA, double precision, dimension( ldb, * ) B, integer LDB)

DLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.

Purpose:

 DLAGTM performs a matrix-vector product of the form
    B := alpha * A * X + beta * B
 where A is a tridiagonal matrix of order N, B and X are N by NRHS
 matrices, and alpha and beta are real scalars, each of which may be
 0., 1., or -1.

Parameters:

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  No transpose, B := alpha * A * X + beta * B
          = 'T':  Transpose,    B := alpha * A'* X + beta * B
          = 'C':  Conjugate transpose = Transpose

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 X and B.

ALPHA

          ALPHA is DOUBLE PRECISION
          The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
          it is assumed to be 0.

DL

          DL is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) sub-diagonal elements of T.

D

          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of T.

DU

          DU is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) super-diagonal elements of T.

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          The N by NRHS matrix X.

LDX

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

BETA

          BETA is DOUBLE PRECISION
          The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
          it is assumed to be 1.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix B.
          On exit, B is overwritten by the matrix expression
          B := alpha * A * X + beta * B.

LDB

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlagv2 (double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( 2 ) ALPHAR, double precision, dimension( 2 ) ALPHAI, double precision, dimension( 2 ) BETA, double precision CSL, double precision SNL, double precision CSR, double precision SNR)

DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.

Purpose:

 DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
 matrix pencil (A,B) where B is upper triangular. This routine
 computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
 SNR such that
 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
    types), then
    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
    [  0  a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
    [ b11 b12 ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ],
 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
    then
    [ a11 a12 ] := [  CSL  SNL ] [ a11 a12 ] [  CSR -SNR ]
    [ a21 a22 ]    [ -SNL  CSL ] [ a21 a22 ] [  SNR  CSR ]
    [ b11  0  ] := [  CSL  SNL ] [ b11 b12 ] [  CSR -SNR ]
    [  0  b22 ]    [ -SNL  CSL ] [  0  b22 ] [  SNR  CSR ]
    where b11 >= b22 > 0.

Parameters:

A

          A is DOUBLE PRECISION array, dimension (LDA, 2)
          On entry, the 2 x 2 matrix A.
          On exit, A is overwritten by the ``A-part'' of the
          generalized Schur form.

LDA

          LDA is INTEGER
          THe leading dimension of the array A.  LDA >= 2.

B

          B is DOUBLE PRECISION array, dimension (LDB, 2)
          On entry, the upper triangular 2 x 2 matrix B.
          On exit, B is overwritten by the ``B-part'' of the
          generalized Schur form.

LDB

          LDB is INTEGER
          THe leading dimension of the array B.  LDB >= 2.

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (2)

ALPHAI

          ALPHAI is DOUBLE PRECISION array, dimension (2)

BETA

          BETA is DOUBLE PRECISION array, dimension (2)
          (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
          pencil (A,B), k=1,2, i = sqrt(-1).  Note that BETA(k) may
          be zero.

CSL

          CSL is DOUBLE PRECISION
          The cosine of the left rotation matrix.

SNL

          SNL is DOUBLE PRECISION
          The sine of the left rotation matrix.

CSR

          CSR is DOUBLE PRECISION
          The cosine of the right rotation matrix.

SNR

          SNR is DOUBLE PRECISION
          The sine of the right rotation matrix.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

subroutine dlahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, integer INFO)

DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

    DLAHQR is an auxiliary routine called by DHSEQR to update the
    eigenvalues and Schur decomposition already computed by DHSEQR, by
    dealing with the Hessenberg submatrix in rows and columns ILO to
    IHI.

Parameters:

WANTT

          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

N

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

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          It is assumed that H is already upper quasi-triangular in
          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
          ILO = 1). DLAHQR works primarily with the Hessenberg
          submatrix in rows and columns ILO to IHI, but applies
          transformations to all of H if WANTT is .TRUE..
          1 <= ILO <= max(1,IHI); IHI <= N.

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
          On entry, the upper Hessenberg matrix H.
          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
          quasi-triangular in rows and columns ILO:IHI, with any
          2-by-2 diagonal blocks in standard form. If INFO is zero
          and WANTT is .FALSE., the contents of H are unspecified on
          exit.  The output state of H if INFO is nonzero is given
          below under the description of INFO.

LDH

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

WR

          WR is DOUBLE PRECISION array, dimension (N)

WI

          WI is DOUBLE PRECISION array, dimension (N)
          The real and imaginary parts, respectively, of the computed
          eigenvalues ILO to IHI are stored in the corresponding
          elements of WR and WI. If two eigenvalues are computed as a
          complex conjugate pair, they are stored in consecutive
          elements of WR and WI, say the i-th and (i+1)th, with
          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
          eigenvalues are stored in the same order as on the diagonal
          of the Schur form returned in H, with WR(i) = H(i,i), and, if
          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE..
          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,N)
          If WANTZ is .TRUE., on entry Z must contain the current
          matrix Z of transformations accumulated by DHSEQR, and on
          exit Z has been updated; transformations are applied only to
          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
          If WANTZ is .FALSE., Z is not referenced.

LDZ

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

INFO

          INFO is INTEGER
           =   0: successful exit
          .GT. 0: If INFO = i, DLAHQR failed to compute all the
                  eigenvalues ILO to IHI in a total of 30 iterations
                  per eigenvalue; elements i+1:ihi of WR and WI
                  contain those eigenvalues which have been
                  successfully computed.
                  If INFO .GT. 0 and WANTT is .FALSE., then on exit,
                  the remaining unconverged eigenvalues are the
                  eigenvalues of the upper Hessenberg matrix rows
                  and columns ILO thorugh INFO of the final, output
                  value of H.
                  If INFO .GT. 0 and WANTT is .TRUE., then on exit
          (*)       (initial value of H)*U  = U*(final value of H)
                  where U is an orthognal matrix.    The final
                  value of H is upper Hessenberg and triangular in
                  rows and columns INFO+1 through IHI.
                  If INFO .GT. 0 and WANTZ is .TRUE., then on exit
                      (final value of Z)  = (initial value of Z)*U
                  where U is the orthogonal matrix in (*)
                  (regardless of the value of WANTT.)

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

     02-96 Based on modifications by
     David Day, Sandia National Laboratory, USA
     12-04 Further modifications by
     Ralph Byers, University of Kansas, USA
     This is a modified version of DLAHQR from LAPACK version 3.0.
     It is (1) more robust against overflow and underflow and
     (2) adopts the more conservative Ahues & Tisseur stopping
     criterion (LAWN 122, 1997).

subroutine dlahr2 (integer N, integer K, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( nb ) TAU, double precision, dimension( ldt, nb ) T, integer LDT, double precision, dimension( ldy, nb ) Y, integer LDY)

DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

 DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
 matrix A so that elements below the k-th subdiagonal are zero. The
 reduction is performed by an orthogonal similarity transformation
 Q**T * A * Q. The routine returns the matrices V and T which determine
 Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
 This is an auxiliary routine called by DGEHRD.

Parameters:

N

          N is INTEGER
          The order of the matrix A.

K

          K is INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
          K < N.

NB

          NB is INTEGER
          The number of columns to be reduced.

A

          A is DOUBLE PRECISION array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a
          product of elementary reflectors. The other columns of A are
          unchanged. See Further Details.

LDA

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

TAU

          TAU is DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further
          Details.

T

          T is DOUBLE PRECISION array, dimension (LDT,NB)
          The upper triangular matrix T.

LDT

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

Y

          Y is DOUBLE PRECISION array, dimension (LDY,NB)
          The n-by-nb matrix Y.

LDY

          LDY is INTEGER
          The leading dimension of the array Y. LDY >= N.

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 nb elementary reflectors
     Q = H(1) H(2) . . . H(nb).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).
  The elements of the vectors v together form the (n-k+1)-by-nb matrix
  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V**T) * (A - Y*V**T).
  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )
  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an
  element of the vector defining H(i).
  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
  incorporating improvements proposed by Quintana-Orti and Van de
  Gejin. Note that the entries of A(1:K,2:NB) differ from those
  returned by the original LAPACK-3.0's DLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)

References:

Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

subroutine dlaic1 (integer JOB, integer J, double precision, dimension( j ) X, double precision SEST, double precision, dimension( j ) W, double precision GAMMA, double precision SESTPR, double precision S, double precision C)

DLAIC1 applies one step of incremental condition estimation.

Purpose:

 DLAIC1 applies one step of incremental condition estimation in
 its simplest version:
 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
 lower triangular matrix L, such that
          twonorm(L*x) = sest
 Then DLAIC1 computes sestpr, s, c such that
 the vector
                 [ s*x ]
          xhat = [  c  ]
 is an approximate singular vector of
                 [ L       0  ]
          Lhat = [ w**T gamma ]
 in the sense that
          twonorm(Lhat*xhat) = sestpr.
 Depending on JOB, an estimate for the largest or smallest singular
 value is computed.
 Note that [s c]**T and sestpr**2 is an eigenpair of the system
     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
                                           [ gamma ]
 where  alpha =  x**T*w.

Parameters:

JOB

          JOB is INTEGER
          = 1: an estimate for the largest singular value is computed.
          = 2: an estimate for the smallest singular value is computed.

J

          J is INTEGER
          Length of X and W

X

          X is DOUBLE PRECISION array, dimension (J)
          The j-vector x.

SEST

          SEST is DOUBLE PRECISION
          Estimated singular value of j by j matrix L

W

          W is DOUBLE PRECISION array, dimension (J)
          The j-vector w.

GAMMA

          GAMMA is DOUBLE PRECISION
          The diagonal element gamma.

SESTPR

          SESTPR is DOUBLE PRECISION
          Estimated singular value of (j+1) by (j+1) matrix Lhat.

S

          S is DOUBLE PRECISION
          Sine needed in forming xhat.

C

          C is DOUBLE PRECISION
          Cosine needed in forming xhat.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaln2 (logical LTRANS, integer NA, integer NW, double precision SMIN, double precision CA, double precision, dimension( lda, * ) A, integer LDA, double precision D1, double precision D2, double precision, dimension( ldb, * ) B, integer LDB, double precision WR, double precision WI, double precision, dimension( ldx, * ) X, integer LDX, double precision SCALE, double precision XNORM, integer INFO)

DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.

Purpose:

 DLALN2 solves a system of the form  (ca A - w D ) X = s B
 or (ca A**T - w D) X = s B   with possible scaling ("s") and
 perturbation of A.  (A**T means A-transpose.)
 A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
 real diagonal matrix, w is a real or complex value, and X and B are
 NA x 1 matrices -- real if w is real, complex if w is complex.  NA
 may be 1 or 2.
 If w is complex, X and B are represented as NA x 2 matrices,
 the first column of each being the real part and the second
 being the imaginary part.
 "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
 so chosen that X can be computed without overflow.  X is further
 scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
 than overflow.
 If both singular values of (ca A - w D) are less than SMIN,
 SMIN*identity will be used instead of (ca A - w D).  If only one
 singular value is less than SMIN, one element of (ca A - w D) will be
 perturbed enough to make the smallest singular value roughly SMIN.
 If both singular values are at least SMIN, (ca A - w D) will not be
 perturbed.  In any case, the perturbation will be at most some small
 multiple of max( SMIN, ulp*norm(ca A - w D) ).  The singular values
 are computed by infinity-norm approximations, and thus will only be
 correct to a factor of 2 or so.
 Note: all input quantities are assumed to be smaller than overflow
 by a reasonable factor.  (See BIGNUM.)

Parameters:

LTRANS

          LTRANS is LOGICAL
          =.TRUE.:  A-transpose will be used.
          =.FALSE.: A will be used (not transposed.)

NA

          NA is INTEGER
          The size of the matrix A.  It may (only) be 1 or 2.

NW

          NW is INTEGER
          1 if "w" is real, 2 if "w" is complex.  It may only be 1
          or 2.

SMIN

          SMIN is DOUBLE PRECISION
          The desired lower bound on the singular values of A.  This
          should be a safe distance away from underflow or overflow,
          say, between (underflow/machine precision) and  (machine
          precision * overflow ).  (See BIGNUM and ULP.)

CA

          CA is DOUBLE PRECISION
          The coefficient c, which A is multiplied by.

A

          A is DOUBLE PRECISION array, dimension (LDA,NA)
          The NA x NA matrix A.

LDA

          LDA is INTEGER
          The leading dimension of A.  It must be at least NA.

D1

          D1 is DOUBLE PRECISION
          The 1,1 element in the diagonal matrix D.

D2

          D2 is DOUBLE PRECISION
          The 2,2 element in the diagonal matrix D.  Not used if NA=1.

B

          B is DOUBLE PRECISION array, dimension (LDB,NW)
          The NA x NW matrix B (right-hand side).  If NW=2 ("w" is
          complex), column 1 contains the real part of B and column 2
          contains the imaginary part.

LDB

          LDB is INTEGER
          The leading dimension of B.  It must be at least NA.

WR

          WR is DOUBLE PRECISION
          The real part of the scalar "w".

WI

          WI is DOUBLE PRECISION
          The imaginary part of the scalar "w".  Not used if NW=1.

X

          X is DOUBLE PRECISION array, dimension (LDX,NW)
          The NA x NW matrix X (unknowns), as computed by DLALN2.
          If NW=2 ("w" is complex), on exit, column 1 will contain
          the real part of X and column 2 will contain the imaginary
          part.

LDX

          LDX is INTEGER
          The leading dimension of X.  It must be at least NA.

SCALE

          SCALE is DOUBLE PRECISION
          The scale factor that B must be multiplied by to insure
          that overflow does not occur when computing X.  Thus,
          (ca A - w D) X  will be SCALE*B, not B (ignoring
          perturbations of A.)  It will be at most 1.

XNORM

          XNORM is DOUBLE PRECISION
          The infinity-norm of X, when X is regarded as an NA x NW
          real matrix.

INFO

          INFO is INTEGER
          An error flag.  It will be set to zero if no error occurs,
          a negative number if an argument is in error, or a positive
          number if  ca A - w D  had to be perturbed.
          The possible values are:
          = 0: No error occurred, and (ca A - w D) did not have to be
                 perturbed.
          = 1: (ca A - w D) had to be perturbed to make its smallest
               (or only) singular value greater than SMIN.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

double precision function dlangt (character NORM, integer N, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU)

DLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.

Purpose:

 DLANGT  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 real tridiagonal matrix A.

Returns:

DLANGT

    DLANGT = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANGT as described
          above.

N

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

DL

          DL is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) sub-diagonal elements of A.

D

          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of A.

DU

          DU is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) super-diagonal elements of A.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

double precision function dlanhs (character NORM, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)

DLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.

Purpose:

 DLANHS  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 Hessenberg matrix A.

Returns:

DLANHS

    DLANHS = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANHS as described
          above.

N

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

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          The n by n upper Hessenberg matrix A; the part of A below the
          first sub-diagonal is not referenced.

LDA

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

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

double precision function dlansb (character NORM, character UPLO, integer N, integer K, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)

DLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.

Purpose:

 DLANSB  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the element of  largest absolute value  of an
 n by n symmetric band matrix A,  with k super-diagonals.

Returns:

DLANSB

    DLANSB = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANSB as described
          above.

UPLO

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

N

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

K

          K is INTEGER
          The number of super-diagonals or sub-diagonals of the
          band matrix A.  K >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first K+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(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

LDAB

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

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,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

double precision function dlansp (character NORM, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) WORK)

DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.

Purpose:

 DLANSP  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 real symmetric matrix A,  supplied in packed form.

Returns:

DLANSP

    DLANSP = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANSP as described
          above.

UPLO

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

N

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

AP

          AP is DOUBLE PRECISION 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)*(2n-j)/2) = A(i,j) for j<=i<=n.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,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

double precision function dlantb (character NORM, character UPLO, character DIAG, integer N, integer K, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)

DLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.

Purpose:

 DLANTB  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the element of  largest absolute value  of an
 n by n triangular band matrix A,  with ( k + 1 ) diagonals.

Returns:

DLANTB

    DLANTB = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANTB as described
          above.

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.  When N = 0, DLANTB is
          set to zero.

K

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

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first k+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(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
          Note that when DIAG = 'U', the elements of the array AB
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

LDAB

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

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

double precision function dlantp (character NORM, character UPLO, character DIAG, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) WORK)

DLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.

Purpose:

 DLANTP  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 triangular matrix A, supplied in packed form.

Returns:

DLANTP

    DLANTP = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANTP as described
          above.

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.  When N = 0, DLANTP is
          set to zero.

AP

          AP is DOUBLE PRECISION 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.
          Note that when DIAG = 'U', the elements of the array AP
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

double precision function dlantr (character NORM, character UPLO, character DIAG, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)

DLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.

Purpose:

 DLANTR  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 trapezoidal or triangular matrix A.

Returns:

DLANTR

    DLANTR = ( 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 consistent matrix norm.

Parameters:

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANTR as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower trapezoidal.
          = 'U':  Upper trapezoidal
          = 'L':  Lower trapezoidal
          Note that A is triangular instead of trapezoidal if M = N.

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A has unit diagonal.
          = 'N':  Non-unit diagonal
          = 'U':  Unit diagonal

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0, and if
          UPLO = 'U', M <= N.  When M = 0, DLANTR is set to zero.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0, and if
          UPLO = 'L', N <= M.  When N = 0, DLANTR is set to zero.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          The trapezoidal matrix A (A is triangular if M = N).
          If UPLO = 'U', the leading m by n upper trapezoidal part of
          the array A contains the upper trapezoidal matrix, and the
          strictly lower triangular part of A is not referenced.
          If UPLO = 'L', the leading m by n lower trapezoidal part of
          the array A contains the lower trapezoidal matrix, and the
          strictly upper triangular part of A is not referenced.  Note
          that when DIAG = 'U', the diagonal elements of A are not
          referenced and are assumed to be one.

LDA

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

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlanv2 (double precision A, double precision B, double precision C, double precision D, double precision RT1R, double precision RT1I, double precision RT2R, double precision RT2I, double precision CS, double precision SN)

DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.

Purpose:

 DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
 matrix in standard form:
      [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
      [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]
 where either
 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
 conjugate eigenvalues.

Parameters:

A

          A is DOUBLE PRECISION

B

          B is DOUBLE PRECISION

C

          C is DOUBLE PRECISION

D

          D is DOUBLE PRECISION
          On entry, the elements of the input matrix.
          On exit, they are overwritten by the elements of the
          standardised Schur form.

RT1R

          RT1R is DOUBLE PRECISION

RT1I

          RT1I is DOUBLE PRECISION

RT2R

          RT2R is DOUBLE PRECISION

RT2I

          RT2I is DOUBLE PRECISION
          The real and imaginary parts of the eigenvalues. If the
          eigenvalues are a complex conjugate pair, RT1I > 0.

CS

          CS is DOUBLE PRECISION

SN

          SN is DOUBLE PRECISION
          Parameters of the rotation matrix.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  Modified by V. Sima, Research Institute for Informatics, Bucharest,
  Romania, to reduce the risk of cancellation errors,
  when computing real eigenvalues, and to ensure, if possible, that
  abs(RT1R) >= abs(RT2R).

subroutine dlapll (integer N, double precision, dimension( * ) X, integer INCX, double precision, dimension( * ) Y, integer INCY, double precision SSMIN)

DLAPLL measures the linear dependence of two vectors.

Purpose:

 Given two column vectors X and Y, let
                      A = ( X Y ).
 The subroutine first computes the QR factorization of A = Q*R,
 and then computes the SVD of the 2-by-2 upper triangular matrix R.
 The smaller singular value of R is returned in SSMIN, which is used
 as the measurement of the linear dependency of the vectors X and Y.

Parameters:

N

          N is INTEGER
          The length of the vectors X and Y.

X

          X is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          On entry, X contains the N-vector X.
          On exit, X is overwritten.

INCX

          INCX is INTEGER
          The increment between successive elements of X. INCX > 0.

Y

          Y is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCY)
          On entry, Y contains the N-vector Y.
          On exit, Y is overwritten.

INCY

          INCY is INTEGER
          The increment between successive elements of Y. INCY > 0.

SSMIN

          SSMIN is DOUBLE PRECISION
          The smallest singular value of the N-by-2 matrix A = ( X Y ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlapmr (logical FORWRD, integer M, integer N, double precision, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)

DLAPMR rearranges rows of a matrix as specified by a permutation vector.

Purpose:

 DLAPMR rearranges the rows of the M by N matrix X as specified
 by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
 If FORWRD = .TRUE.,  forward permutation:
      X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
 If FORWRD = .FALSE., backward permutation:
      X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.

Parameters:

FORWRD

          FORWRD is LOGICAL
          = .TRUE., forward permutation
          = .FALSE., backward permutation

M

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

N

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

X

          X is DOUBLE PRECISION array, dimension (LDX,N)
          On entry, the M by N matrix X.
          On exit, X contains the permuted matrix X.

LDX

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

K

          K is INTEGER array, dimension (M)
          On entry, K contains the permutation vector. K is used as
          internal workspace, but reset to its original value on
          output.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlapmt (logical FORWRD, integer M, integer N, double precision, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)

DLAPMT performs a forward or backward permutation of the columns of a matrix.

Purpose:

 DLAPMT rearranges the columns of the M by N matrix X as specified
 by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
 If FORWRD = .TRUE.,  forward permutation:
      X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.
 If FORWRD = .FALSE., backward permutation:
      X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

Parameters:

FORWRD

          FORWRD is LOGICAL
          = .TRUE., forward permutation
          = .FALSE., backward permutation

M

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

N

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

X

          X is DOUBLE PRECISION array, dimension (LDX,N)
          On entry, the M by N matrix X.
          On exit, X contains the permuted matrix X.

LDX

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

K

          K is INTEGER array, dimension (N)
          On entry, K contains the permutation vector. K is used as
          internal workspace, but reset to its original value on
          output.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaqp2 (integer M, integer N, integer OFFSET, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, double precision, dimension( * ) TAU, double precision, dimension( * ) VN1, double precision, dimension( * ) VN2, double precision, dimension( * ) WORK)

DLAQP2 computes a QR factorization with column pivoting of the matrix block.

Purpose:

 DLAQP2 computes a QR factorization with column pivoting of
 the block A(OFFSET+1:M,1:N).
 The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

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.

OFFSET

          OFFSET is INTEGER
          The number of rows of the matrix A that must be pivoted
          but no factorized. OFFSET >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
          the triangular factor obtained; the elements in block
          A(OFFSET+1:M,1:N) below the diagonal, together with the
          array TAU, represent the orthogonal matrix Q as a product of
          elementary reflectors. Block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.

LDA

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

JPVT

          JPVT is INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(i) = 0,
          the i-th column of A is a free column.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

TAU

          TAU is DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

VN1

          VN1 is DOUBLE PRECISION array, dimension (N)
          The vector with the partial column norms.

VN2

          VN2 is DOUBLE PRECISION array, dimension (N)
          The vector with the exact column norms.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176

subroutine dlaqps (integer M, integer N, integer OFFSET, integer NB, integer KB, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, double precision, dimension( * ) TAU, double precision, dimension( * ) VN1, double precision, dimension( * ) VN2, double precision, dimension( * ) AUXV, double precision, dimension( ldf, * ) F, integer LDF)

DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.

Purpose:

 DLAQPS computes a step of QR factorization with column pivoting
 of a real M-by-N matrix A by using Blas-3.  It tries to factorize
 NB columns from A starting from the row OFFSET+1, and updates all
 of the matrix with Blas-3 xGEMM.
 In some cases, due to catastrophic cancellations, it cannot
 factorize NB columns.  Hence, the actual number of factorized
 columns is returned in KB.
 Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

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

OFFSET

          OFFSET is INTEGER
          The number of rows of A that have been factorized in
          previous steps.

NB

          NB is INTEGER
          The number of columns to factorize.

KB

          KB is INTEGER
          The number of columns actually factorized.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, block A(OFFSET+1:M,1:KB) is the triangular
          factor obtained and block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.
          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
          been updated.

LDA

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

JPVT

          JPVT is INTEGER array, dimension (N)
          JPVT(I) = K <==> Column K of the full matrix A has been
          permuted into position I in AP.

TAU

          TAU is DOUBLE PRECISION array, dimension (KB)
          The scalar factors of the elementary reflectors.

VN1

          VN1 is DOUBLE PRECISION array, dimension (N)
          The vector with the partial column norms.

VN2

          VN2 is DOUBLE PRECISION array, dimension (N)
          The vector with the exact column norms.

AUXV

          AUXV is DOUBLE PRECISION array, dimension (NB)
          Auxiliar vector.

F

          F is DOUBLE PRECISION array, dimension (LDF,NB)
          Matrix F**T = L*Y**T*A.

LDF

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176

subroutine dlaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

    DLAQR0 computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
    Schur form), and Z is the orthogonal matrix of Schur vectors.
    Optionally Z may be postmultiplied into an input orthogonal
    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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

Parameters:

WANTT

          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

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 and, if ILO.GT.1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to DGEBAL, and then passed to DGEHRD when the
           matrix output by DGEBAL 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 DOUBLE PRECISION array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
           the upper quasi-triangular matrix T from the Schur
           decomposition (the Schur form); 2-by-2 diagonal blocks
           (corresponding to complex conjugate pairs of eigenvalues)
           are returned in standard form, with H(i,i) = H(i+1,i+1)
           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
           .FALSE., then 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.)
           This subroutine may explicitly set 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).

WR

          WR is DOUBLE PRECISION array, dimension (IHI)

WI

          WI is DOUBLE PRECISION array, dimension (IHI)
           The real and imaginary parts, respectively, of the computed
           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
           and WI(ILO:IHI). If two eigenvalues are computed as a
           complex conjugate pair, they are stored in consecutive
           elements of WR and WI, say the i-th and (i+1)th, with
           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
           the eigenvalues are stored in the same order as on the
           diagonal of the Schur form returned in H, with
           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (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 WANTZ is .TRUE.
           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

WORK

          WORK is DOUBLE PRECISION array, dimension LWORK
           On exit, if LWORK = -1, 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, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.
           If LWORK = -1, then DLAQR0 does a workspace query.
           In this case, DLAQR0 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
           .GT. 0:  if INFO = i, DLAQR0 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 WANT is .FALSE., 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 WANTT is .TRUE., then on exit
           (*)  (initial value of H)*U  = U*(final value of H)
                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.
                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
                where U is the orthogonal matrix in (*) (regard-
                less of the value of WANTT.)
                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
                accessed.

Contributors:

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

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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaqr1 (integer N, double precision, dimension( ldh, * ) H, integer LDH, double precision SR1, double precision SI1, double precision SR2, double precision SI2, double precision, dimension( * ) V)

DLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.

Purpose:

      Given a 2-by-2 or 3-by-3 matrix H, DLAQR1 sets v to a
      scalar multiple of the first column of the product
      (*)  K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
      scaling to avoid overflows and most underflows. It
      is assumed that either
              1) sr1 = sr2 and si1 = -si2
          or
              2) si1 = si2 = 0.
      This is useful for starting double implicit shift bulges
      in the QR algorithm.

Parameters:

N

          N is INTEGER
              Order of the matrix H. N must be either 2 or 3.

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
              The 2-by-2 or 3-by-3 matrix H in (*).

LDH

          LDH is INTEGER
              The leading dimension of H as declared in
              the calling procedure.  LDH.GE.N

SR1

          SR1 is DOUBLE PRECISION

SI1

          SI1 is DOUBLE PRECISION

SR2

          SR2 is DOUBLE PRECISION

SI2

          SI2 is DOUBLE PRECISION
              The shifts in (*).

V

          V is DOUBLE PRECISION array, dimension (N)
              A scalar multiple of the first column of the
              matrix K in (*).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Contributors:

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

subroutine dlaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, double precision, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, double precision, dimension( * ) SR, double precision, dimension( * ) SI, double precision, dimension( ldv, * ) V, integer LDV, integer NH, double precision, dimension( ldt, * ) T, integer LDT, integer NV, double precision, dimension( ldwv, * ) WV, integer LDWV, double precision, dimension( * ) WORK, integer LWORK)

DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Purpose:

    DLAQR2 is identical to DLAQR3 except that it avoids
    recursion by calling DLAHQR instead of DLAQR4.
    Aggressive early deflation:
    This subroutine accepts as input an upper Hessenberg matrix
    H and performs an orthogonal similarity transformation
    designed to detect and deflate fully converged eigenvalues from
    a trailing principal submatrix.  On output H has been over-
    written by a new Hessenberg matrix that is a perturbation of
    an orthogonal similarity transformation of H.  It is to be
    hoped that the final version of H has many zero subdiagonal
    entries.

Parameters:

WANTT

          WANTT is LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

WANTZ

          WANTZ is LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

N

          N is INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

KTOP

          KTOP is INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

NW

          NW is INTEGER
          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

LDH

          LDH is INTEGER
          Leading dimension of H just as declared in the calling
          subroutine.  N .LE. LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,N)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
          The leading dimension of Z just as declared in the
          calling subroutine.  1 .LE. LDZ.

NS

          NS is INTEGER
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

ND

          ND is INTEGER
          The number of converged eigenvalues uncovered by this
          subroutine.

SR

          SR is DOUBLE PRECISION array, dimension (KBOT)

SI

          SI is DOUBLE PRECISION array, dimension (KBOT)
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

V

          V is DOUBLE PRECISION array, dimension (LDV,NW)
          An NW-by-NW work array.

LDV

          LDV is INTEGER
          The leading dimension of V just as declared in the
          calling subroutine.  NW .LE. LDV

NH

          NH is INTEGER
          The number of columns of T.  NH.GE.NW.

T

          T is DOUBLE PRECISION array, dimension (LDT,NW)

LDT

          LDT is INTEGER
          The leading dimension of T just as declared in the
          calling subroutine.  NW .LE. LDT

NV

          NV is INTEGER
          The number of rows of work array WV available for
          workspace.  NV.GE.NW.

WV

          WV is DOUBLE PRECISION array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER
          The leading dimension of W just as declared in the
          calling subroutine.  NW .LE. LDV

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.
          If LWORK = -1, then a workspace query is assumed; DLAQR2
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Contributors:

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

subroutine dlaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, double precision, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, double precision, dimension( * ) SR, double precision, dimension( * ) SI, double precision, dimension( ldv, * ) V, integer LDV, integer NH, double precision, dimension( ldt, * ) T, integer LDT, integer NV, double precision, dimension( ldwv, * ) WV, integer LDWV, double precision, dimension( * ) WORK, integer LWORK)

DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Purpose:

    Aggressive early deflation:
    DLAQR3 accepts as input an upper Hessenberg matrix
    H and performs an orthogonal similarity transformation
    designed to detect and deflate fully converged eigenvalues from
    a trailing principal submatrix.  On output H has been over-
    written by a new Hessenberg matrix that is a perturbation of
    an orthogonal similarity transformation of H.  It is to be
    hoped that the final version of H has many zero subdiagonal
    entries.

Parameters:

WANTT

          WANTT is LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

WANTZ

          WANTZ is LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

N

          N is INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

KTOP

          KTOP is INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

NW

          NW is INTEGER
          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

LDH

          LDH is INTEGER
          Leading dimension of H just as declared in the calling
          subroutine.  N .LE. LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,N)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
          The leading dimension of Z just as declared in the
          calling subroutine.  1 .LE. LDZ.

NS

          NS is INTEGER
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

ND

          ND is INTEGER
          The number of converged eigenvalues uncovered by this
          subroutine.

SR

          SR is DOUBLE PRECISION array, dimension (KBOT)

SI

          SI is DOUBLE PRECISION array, dimension (KBOT)
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

V

          V is DOUBLE PRECISION array, dimension (LDV,NW)
          An NW-by-NW work array.

LDV

          LDV is INTEGER
          The leading dimension of V just as declared in the
          calling subroutine.  NW .LE. LDV

NH

          NH is INTEGER
          The number of columns of T.  NH.GE.NW.

T

          T is DOUBLE PRECISION array, dimension (LDT,NW)

LDT

          LDT is INTEGER
          The leading dimension of T just as declared in the
          calling subroutine.  NW .LE. LDT

NV

          NV is INTEGER
          The number of rows of work array WV available for
          workspace.  NV.GE.NW.

WV

          WV is DOUBLE PRECISION array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER
          The leading dimension of W just as declared in the
          calling subroutine.  NW .LE. LDV

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.
          If LWORK = -1, then a workspace query is assumed; DLAQR3
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

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

subroutine dlaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

    DLAQR4 implements one level of recursion for DLAQR0.
    It is a complete implementation of the small bulge multi-shift
    QR algorithm.  It may be called by DLAQR0 and, for large enough
    deflation window size, it may be called by DLAQR3.  This
    subroutine is identical to DLAQR0 except that it calls DLAQR2
    instead of DLAQR3.
    DLAQR4 computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
    Schur form), and Z is the orthogonal matrix of Schur vectors.
    Optionally Z may be postmultiplied into an input orthogonal
    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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

Parameters:

WANTT

          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

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 and, if ILO.GT.1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to DGEBAL, and then passed to DGEHRD when the
           matrix output by DGEBAL 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 DOUBLE PRECISION array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
           the upper quasi-triangular matrix T from the Schur
           decomposition (the Schur form); 2-by-2 diagonal blocks
           (corresponding to complex conjugate pairs of eigenvalues)
           are returned in standard form, with H(i,i) = H(i+1,i+1)
           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
           .FALSE., then 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.)
           This subroutine may explicitly set 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).

WR

          WR is DOUBLE PRECISION array, dimension (IHI)

WI

          WI is DOUBLE PRECISION array, dimension (IHI)
           The real and imaginary parts, respectively, of the computed
           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
           and WI(ILO:IHI). If two eigenvalues are computed as a
           complex conjugate pair, they are stored in consecutive
           elements of WR and WI, say the i-th and (i+1)th, with
           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
           the eigenvalues are stored in the same order as on the
           diagonal of the Schur form returned in H, with
           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (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 WANTZ is .TRUE.
           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

WORK

          WORK is DOUBLE PRECISION array, dimension LWORK
           On exit, if LWORK = -1, 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, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.
           If LWORK = -1, then DLAQR4 does a workspace query.
           In this case, DLAQR4 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
           .GT. 0:  if INFO = i, DLAQR4 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 WANT is .FALSE., 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 WANTT is .TRUE., then on exit
           (*)  (initial value of H)*U  = U*(final value of H)
                where U is a orthogonal matrix.  The final
                value of  H is upper Hessenberg and triangular in
                rows and columns INFO+1 through IHI.
                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
                where U is the orthogonal matrix in (*) (regard-
                less of the value of WANTT.)
                If INFO .GT. 0 and WANTZ is .FALSE., 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

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 dlaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP, integer KBOT, integer NSHFTS, double precision, dimension( * ) SR, double precision, dimension( * ) SI, double precision, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldu, * ) U, integer LDU, integer NV, double precision, dimension( ldwv, * ) WV, integer LDWV, integer NH, double precision, dimension( ldwh, * ) WH, integer LDWH)

DLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:

    DLAQR5, called by DLAQR0, performs a
    single small-bulge multi-shift QR sweep.

Parameters:

WANTT

          WANTT is LOGICAL
             WANTT = .true. if the quasi-triangular Schur factor
             is being computed.  WANTT is set to .false. otherwise.

WANTZ

          WANTZ is LOGICAL
             WANTZ = .true. if the orthogonal Schur factor is being
             computed.  WANTZ is set to .false. otherwise.

KACC22

          KACC22 is INTEGER with value 0, 1, or 2.
             Specifies the computation mode of far-from-diagonal
             orthogonal updates.
        = 0: DLAQR5 does not accumulate reflections and does not
             use matrix-matrix multiply to update far-from-diagonal
             matrix entries.
        = 1: DLAQR5 accumulates reflections and uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries.
        = 2: DLAQR5 accumulates reflections, uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries,
             and takes advantage of 2-by-2 block structure during
             matrix multiplies.

N

          N is INTEGER
             N is the order of the Hessenberg matrix H upon which this
             subroutine operates.

KTOP

          KTOP is INTEGER

KBOT

          KBOT is INTEGER
             These are the first and last rows and columns of an
             isolated diagonal block upon which the QR sweep is to be
             applied. It is assumed without a check that
                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0
             and
                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS

          NSHFTS is INTEGER
             NSHFTS gives the number of simultaneous shifts.  NSHFTS
             must be positive and even.

SR

          SR is DOUBLE PRECISION array, dimension (NSHFTS)

SI

          SI is DOUBLE PRECISION array, dimension (NSHFTS)
             SR contains the real parts and SI contains the imaginary
             parts of the NSHFTS shifts of origin that define the
             multi-shift QR sweep.  On output SR and SI may be
             reordered.

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
             On input H contains a Hessenberg matrix.  On output a
             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
             to the isolated diagonal block in rows and columns KTOP
             through KBOT.

LDH

          LDH is INTEGER
             LDH is the leading dimension of H just as declared in the
             calling procedure.  LDH.GE.MAX(1,N).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
             Specify the rows of Z to which transformations must be
             applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,IHIZ)
             If WANTZ = .TRUE., then the QR Sweep orthogonal
             similarity transformation is accumulated into
             Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
             If WANTZ = .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
             LDA is the leading dimension of Z just as declared in
             the calling procedure. LDZ.GE.N.

V

          V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)

LDV

          LDV is INTEGER
             LDV is the leading dimension of V as declared in the
             calling procedure.  LDV.GE.3.

U

          U is DOUBLE PRECISION array, dimension (LDU,3*NSHFTS-3)

LDU

          LDU is INTEGER
             LDU is the leading dimension of U just as declared in the
             in the calling subroutine.  LDU.GE.3*NSHFTS-3.

NH

          NH is INTEGER
             NH is the number of columns in array WH available for
             workspace. NH.GE.1.

WH

          WH is DOUBLE PRECISION array, dimension (LDWH,NH)

LDWH

          LDWH is INTEGER
             Leading dimension of WH just as declared in the
             calling procedure.  LDWH.GE.3*NSHFTS-3.

NV

          NV is INTEGER
             NV is the number of rows in WV agailable for workspace.
             NV.GE.1.

WV

          WV is DOUBLE PRECISION array, dimension (LDWV,3*NSHFTS-3)

LDWV

          LDWV is INTEGER
             LDWV is the leading dimension of WV as declared in the
             in the calling subroutine.  LDWV.GE.NV.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

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

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.

subroutine dlaqsb (character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)

DLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.

Purpose:

 DLAQSB equilibrates a symmetric band matrix A using the scaling
 factors in the vector S.

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.

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 DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric 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**T*U or A = L*L**T 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.

S

          S is DOUBLE PRECISION array, dimension (N)
          The scale factors for A.

SCOND

          SCOND is DOUBLE PRECISION
          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).

Internal Parameters:

  THRESH is a threshold value used to decide if scaling should be done
  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.
  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaqsp (character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)

DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.

Purpose:

 DLAQSP equilibrates a symmetric matrix A using the scaling factors
 in the vector S.

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.

AP

          AP is DOUBLE PRECISION 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 equilibrated matrix:  diag(S) * A * diag(S), in
          the same storage format as A.

S

          S is DOUBLE PRECISION array, dimension (N)
          The scale factors for A.

SCOND

          SCOND is DOUBLE PRECISION
          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).

Internal Parameters:

  THRESH is a threshold value used to decide if scaling should be done
  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.
  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaqtr (logical LTRAN, logical LREAL, integer N, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( * ) B, double precision W, double precision SCALE, double precision, dimension( * ) X, double precision, dimension( * ) WORK, integer INFO)

DLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.

Purpose:

 DLAQTR solves the real quasi-triangular system
              op(T)*p = scale*c,               if LREAL = .TRUE.
 or the complex quasi-triangular systems
            op(T + iB)*(p+iq) = scale*(c+id),  if LREAL = .FALSE.
 in real arithmetic, where T is upper quasi-triangular.
 If LREAL = .FALSE., then the first diagonal block of T must be
 1 by 1, B is the specially structured matrix
                B = [ b(1) b(2) ... b(n) ]
                    [       w            ]
                    [           w        ]
                    [              .     ]
                    [                 w  ]
 op(A) = A or A**T, A**T denotes the transpose of
 matrix A.
 On input, X = [ c ].  On output, X = [ p ].
               [ d ]                  [ q ]
 This subroutine is designed for the condition number estimation
 in routine DTRSNA.

Parameters:

LTRAN

          LTRAN is LOGICAL
          On entry, LTRAN specifies the option of conjugate transpose:
             = .FALSE.,    op(T+i*B) = T+i*B,
             = .TRUE.,     op(T+i*B) = (T+i*B)**T.

LREAL

          LREAL is LOGICAL
          On entry, LREAL specifies the input matrix structure:
             = .FALSE.,    the input is complex
             = .TRUE.,     the input is real

N

          N is INTEGER
          On entry, N specifies the order of T+i*B. N >= 0.

T

          T is DOUBLE PRECISION array, dimension (LDT,N)
          On entry, T contains a matrix in Schur canonical form.
          If LREAL = .FALSE., then the first diagonal block of T mu
          be 1 by 1.

LDT

          LDT is INTEGER
          The leading dimension of the matrix T. LDT >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (N)
          On entry, B contains the elements to form the matrix
          B as described above.
          If LREAL = .TRUE., B is not referenced.

W

          W is DOUBLE PRECISION
          On entry, W is the diagonal element of the matrix B.
          If LREAL = .TRUE., W is not referenced.

SCALE

          SCALE is DOUBLE PRECISION
          On exit, SCALE is the scale factor.

X

          X is DOUBLE PRECISION array, dimension (2*N)
          On entry, X contains the right hand side of the system.
          On exit, X is overwritten by the solution.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER
          On exit, INFO is set to
             0: successful exit.
               1: the some diagonal 1 by 1 block has been perturbed by
                  a small number SMIN to keep nonsingularity.
               2: the some diagonal 2 by 2 block has been perturbed by
                  a small number in DLALN2 to keep nonsingularity.
          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlar1v (integer N, integer B1, integer BN, double precision LAMBDA, double precision, dimension( * ) D, double precision, dimension( * ) L, double precision, dimension( * ) LD, double precision, dimension( * ) LLD, double precision PIVMIN, double precision GAPTOL, double precision, dimension( * ) Z, logical WANTNC, integer NEGCNT, double precision ZTZ, double precision MINGMA, integer R, integer, dimension( * ) ISUPPZ, double precision NRMINV, double precision RESID, double precision RQCORR, double precision, dimension( * ) WORK)

DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

Purpose:

 DLAR1V computes the (scaled) r-th column of the inverse of
 the sumbmatrix in rows B1 through BN of the tridiagonal matrix
 L D L**T - sigma I. When sigma is close to an eigenvalue, the
 computed vector is an accurate eigenvector. Usually, r corresponds
 to the index where the eigenvector is largest in magnitude.
 The following steps accomplish this computation :
 (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
 (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
 (c) Computation of the diagonal elements of the inverse of
     L D L**T - sigma I by combining the above transforms, and choosing
     r as the index where the diagonal of the inverse is (one of the)
     largest in magnitude.
 (d) Computation of the (scaled) r-th column of the inverse using the
     twisted factorization obtained by combining the top part of the
     the stationary and the bottom part of the progressive transform.

Parameters:

N

          N is INTEGER
           The order of the matrix L D L**T.

B1

          B1 is INTEGER
           First index of the submatrix of L D L**T.

BN

          BN is INTEGER
           Last index of the submatrix of L D L**T.

LAMBDA

          LAMBDA is DOUBLE PRECISION
           The shift. In order to compute an accurate eigenvector,
           LAMBDA should be a good approximation to an eigenvalue
           of L D L**T.

L

          L is DOUBLE PRECISION array, dimension (N-1)
           The (n-1) subdiagonal elements of the unit bidiagonal matrix
           L, in elements 1 to N-1.

D

          D is DOUBLE PRECISION array, dimension (N)
           The n diagonal elements of the diagonal matrix D.

LD

          LD is DOUBLE PRECISION array, dimension (N-1)
           The n-1 elements L(i)*D(i).

LLD

          LLD is DOUBLE PRECISION array, dimension (N-1)
           The n-1 elements L(i)*L(i)*D(i).

PIVMIN

          PIVMIN is DOUBLE PRECISION
           The minimum pivot in the Sturm sequence.

GAPTOL

          GAPTOL is DOUBLE PRECISION
           Tolerance that indicates when eigenvector entries are negligible
           w.r.t. their contribution to the residual.

Z

          Z is DOUBLE PRECISION array, dimension (N)
           On input, all entries of Z must be set to 0.
           On output, Z contains the (scaled) r-th column of the
           inverse. The scaling is such that Z(R) equals 1.

WANTNC

          WANTNC is LOGICAL
           Specifies whether NEGCNT has to be computed.

NEGCNT

          NEGCNT is INTEGER
           If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
           in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.

ZTZ

          ZTZ is DOUBLE PRECISION
           The square of the 2-norm of Z.

MINGMA

          MINGMA is DOUBLE PRECISION
           The reciprocal of the largest (in magnitude) diagonal
           element of the inverse of L D L**T - sigma I.

R

          R is INTEGER
           The twist index for the twisted factorization used to
           compute Z.
           On input, 0 <= R <= N. If R is input as 0, R is set to
           the index where (L D L**T - sigma I)^{-1} is largest
           in magnitude. If 1 <= R <= N, R is unchanged.
           On output, R contains the twist index used to compute Z.
           Ideally, R designates the position of the maximum entry in the
           eigenvector.

ISUPPZ

          ISUPPZ is INTEGER array, dimension (2)
           The support of the vector in Z, i.e., the vector Z is
           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

NRMINV

          NRMINV is DOUBLE PRECISION
           NRMINV = 1/SQRT( ZTZ )

RESID

          RESID is DOUBLE PRECISION
           The residual of the FP vector.
           RESID = ABS( MINGMA )/SQRT( ZTZ )

RQCORR

          RQCORR is DOUBLE PRECISION
           The Rayleigh Quotient correction to LAMBDA.
           RQCORR = MINGMA*TMP

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 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 dlar2v (integer N, double precision, dimension( * ) X, double precision, dimension( * ) Y, double precision, dimension( * ) Z, integer INCX, double precision, dimension( * ) C, double precision, dimension( * ) S, integer INCC)

DLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

Purpose:

 DLAR2V applies a vector of real plane rotations from both sides to
 a sequence of 2-by-2 real symmetric matrices, defined by the elements
 of the vectors x, y and z. For i = 1,2,...,n
    ( x(i)  z(i) ) := (  c(i)  s(i) ) ( x(i)  z(i) ) ( c(i) -s(i) )
    ( z(i)  y(i) )    ( -s(i)  c(i) ) ( z(i)  y(i) ) ( s(i)  c(i) )

Parameters:

N

          N is INTEGER
          The number of plane rotations to be applied.

X

          X is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector x.

Y

          Y is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector y.

Z

          Z is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector z.

INCX

          INCX is INTEGER
          The increment between elements of X, Y and Z. INCX > 0.

C

          C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

S

          S is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

INCC

          INCC is INTEGER
          The increment between elements of C and S. INCC > 0.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlarf (character SIDE, integer M, integer N, double precision, dimension( * ) V, integer INCV, double precision TAU, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) WORK)

DLARF applies an elementary reflector to a general rectangular matrix.

Purpose:

 DLARF applies a real elementary reflector H to a real m by n matrix
 C, from either the left or the right. H is represented in the form
       H = I - tau * v * v**T
 where tau is a real scalar and v is a real vector.
 If tau = 0, then H is taken to be the unit matrix.

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.

V

          V is DOUBLE PRECISION array, dimension
                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
          The vector v in the representation of H. V is not used if
          TAU = 0.

INCV

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

TAU

          TAU is DOUBLE PRECISION
          The value tau in the representation of H.

C

          C is DOUBLE PRECISION 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 DOUBLE PRECISION 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

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

DLARFB applies a block reflector or its transpose to a general rectangular matrix.

Purpose:

 DLARFB applies a real block reflector H or its transpose H**T to a
 real m by n matrix C, from either the left or the right.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply H or H**T from the Left
          = 'R': apply H or H**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply H (No transpose)
          = 'T': apply H**T (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)
          = '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
          = '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).

V

          V is DOUBLE PRECISION array, dimension
                                (LDV,K) if STOREV = 'C'
                                (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                (LDV,N) if STOREV = 'R' and SIDE = 'R'
          The matrix V. See Further Details.

LDV

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

T

          T is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.

LDC

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

WORK

          WORK is DOUBLE PRECISION 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:

June 2013

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 = (  1       )                 V = (  1 v1 v1 v1 v1 )
                   ( v1  1    )                     (     1 v2 v2 v2 )
                   ( v1 v2  1 )                     (        1 v3 v3 )
                   ( v1 v2 v3 )
                   ( v1 v2 v3 )
  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                   (     1 v3 )
                   (        1 )

subroutine dlarfg (integer N, double precision ALPHA, double precision, dimension( * ) X, integer INCX, double precision TAU)

DLARFG generates an elementary reflector (Householder matrix).

Purpose:

 DLARFG generates a real elementary reflector H of order n, such
 that
       H * ( alpha ) = ( beta ),   H**T * H = I.
           (   x   )   (   0  )
 where alpha and beta are scalars, and x is an (n-1)-element real
 vector. H is represented in the form
       H = I - tau * ( 1 ) * ( 1 v**T ) ,
                     ( v )
 where tau is a real scalar and v is a real (n-1)-element
 vector.
 If the elements of x are all zero, then tau = 0 and H is taken to be
 the unit matrix.
 Otherwise  1 <= tau <= 2.

Parameters:

N

          N is INTEGER
          The order of the elementary reflector.

ALPHA

          ALPHA is DOUBLE PRECISION
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.

X

          X is DOUBLE PRECISION array, dimension
                         (1+(N-2)*abs(INCX))
          On entry, the vector x.
          On exit, it is overwritten with the vector v.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

TAU

          TAU is DOUBLE PRECISION
          The value tau.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

subroutine dlarfgp (integer N, double precision ALPHA, double precision, dimension( * ) X, integer INCX, double precision TAU)

DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

Purpose:

 DLARFGP generates a real elementary reflector H of order n, such
 that
       H * ( alpha ) = ( beta ),   H**T * H = I.
           (   x   )   (   0  )
 where alpha and beta are scalars, beta is non-negative, and x is
 an (n-1)-element real vector.  H is represented in the form
       H = I - tau * ( 1 ) * ( 1 v**T ) ,
                     ( v )
 where tau is a real scalar and v is a real (n-1)-element
 vector.
 If the elements of x are all zero, then tau = 0 and H is taken to be
 the unit matrix.

Parameters:

N

          N is INTEGER
          The order of the elementary reflector.

ALPHA

          ALPHA is DOUBLE PRECISION
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.

X

          X is DOUBLE PRECISION array, dimension
                         (1+(N-2)*abs(INCX))
          On entry, the vector x.
          On exit, it is overwritten with the vector v.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

TAU

          TAU is DOUBLE PRECISION
          The value tau.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

subroutine dlarft (character DIRECT, character STOREV, integer N, integer K, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( * ) TAU, double precision, dimension( ldt, * ) T, integer LDT)

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

Purpose:

 DLARFT forms the triangular factor T of a real 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**T
 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**T * T * V

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)
          = '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
          = '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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

T

          T is DOUBLE PRECISION 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

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.
  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                   ( v1  1    )                     (     1 v2 v2 v2 )
                   ( v1 v2  1 )                     (        1 v3 v3 )
                   ( v1 v2 v3 )
                   ( v1 v2 v3 )
  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                   (     1 v3 )
                   (        1 )

subroutine dlarfx (character SIDE, integer M, integer N, double precision, dimension( * ) V, double precision TAU, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) WORK)

DLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.

Purpose:

 DLARFX applies a real elementary reflector H to a real m by n
 matrix C, from either the left or the right. H is represented in the
 form
       H = I - tau * v * v**T
 where tau is a real scalar and v is a real vector.
 If tau = 0, then H is taken to be the unit matrix
 This version uses inline code if H has order < 11.

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.

V

          V is DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
                                     or (N) if SIDE = 'R'
          The vector v in the representation of H.

TAU

          TAU is DOUBLE PRECISION
          The value tau in the representation of H.

C

          C is DOUBLE PRECISION 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. LDA >= (1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                      (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'
          WORK is not referenced if H has order < 11.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlargv (integer N, double precision, dimension( * ) X, integer INCX, double precision, dimension( * ) Y, integer INCY, double precision, dimension( * ) C, integer INCC)

DLARGV generates a vector of plane rotations with real cosines and real sines.

Purpose:

 DLARGV generates a vector of real plane rotations, determined by
 elements of the real vectors x and y. For i = 1,2,...,n
    (  c(i)  s(i) ) ( x(i) ) = ( a(i) )
    ( -s(i)  c(i) ) ( y(i) ) = (   0  )

Parameters:

N

          N is INTEGER
          The number of plane rotations to be generated.

X

          X is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          On entry, the vector x.
          On exit, x(i) is overwritten by a(i), for i = 1,...,n.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

Y

          Y is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCY)
          On entry, the vector y.
          On exit, the sines of the plane rotations.

INCY

          INCY is INTEGER
          The increment between elements of Y. INCY > 0.

C

          C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

INCC

          INCC is INTEGER
          The increment between elements of C. INCC > 0.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlarrv (integer N, double precision VL, double precision VU, double precision, dimension( * ) D, double precision, dimension( * ) L, double precision PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, double precision MINRGP, double precision RTOL1, double precision RTOL2, double precision, dimension( * ) W, double precision, dimension( * ) WERR, double precision, dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, double precision, dimension( * ) GERS, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

 DLARRV computes the eigenvectors of the tridiagonal matrix
 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
 The input eigenvalues should have been computed by DLARRE.

Parameters:

N

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

VL

          VL is DOUBLE PRECISION
          Lower bound of the interval that contains the desired
          eigenvalues. VL < VU. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE.

VU

          VU is DOUBLE PRECISION
          Upper bound of the interval that contains the desired
          eigenvalues. VL < VU. 
          Note: VU is currently not used by this implementation of DLARRV, VU is
          passed to DLARRV because it could be used compute gaps on the right end
          of the extremal eigenvalues. However, with not much initial accuracy in
          LAMBDA and VU, the formula can lead to an overestimation of the right gap
          and thus to inadequately early RQI 'convergence'. This is currently
          prevented this by forcing a small right gap. And so it turns out that VU
          is currently not used by this implementation of DLARRV.

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D.
          On exit, D may be overwritten.

L

          L is DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not split.) At the end of each block
          is stored the corresponding shift as given by DLARRE.
          On exit, L is overwritten.

PIVMIN

          PIVMIN is DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence.

ISPLIT

          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.

M

          M is INTEGER
          The total number of input eigenvalues.  0 <= M <= N.

DOL

          DOL is INTEGER

DOU

          DOU is INTEGER
          If the user wants to compute only selected eigenvectors from all
          the eigenvalues supplied, he can specify an index range DOL:DOU.
          Or else the setting DOL=1, DOU=M should be applied.
          Note that DOL and DOU refer to the order in which the eigenvalues
          are stored in W.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
          computed eigenvectors. All other columns of Z are set to zero.

MINRGP

          MINRGP is DOUBLE PRECISION

RTOL1

          RTOL1 is DOUBLE PRECISION

RTOL2

          RTOL2 is DOUBLE PRECISION
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

W

          W is DOUBLE PRECISION array, dimension (N)
          The first M elements of W contain the APPROXIMATE 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 DLARRE is expected here ). Furthermore, they are with
          respect to the shift of the corresponding root representation
          for their block. On exit, W holds the eigenvalues of the
          UNshifted matrix.

WERR

          WERR is DOUBLE PRECISION array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W

WGAP

          WGAP is DOUBLE PRECISION array, dimension (N)
          The separation from the right neighbor eigenvalue in W.

IBLOCK

          IBLOCK is INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc.

INDEXW

          INDEXW is INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

GERS

          GERS is DOUBLE PRECISION array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
          be computed from the original UNshifted matrix.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', 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 eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I ).

WORK

          WORK is DOUBLE PRECISION array, dimension (12*N)

IWORK

          IWORK is INTEGER array, dimension (7*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          > 0:  A problem occurred in DLARRV.
          < 0:  One of the called subroutines signaled an internal problem.
                Needs inspection of the corresponding parameter IINFO
                for further information.
          =-1:  Problem in DLARRB when refining a child's eigenvalues.
          =-2:  Problem in DLARRF when computing the RRR of a child.
                When a child is inside a tight cluster, it can be difficult
                to find an RRR. A partial remedy from the user's point of
                view is to make the parameter MINRGP smaller and recompile.
                However, as the orthogonality of the computed vectors is
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINRGP.
          =-3:  Problem in DLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps.

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 dlartv (integer N, double precision, dimension( * ) X, integer INCX, double precision, dimension( * ) Y, integer INCY, double precision, dimension( * ) C, double precision, dimension( * ) S, integer INCC)

DLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors.

Purpose:

 DLARTV applies a vector of real plane rotations to elements of the
 real vectors x and y. For i = 1,2,...,n
    ( x(i) ) := (  c(i)  s(i) ) ( x(i) )
    ( y(i) )    ( -s(i)  c(i) ) ( y(i) )

Parameters:

N

          N is INTEGER
          The number of plane rotations to be applied.

X

          X is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCX)
          The vector x.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

Y

          Y is DOUBLE PRECISION array,
                         dimension (1+(N-1)*INCY)
          The vector y.

INCY

          INCY is INTEGER
          The increment between elements of Y. INCY > 0.

C

          C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

S

          S is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

INCC

          INCC is INTEGER
          The increment between elements of C and S. INCC > 0.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlaswp (integer N, double precision, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX)

DLASWP performs a series of row interchanges on a general rectangular matrix.

Purpose:

 DLASWP performs a series of row interchanges on the matrix A.
 One row interchange is initiated for each of rows K1 through K2 of A.

Parameters:

N

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

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the matrix of column dimension N to which the row
          interchanges will be applied.
          On exit, the permuted matrix.

LDA

          LDA is INTEGER
          The leading dimension of the array A.

K1

          K1 is INTEGER
          The first element of IPIV for which a row interchange will
          be done.

K2

          K2 is INTEGER
          (K2-K1+1) is the number of elements of IPIV for which a row
          interchange will be done.

IPIV

          IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
          The vector of pivot indices. Only the elements in positions
          K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
          IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
          interchanged.

INCX

          INCX is INTEGER
          The increment between successive values of IPIV. If INCX
          is negative, the pivots are applied in reverse order.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

Further Details:

  Modified by
   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

subroutine dlat2s (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, real, dimension( ldsa, * ) SA, integer LDSA, integer INFO)

DLAT2S converts a double-precision triangular matrix to a single-precision triangular matrix.

Purpose:

 DLAT2S converts a DOUBLE PRECISION triangular matrix, SA, to a SINGLE
 PRECISION triangular matrix, A.
 RMAX is the overflow for the SINGLE PRECISION arithmetic
 DLAS2S checks that all the entries of A are between -RMAX and
 RMAX. If not the conversion is aborted and a flag is raised.
 This is an auxiliary routine so there is no argument checking.

Parameters:

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

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

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N triangular coefficient matrix A.

LDA

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

SA

          SA is REAL array, dimension (LDSA,N)
          Only the UPLO part of SA is referenced.  On exit, if INFO=0,
          the N-by-N coefficient matrix SA; if INFO>0, the content of
          the UPLO part of SA is unspecified.

LDSA

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

INFO

          INFO is INTEGER
          = 0:  successful exit.
          = 1:  an entry of the matrix A is greater than the SINGLE
                PRECISION overflow threshold, in this case, the content
                of the UPLO part of SA in exit is unspecified.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine dlatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)

DLATBS solves a triangular banded system of equations.

Purpose:

 DLATBS solves one of the triangular systems
    A *x = s*b  or  A**T*x = s*b
 with scaling to prevent overflow, where A is an upper or lower
 triangular band matrix.  Here A**T denotes the transpose of A, x and b
 are n-element vectors, and s is a scaling factor, usually less than
 or equal to 1, chosen so that the components of x will be less than
 the overflow threshold.  If the unscaled problem will not cause
 overflow, the Level 2 BLAS routine DTBSV is called.  If the matrix A
 is singular (A(j,j) = 0 for some j), then s is set to 0 and a
 non-trivial solution to A*x = 0 is returned.

Parameters:

UPLO

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

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A**T* x = s*b  (Transpose)
          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

N

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

KD

          KD is INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

AB

          AB is DOUBLE PRECISION 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).

LDAB

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

X

          X is DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b  or  A**T* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is DOUBLE PRECISION array, dimension (N)
          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.
          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

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

Further Details:

  A rough bound on x is computed; if that is less than overflow, DTBSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.
  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is
       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end
  Define bounds on the components of x after j iterations of the loop:
     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence
     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and
     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                   1<=i< j
  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).
  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
  algorithm for A upper triangular is
       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
       end
  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j
  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is
       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j
  and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

subroutine dlatdf (integer IJOB, integer N, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) RHS, double precision RDSUM, double precision RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV)

DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.

Purpose:

 DLATDF uses the LU factorization of the n-by-n matrix Z computed by
 DGETC2 and computes a contribution to the reciprocal Dif-estimate
 by solving Z * x = b for x, and choosing the r.h.s. b such that
 the norm of x is as large as possible. On entry RHS = b holds the
 contribution from earlier solved sub-systems, and on return RHS = x.
 The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
 where P and Q are permutation matrices. L is lower triangular with
 unit diagonal elements and U is upper triangular.

Parameters:

IJOB

          IJOB is INTEGER
          IJOB = 2: First compute an approximative null-vector e
              of Z using DGECON, e is normalized and solve for
              Zx = +-e - f with the sign giving the greater value
              of 2-norm(x). About 5 times as expensive as Default.
          IJOB .ne. 2: Local look ahead strategy where all entries of
              the r.h.s. b is chosen as either +1 or -1 (Default).

N

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

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, the LU part of the factorization of the n-by-n
          matrix Z computed by DGETC2:  Z = P * L * U * Q

LDZ

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

RHS

          RHS is DOUBLE PRECISION array, dimension (N)
          On entry, RHS contains contributions from other subsystems.
          On exit, RHS contains the solution of the subsystem with
          entries acoording to the value of IJOB (see above).

RDSUM

          RDSUM is DOUBLE PRECISION
          On entry, the sum of squares of computed contributions to
          the Dif-estimate under computation by DTGSYL, where the
          scaling factor RDSCAL (see below) has been factored out.
          On exit, the corresponding sum of squares updated with the
          contributions from the current sub-system.
          If TRANS = 'T' RDSUM is not touched.
          NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.

RDSCAL

          RDSCAL is DOUBLE PRECISION
          On entry, scaling factor used to prevent overflow in RDSUM.
          On exit, RDSCAL is updated w.r.t. the current contributions
          in RDSUM.
          If TRANS = 'T', RDSCAL is not touched.
          NOTE: RDSCAL only makes sense when DTGSY2 is called by
                DTGSYL.

IPIV

          IPIV is INTEGER array, dimension (N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension (N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

  [1] Bo Kagstrom and Lars Westin,
      Generalized Schur Methods with Condition Estimators for
      Solving the Generalized Sylvester Equation, IEEE Transactions
      on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
  [2] Peter Poromaa,
      On Efficient and Robust Estimators for the Separation
      between two Regular Matrix Pairs with Applications in
      Condition Estimation. Report IMINF-95.05, Departement of
      Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

subroutine dlatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)

DLATPS solves a triangular system of equations with the matrix held in packed storage.

Purpose:

 DLATPS solves one of the triangular systems
    A *x = s*b  or  A**T*x = s*b
 with scaling to prevent overflow, where A is an upper or lower
 triangular matrix stored in packed form.  Here A**T denotes the
 transpose of A, x and b are n-element vectors, and s is a scaling
 factor, usually less than or equal to 1, chosen so that the
 components of x will be less than the overflow threshold.  If the
 unscaled problem will not cause overflow, the Level 2 BLAS routine
 DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
 then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

Parameters:

UPLO

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

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A**T* x = s*b  (Transpose)
          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

N

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

AP

          AP is DOUBLE PRECISION 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.

X

          X is DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b  or  A**T* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is DOUBLE PRECISION array, dimension (N)
          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.
          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

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

Further Details:

  A rough bound on x is computed; if that is less than overflow, DTPSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.
  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is
       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end
  Define bounds on the components of x after j iterations of the loop:
     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence
     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and
     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                   1<=i< j
  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).
  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
  algorithm for A upper triangular is
       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
       end
  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j
  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is
       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j
  and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

subroutine dlatrd (character UPLO, integer N, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, double precision, dimension( * ) TAU, double precision, dimension( ldw, * ) W, integer LDW)

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Purpose:

 DLATRD reduces NB rows and columns of a real symmetric matrix A to
 symmetric tridiagonal form by an orthogonal similarity
 transformation Q**T * A * Q, and returns the matrices V and W which are
 needed to apply the transformation to the unreduced part of A.
 If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.
 This is an auxiliary routine called by DSYTRD.

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.

NB

          NB is INTEGER
          The number of rows and columns to be reduced.

A

          A is DOUBLE PRECISION 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 UPLO = 'U', the last NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements above the diagonal
            with the array TAU, represent the orthogonal matrix Q as a
            product of elementary reflectors;
          if UPLO = 'L', the first NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements below the diagonal
            with the array TAU, represent the  orthogonal matrix Q as a
            product of elementary reflectors.
          See Further Details.

LDA

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

E

          E is DOUBLE PRECISION array, dimension (N-1)
          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
          elements of the last NB columns of the reduced matrix;
          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
          the first NB columns of the reduced matrix.

TAU

          TAU is DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors, stored in
          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
          See Further Details.

W

          W is DOUBLE PRECISION array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.

LDW

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

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) H(n-1) . . . H(n-nb+1).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
  and tau in TAU(i-1).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(nb).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
  and tau in TAU(i).
  The elements of the vectors v together form the n-by-nb matrix V
  which is needed, with W, to apply the transformation to the unreduced
  part of the matrix, using a symmetric rank-2k update of the form:
  A := A - V*W**T - W*V**T.
  The contents of A on exit are illustrated by the following examples
  with n = 5 and nb = 2:
  if UPLO = 'U':                       if UPLO = 'L':
    (  a   a   a   v4  v5 )              (  d                  )
    (      a   a   v4  v5 )              (  1   d              )
    (          a   1   v5 )              (  v1  1   a          )
    (              d   1  )              (  v1  v2  a   a      )
    (                  d  )              (  v1  v2  a   a   a  )
  where d denotes a diagonal element of the reduced matrix, a denotes
  an element of the original matrix that is unchanged, and vi denotes
  an element of the vector defining H(i).

subroutine dlatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)

DLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

 DLATRS solves one of the triangular systems
    A *x = s*b  or  A**T *x = s*b
 with scaling to prevent overflow.  Here A is an upper or lower
 triangular matrix, A**T denotes the transpose of A, x and b are
 n-element vectors, and s is a scaling factor, usually less than
 or equal to 1, chosen so that the components of x will be less than
 the overflow threshold.  If the unscaled problem will not cause
 overflow, the Level 2 BLAS routine DTRSV is called.  If the matrix A
 is singular (A(j,j) = 0 for some j), then s is set to 0 and a
 non-trivial solution to A*x = 0 is returned.

Parameters:

UPLO

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

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b  (No transpose)
          = 'T':  Solve A**T* x = s*b  (Transpose)
          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

N

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

A

          A is DOUBLE PRECISION 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).

X

          X is DOUBLE PRECISION array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is DOUBLE PRECISION
          The scaling factor s for the triangular system
             A * x = s*b  or  A**T* x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is DOUBLE PRECISION array, dimension (N)
          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.
          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

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

Further Details:

  A rough bound on x is computed; if that is less than overflow, DTRSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.
  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is
       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end
  Define bounds on the components of x after j iterations of the loop:
     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence
     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and
     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                   1<=i< j
  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).
  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
  algorithm for A upper triangular is
       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
       end
  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j
  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is
       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j
  and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

subroutine dlauu2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)

DLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).

Purpose:

 DLAUU2 computes the product U * U**T or L**T * L, where the triangular
 factor U or L is stored in the upper or lower triangular part of
 the array A.
 If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
 overwriting the factor U in A.
 If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
 overwriting the factor L in A.
 This is the unblocked form of the algorithm, calling Level 2 BLAS.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the triangular factor U or L.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U**T;
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L**T * L.

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 dlauum (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)

DLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).

Purpose:

 DLAUUM computes the product U * U**T or L**T * L, where the triangular
 factor U or L is stored in the upper or lower triangular part of
 the array A.
 If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
 overwriting the factor U in A.
 If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
 overwriting the factor L in A.
 This is the blocked form of the algorithm, calling Level 3 BLAS.

Parameters:

UPLO

          UPLO is CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the triangular factor U or L.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U**T;
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L**T * L.

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 drscl (integer N, double precision SA, double precision, dimension( * ) SX, integer INCX)

DRSCL multiplies a vector by the reciprocal of a real scalar.

Purpose:

 DRSCL multiplies an n-element real vector x by the real scalar 1/a.
 This is done without overflow or underflow as long as
 the final result x/a does not overflow or underflow.

Parameters:

N

          N is INTEGER
          The number of components of the vector x.

SA

          SA is DOUBLE PRECISION
          The scalar a which is used to divide each component of x.
          SA must be >= 0, or the subroutine will divide by zero.

SX

          SX is DOUBLE PRECISION array, dimension
                         (1+(N-1)*abs(INCX))
          The n-element vector x.

INCX

          INCX is INTEGER
          The increment between successive values of the vector SX.
          > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2017

subroutine dtprfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldwork, * ) WORK, integer LDWORK)

DTPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.

Purpose:

 DTPRFB applies a real "triangular-pentagonal" block reflector H or its
 transpose H**T to a real matrix C, which is composed of two
 blocks A and B, either from the left or right.

Parameters:

SIDE

          SIDE is CHARACTER*1
          = 'L': apply H or H**T from the Left
          = 'R': apply H or H**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply H (No transpose)
          = 'T': apply H**T (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)
          = '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': Columns
          = 'R': Rows

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 order of the matrix T, i.e. the number of elementary
          reflectors whose product defines the block reflector.
          K >= 0.

L

          L is INTEGER
          The order of the trapezoidal part of V.
          K >= L >= 0.  See Further Details.

V

          V is DOUBLE PRECISION array, dimension
                                (LDV,K) if STOREV = 'C'
                                (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                (LDV,N) if STOREV = 'R' and SIDE = 'R'
          The pentagonal matrix V, which contains the elementary reflectors
          H(1), H(2), ..., H(K).  See Further Details.

LDV

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

T

          T is DOUBLE PRECISION 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.

A

          A is DOUBLE PRECISION 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
          H*C or H**T*C or C*H or C*H**T.  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 DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the M-by-N matrix B.
          On exit, B is overwritten by the corresponding block of
          H*C or H**T*C or C*H or C*H**T.  See Further Details.

LDB

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

WORK

          WORK is DOUBLE PRECISION array, dimension
          (LDWORK,N) if SIDE = 'L',
          (LDWORK,K) if SIDE = 'R'.

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= K;
          if SIDE = 'R', LDWORK >= M.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  The matrix C is a composite matrix formed from blocks A and B.
  The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
  and if SIDE = 'L', A is of size K-by-N.
  If SIDE = 'R' and DIRECT = 'F', C = [A B].
  If SIDE = 'L' and DIRECT = 'F', C = [A]
                                      [B].
  If SIDE = 'R' and DIRECT = 'B', C = [B A].
  If SIDE = 'L' and DIRECT = 'B', C = [B]
                                      [A].
  The pentagonal matrix V is composed of a rectangular block V1 and a
  trapezoidal block V2.  The size of the trapezoidal block is determined by
  the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;
  if L=0, there is no trapezoidal block, thus V = V1 is rectangular.
  If DIRECT = 'F' and STOREV = 'C':  V = [V1]
                                         [V2]
     - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)
  If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]
     - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)
  If DIRECT = 'B' and STOREV = 'C':  V = [V2]
                                         [V1]
     - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)
  If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]
     - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)
  If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
  If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
  If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
  If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.

subroutine slatrd (character UPLO, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) E, real, dimension( * ) TAU, real, dimension( ldw, * ) W, integer LDW)

SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Purpose:

 SLATRD reduces NB rows and columns of a real symmetric matrix A to
 symmetric tridiagonal form by an orthogonal similarity
 transformation Q**T * A * Q, and returns the matrices V and W which are
 needed to apply the transformation to the unreduced part of A.
 If UPLO = 'U', SLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', SLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.
 This is an auxiliary routine called by SSYTRD.

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.

NB

          NB is INTEGER
          The number of rows and columns to be reduced.

A

          A is REAL 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 UPLO = 'U', the last NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements above the diagonal
            with the array TAU, represent the orthogonal matrix Q as a
            product of elementary reflectors;
          if UPLO = 'L', the first NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements below the diagonal
            with the array TAU, represent the  orthogonal matrix Q as a
            product of elementary reflectors.
          See Further Details.

LDA

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

E

          E is REAL array, dimension (N-1)
          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
          elements of the last NB columns of the reduced matrix;
          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
          the first NB columns of the reduced matrix.

TAU

          TAU is REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors, stored in
          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
          See Further Details.

W

          W is REAL array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.

LDW

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

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) H(n-1) . . . H(n-nb+1).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
  and tau in TAU(i-1).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(nb).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
  and tau in TAU(i).
  The elements of the vectors v together form the n-by-nb matrix V
  which is needed, with W, to apply the transformation to the unreduced
  part of the matrix, using a symmetric rank-2k update of the form:
  A := A - V*W**T - W*V**T.
  The contents of A on exit are illustrated by the following examples
  with n = 5 and nb = 2:
  if UPLO = 'U':                       if UPLO = 'L':
    (  a   a   a   v4  v5 )              (  d                  )
    (      a   a   v4  v5 )              (  1   d              )
    (          a   1   v5 )              (  v1  1   a          )
    (              d   1  )              (  v1  v2  a   a      )
    (                  d  )              (  v1  v2  a   a   a  )
  where d denotes a diagonal element of the reduced matrix, a denotes
  an element of the original matrix that is unchanged, and vi denotes
  an element of the vector defining H(i).

Author

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