## table of contents

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

**dlarfy**(UPLO, N, V, INCV, TAU, C, LDC, WORK)

**DLARFY**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

**References:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**References:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 > 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 > 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO > 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 triangular in rows and columns INFO+1 through IHI. If INFO > 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 (<= 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**References:**

## 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) Auxiliary 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**References:**

## 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 >= 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 > 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 > 0, then 1 <= ILO <= IHI <= 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) < 0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i > 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 >= 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) > 0 and WI(i+1) < 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ). Otherwise, LDZ >= 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 >= 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 > 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not accessed.

**Contributors:**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 >= 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 <= NW <= (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 <= 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 <= ILOZ <= IHIZ <= 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 <= 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 <= LDV

*NH*

NH is INTEGER The number of columns of T. NH >= 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 <= LDT

*NV*

NV is INTEGER The number of rows of work array WV available for workspace. NV >= 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 <= 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 <= NW <= (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 <= 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 <= ILOZ <= IHIZ <= 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 <= 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 <= LDV

*NH*

NH is INTEGER The number of columns of T. NH >= 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 <= LDT

*NV*

NV is INTEGER The number of rows of work array WV available for workspace. NV >= 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 <= 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 >= 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 > 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 > 0, then 1 <= ILO <= IHI <= 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) < 0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i > 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 >= 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) > 0 and WI(i+1) < 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ). Otherwise, LDZ >= 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 >= 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 > 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not accessed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**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 >= 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 <= ILOZ <= IHIZ <= 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 >= 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 >= 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 >= 3*NSHFTS-3.

*NV*

NV is INTEGER NV is the number of rows in WV agailable for workspace. NV >= 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 >= NV.

*NH*

NH is INTEGER NH is the number of columns in array WH available for workspace. NH >= 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 >= 3*NSHFTS-3.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**References:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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). If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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. LDC >= (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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dlarfy (character UPLO, integer N, double precision, dimension( * ) V, integer INCV, double precision TAU, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) WORK)¶

**DLARFY**

**Purpose:**

DLARFY applies an elementary reflector, or Householder matrix, H, to an n x n symmetric matrix C, from both the left and the right. H is represented in the form H = I - tau * v * v' where tau is a scalar and v is a vector. If tau is zero, then H is taken to be the unit matrix.

**Parameters**

*UPLO*

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

*N*

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

*V*

V is DOUBLE PRECISION array, dimension (1 + (N-1)*abs(INCV)) The vector v as described above.

*INCV*

INCV is INTEGER The increment between successive elements of v. INCV must not be zero.

*TAU*

TAU is DOUBLE PRECISION The value tau as described above.

*C*

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

*LDC*

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

*WORK*

WORK is DOUBLE PRECISION array, dimension (N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 < 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 according 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

**Contributors:**

**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*

*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*

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

*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*

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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', LDA >= max(1,K); If SIDE = 'R', LDA >= 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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).

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