dlals0(3)

double

Section 3 liblapack-doc bookworm source

Description

doubleOTHERcomputational

NAME

doubleOTHERcomputational - double

SYNOPSIS

Functions

subroutine ctplqt (M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
CTPLQT
subroutine ctplqt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
CTPLQT2
subroutine ctpmlqt (SIDE, TRANS, M, N, K, L, MB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
CTPMLQT
subroutine dbbcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, WORK, LWORK, INFO)
DBBCSD
subroutine dgetsqrhrt (M, N, MB1, NB1, NB2, A, LDA, T, LDT, WORK, LWORK, INFO)
DGETSQRHRT
subroutine dgghd3 (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DGGHD3
subroutine dgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
DGGHRD
subroutine dggqrf (N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
DGGQRF
subroutine dggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
DGGRQF
subroutine dggsvp3 (JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
DGGSVP3
subroutine dgsvj0 (JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
DGSVJ0 pre-processor for the routine dgesvj.
subroutine dgsvj1 (JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots.
subroutine dhsein (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO)
DHSEIN
subroutine dhseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)
DHSEQR
subroutine dla_lin_berr (N, NZ, NRHS, RES, AYB, BERR)
DLA_LIN_BERR computes a component-wise relative backward error.
subroutine dla_wwaddw (N, X, Y, W)
DLA_WWADDW adds a vector into a doubled-single vector.
subroutine dlals0 (ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO)
DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine dlalsa (ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)
DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine dlalsd (UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, IWORK, INFO)
DLALSD uses the singular value decomposition of A to solve the least squares problem.
double precision function dlansf (NORM, TRANSR, UPLO, N, A, WORK)
DLANSF 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 in RFP format.
subroutine dlarscl2 (M, N, D, X, LDX)
DLARSCL2 performs reciprocal diagonal scaling on a matrix.
subroutine dlarz (SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
subroutine dlarzb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
DLARZB applies a block reflector or its transpose to a general matrix.
subroutine dlarzt (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
DLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine dlascl2 (M, N, D, X, LDX)
DLASCL2 performs diagonal scaling on a matrix.
subroutine dlatrz (M, N, L, A, LDA, TAU, WORK)
DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine dopgtr (UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
DOPGTR
subroutine dopmtr (SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
DOPMTR
subroutine dorbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
DORBDB
subroutine dorbdb1 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
DORBDB1
subroutine dorbdb2 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
DORBDB2
subroutine dorbdb3 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
DORBDB3
subroutine dorbdb4 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO)
DORBDB4
subroutine dorbdb5 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
DORBDB5
subroutine dorbdb6 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
DORBDB6
recursive subroutine dorcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK, IWORK, INFO)
DORCSD
subroutine dorcsd2by1 (JOBU1, JOBU2, JOBV1T, M, P, Q, X11, LDX11, X21, LDX21, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, WORK, LWORK, IWORK, INFO)
DORCSD2BY1
subroutine dorg2l (M, N, K, A, LDA, TAU, WORK, INFO)
DORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine dorg2r (M, N, K, A, LDA, TAU, WORK, INFO)
DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine dorghr (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
DORGHR
subroutine dorgl2 (M, N, K, A, LDA, TAU, WORK, INFO)
DORGL2
subroutine dorglq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGLQ
subroutine dorgql (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQL
subroutine dorgqr (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
subroutine dorgr2 (M, N, K, A, LDA, TAU, WORK, INFO)
DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
subroutine dorgrq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGRQ
subroutine dorgtr (UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
DORGTR
subroutine dorgtsqr (M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO)
DORGTSQR
subroutine dorgtsqr_row (M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO)
DORGTSQR_ROW
subroutine dorhr_col (M, N, NB, A, LDA, T, LDT, D, INFO)
DORHR_COL
subroutine dorm2l (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine dorm2r (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine dormbr (VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMBR
subroutine dormhr (SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMHR
subroutine dorml2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
subroutine dormlq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMLQ
subroutine dormql (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQL
subroutine dormqr (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
subroutine dormr2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
subroutine dormr3 (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, INFO)
DORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
subroutine dormrq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMRQ
subroutine dormrz (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMRZ
subroutine dormtr (SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMTR
subroutine dpbcon (UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, IWORK, INFO)
DPBCON
subroutine dpbequ (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
DPBEQU
subroutine dpbrfs (UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DPBRFS
subroutine dpbstf (UPLO, N, KD, AB, LDAB, INFO)
DPBSTF
subroutine dpbtf2 (UPLO, N, KD, AB, LDAB, INFO)
DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
subroutine dpbtrf (UPLO, N, KD, AB, LDAB, INFO)
DPBTRF
subroutine dpbtrs (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
DPBTRS
subroutine dpftrf (TRANSR, UPLO, N, A, INFO)
DPFTRF
subroutine dpftri (TRANSR, UPLO, N, A, INFO)
DPFTRI
subroutine dpftrs (TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
DPFTRS
subroutine dppcon (UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO)
DPPCON
subroutine dppequ (UPLO, N, AP, S, SCOND, AMAX, INFO)
DPPEQU
subroutine dpprfs (UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DPPRFS
subroutine dpptrf (UPLO, N, AP, INFO)
DPPTRF
subroutine dpptri (UPLO, N, AP, INFO)
DPPTRI
subroutine dpptrs (UPLO, N, NRHS, AP, B, LDB, INFO)
DPPTRS
subroutine dpstf2 (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine dpstrf (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine dsbgst (VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO)
DSBGST
subroutine dsbtrd (VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
DSBTRD
subroutine dsfrk (TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C)
DSFRK performs a symmetric rank-k operation for matrix in RFP format.
subroutine dspcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSPCON
subroutine dspgst (ITYPE, UPLO, N, AP, BP, INFO)
DSPGST
subroutine dsprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DSPRFS
subroutine dsptrd (UPLO, N, AP, D, E, TAU, INFO)
DSPTRD
subroutine dsptrf (UPLO, N, AP, IPIV, INFO)
DSPTRF
subroutine dsptri (UPLO, N, AP, IPIV, WORK, INFO)
DSPTRI
subroutine dsptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
DSPTRS
subroutine dstegr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEGR
subroutine dstein (N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEIN
subroutine dstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEMR
subroutine dtbcon (NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, IWORK, INFO)
DTBCON
subroutine dtbrfs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DTBRFS
subroutine dtbtrs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
DTBTRS
subroutine dtfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine dtftri (TRANSR, UPLO, DIAG, N, A, INFO)
DTFTRI
subroutine dtfttp (TRANSR, UPLO, N, ARF, AP, INFO)
DTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
subroutine dtfttr (TRANSR, UPLO, N, ARF, A, LDA, INFO)
DTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
subroutine dtgsen (IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
DTGSEN
subroutine dtgsja (JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
DTGSJA
subroutine dtgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
DTGSNA
subroutine dtpcon (NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK, INFO)
DTPCON
subroutine dtplqt (M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
DTPLQT
subroutine dtplqt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
DTPLQT2 computes a LQ factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine dtpmlqt (SIDE, TRANS, M, N, K, L, MB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
DTPMLQT
subroutine dtpmqrt (SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
DTPMQRT
subroutine dtpqrt (M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
DTPQRT
subroutine dtpqrt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
DTPQRT2 computes a QR factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine dtprfs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DTPRFS
subroutine dtptri (UPLO, DIAG, N, AP, INFO)
DTPTRI
subroutine dtptrs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO)
DTPTRS
subroutine dtpttf (TRANSR, UPLO, N, AP, ARF, INFO)
DTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
subroutine dtpttr (UPLO, N, AP, A, LDA, INFO)
DTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
subroutine dtrcon (NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, IWORK, INFO)
DTRCON
subroutine dtrevc (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
DTREVC
subroutine dtrevc3 (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, INFO)
DTREVC3
subroutine dtrexc (COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
DTREXC
subroutine dtrrfs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DTRRFS
subroutine dtrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO)
DTRSEN
subroutine dtrsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO)
DTRSNA
subroutine dtrti2 (UPLO, DIAG, N, A, LDA, INFO)
DTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
subroutine dtrtri (UPLO, DIAG, N, A, LDA, INFO)
DTRTRI
subroutine dtrtrs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
DTRTRS
subroutine dtrttf (TRANSR, UPLO, N, A, LDA, ARF, INFO)
DTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
subroutine dtrttp (UPLO, N, A, LDA, AP, INFO)
DTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
subroutine dtzrzf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DTZRZF
subroutine stplqt (M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
STPLQT
subroutine stplqt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
STPLQT2 computes a LQ factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine stpmlqt (SIDE, TRANS, M, N, K, L, MB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
STPMLQT
subroutine ztplqt (M, N, L, MB, A, LDA, B, LDB, T, LDT, WORK, INFO)
ZTPLQT
subroutine ztplqt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
ZTPLQT2 computes a LQ factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine ztpmlqt (SIDE, TRANS, M, N, K, L, MB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO)
ZTPMLQT

Detailed Description

This is the group of double other Computational routines

Function Documentation

subroutine ctplqt (integer M, integer N, integer L, integer MB, complex,dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integerLDB, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( * )WORK, integer INFO)

CTPLQT

Purpose:

CTPLQT computes a blocked LQ factorization of a complex
’triangular-pentagonal’ matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.

Parameters

M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.

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

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

MB is INTEGER
The block size to be used in the blocked QR. M >= MB >= 1.

A is COMPLEX array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

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

B is COMPLEX array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.

LDB

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

T is COMPLEX array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.

LDT

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

WORK

WORK is COMPLEX array, dimension (MB*M)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
upper trapezoidal matrix B2:
[ B ] = [ B1 ] [ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
[ C ] = [ A ] [ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
[ V ] = [ V1 ] [ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)’s.

The number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The MB-by-MB (and IB-by-IB
for the last block) T’s are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].

subroutine ctplqt2 (integer M, integer N, integer L, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex,dimension( ldt, * ) T, integer LDT, integer INFO)

CTPLQT2

Purpose:

CTPLQT2 computes a LQ a factorization of a complex ’triangular-pentagonal’
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.

Parameters

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

N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

A is COMPLEX array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

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

LDB

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

T is COMPLEX array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.

LDT

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ][ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:

B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C

C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,

W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)’s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W**T * T * W

where WˆH is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.

subroutine ctpmlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer L, integer MB, complex, dimension( ldv, * ) V, integerLDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex,dimension( * ) WORK, integer INFO)

CTPMLQT

Purpose:

CTPMLQT applies a complex unitary matrix Q obtained from a
’triangular-pentagonal’ complex block reflector H to a general
complex matrix C, which consists of two blocks A and B.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**H from the Left;
= ’R’: apply Q or Q**H from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Conjugate transpose, apply Q**H.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.

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

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in CTPLQT.

V is COMPLEX array, dimension (LDV,K)
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
CTPLQT in B. See Further Details.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= K.

T is COMPLEX array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by CTPLQT, stored as a MB-by-K matrix.

LDT

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

A is COMPLEX array, dimension
(LDA,N) if SIDE = ’L’ or
(LDA,K) if SIDE = ’R’
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDA >= max(1,K);
If SIDE = ’R’, LDA >= max(1,M).

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

LDB

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

WORK

WORK is COMPLEX array. The dimension of WORK is
N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:

V = [V1] [V2].

The size of the trapezoidal block V2 is determined by the parameter L,
where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L
rows of a K-by-K upper triangular matrix. If L=K, V2 is lower triangular;
if L=0, there is no trapezoidal block, hence V = V1 is rectangular.

If SIDE = ’L’: C = [A] where A is K-by-N, B is M-by-N and V is K-by-M.
[B]

If SIDE = ’R’: C = [A B] where A is M-by-K, B is M-by-N and V is K-by-N.

The complex unitary matrix Q is formed from V and T.

If TRANS=’N’ and SIDE=’L’, C is on exit replaced with Q * C.

If TRANS=’C’ and SIDE=’L’, C is on exit replaced with Q**H * C.

If TRANS=’N’ and SIDE=’R’, C is on exit replaced with C * Q.

If TRANS=’C’ and SIDE=’R’, C is on exit replaced with C * Q**H.

subroutine dbbcsd (character JOBU1, character JOBU2, character JOBV1T,character JOBV2T, character TRANS, integer M, integer P, integer Q, doubleprecision, dimension( * ) THETA, double precision, dimension( * ) PHI,double precision, dimension( ldu1, * ) U1, integer LDU1, double precision,dimension( ldu2, * ) U2, integer LDU2, double precision, dimension( ldv1t,* ) V1T, integer LDV1T, double precision, dimension( ldv2t, * ) V2T,integer LDV2T, double precision, dimension( * ) B11D, double precision,dimension( * ) B11E, double precision, dimension( * ) B12D, doubleprecision, dimension( * ) B12E, double precision, dimension( * ) B21D,double precision, dimension( * ) B21E, double precision, dimension( * )B22D, double precision, dimension( * ) B22E, double precision, dimension( *) WORK, integer LWORK, integer INFO)

DBBCSD

Purpose:

DBBCSD computes the CS decomposition of an orthogonal matrix in
bidiagonal-block form,

[ B11 | B12 0 0 ]
[ 0 | 0 -I 0 ]
X = [----------------]
[ B21 | B22 0 0 ]
[ 0 | 0 0 I ]

[ C | -S 0 0 ]
[ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T
= [---------] [---------------] [---------] .
[ | U2 ] [ S | C 0 0 ] [ | V2 ]
[ 0 | 0 0 I ]

X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
transposed and/or permuted. This can be done in constant time using
the TRANS and SIGNS options. See DORCSD for details.)

The bidiagonal matrices B11, B12, B21, and B22 are represented
implicitly by angles THETA(1:Q) and PHI(1:Q-1).

The orthogonal matrices U1, U2, V1T, and V2T are input/output.
The input matrices are pre- or post-multiplied by the appropriate
singular vector matrices.

Parameters

JOBU1

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

JOBU2

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

JOBV1T

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

JOBV2T

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

TRANS

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

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

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

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

THETA

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

PHI

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

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

LDU1

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

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

LDU2

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

V1T

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

LDV1T

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

V2T

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

LDV2T

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

B11D

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

B11E

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

B12D

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

B12E

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

B21D

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

B21E

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

B22D

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

B22E

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= MAX(1,8*Q).

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

INFO

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

Internal Parameters:

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

References:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgetsqrhrt (integer M, integer N, integer MB1, integer NB1,integer NB2, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( ldt, * ) T, integer LDT, double precision, dimension(* ) WORK, integer LWORK, integer INFO)

DGETSQRHRT

Purpose:

DGETSQRHRT computes a NB2-sized column blocked QR-factorization
of a real M-by-N matrix A with M >= N,

A = Q * R.

The routine uses internally a NB1-sized column blocked and MB1-sized
row blocked TSQR-factorization and perfors the reconstruction
of the Householder vectors from the TSQR output. The routine also
converts the R_tsqr factor from the TSQR-factorization output into
the R factor that corresponds to the Householder QR-factorization,

A = Q_tsqr * R_tsqr = Q * R.

The output Q and R factors are stored in the same format as in DGEQRT
(Q is in blocked compact WY-representation). See the documentation
of DGEQRT for more details on the format.

Parameters

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

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

MB1

MB1 is INTEGER
The row block size to be used in the blocked TSQR.
MB1 > N.

NB1

NB1 is INTEGER
The column block size to be used in the blocked TSQR.
N >= NB1 >= 1.

NB2

NB2 is INTEGER
The block size to be used in the blocked QR that is
output. NB2 >= 1.

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry: an M-by-N matrix A.

On exit:
a) the elements on and above the diagonal
of the array contain the N-by-N upper-triangular
matrix R corresponding to the Householder QR;
b) the elements below the diagonal represent Q by
the columns of blocked V (compact WY-representation).

LDA

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

T is DOUBLE PRECISION array, dimension (LDT,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= NB2.

WORK

(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

The dimension of the array WORK.
LWORK >= MAX( LWT + LW1, MAX( LWT+N*N+LW2, LWT+N*N+N ) ),
where
NUM_ALL_ROW_BLOCKS = CEIL((M-N)/(MB1-N)),
NB1LOCAL = MIN(NB1,N).
LWT = NUM_ALL_ROW_BLOCKS * N * NB1LOCAL,
LW1 = NB1LOCAL * N,
LW2 = NB1LOCAL * MAX( NB1LOCAL, ( N - NB1LOCAL ) ),
If LWORK = -1, then a workspace query is assumed.
The routine only calculates the optimal size of the WORK
array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued
by XERBLA.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2020, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dgghd3 (character COMPQ, character COMPZ, integer N, integer ILO,integer IHI, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( ldb, * ) B, integer LDB, double precision, dimension(ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integerLDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGGHD3

Purpose:

DGGHD3 reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.

This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.

The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that

Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then DGGHD3 reduces the original
problem to generalized Hessenberg form.

This is a blocked variant of DGGHRD, using matrix-matrix
multiplications for parts of the computation to enhance performance.

Parameters

COMPQ

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

COMPZ

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

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

ILO

ILO is INTEGER

IHI

IHI is INTEGER

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

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

LDA

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

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

LDB

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

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

LDQ

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

Z is DOUBLE PRECISION array, dimension (LDZ, N)
On entry, if COMPZ = ’V’, the orthogonal matrix Z1.
On exit, if COMPZ=’I’, the orthogonal matrix Z, and if
COMPZ = ’V’, the product Z1*Z.
Not referenced if COMPZ=’N’.

LDZ

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

WORK

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

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= 1.
For optimum performance LWORK >= 6*N*NB, where NB is the
optimal blocksize.

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

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

subroutine dgghrd (character COMPQ, character COMPZ, integer N, integer ILO,integer IHI, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( ldb, * ) B, integer LDB, double precision, dimension(ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integerLDZ, integer INFO)

DGGHRD

Purpose:

DGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.

The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that

Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then DGGHRD reduces the original
problem to generalized Hessenberg form.

Parameters

COMPQ

COMPZ

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

ILO

ILO is INTEGER

IHI

IHI is INTEGER

LDA

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

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

LDB

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

LDQ

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

LDZ

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

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

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

DGGQRF

Purpose:

DGGQRF computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B:

A = Q*R, B = Q*T*Z,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:

if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
( 0 ) N-M N M-N
M

where R11 is upper triangular, and

if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
P-N N ( T21 ) P
P

where T12 or T21 is upper triangular.

In particular, if B is square and nonsingular, the GQR factorization
of A and B implicitly gives the QR factorization of inv(B)*A:

inv(B)*A = Z**T*(inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.

Parameters

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

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

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

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

LDA

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

TAUA

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

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

LDB

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

TAUB

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N,M,P).
For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
where NB1 is the optimal blocksize for the QR factorization
of an N-by-M matrix, NB2 is the optimal blocksize for the
RQ factorization of an N-by-P matrix, and NB3 is the optimal
blocksize for a call of DORMQR.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(n,m).

Each H(i) has the form

H(i) = I - taua * v * v**T

where taua is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine DORGQR.
To use Q to update another matrix, use LAPACK subroutine DORMQR.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(n,p).

Each H(i) has the form

H(i) = I - taub * v * v**T

where taub is a real scalar, and v is a real vector with
v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine DORGRQ.
To use Z to update another matrix, use LAPACK subroutine DORMRQ.

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

DGGRQF

Purpose:

DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:

A = R*Q, B = Z*T*Q,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:

if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):

A*inv(B) = (R*inv(T))*Z**T

where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.

Parameters

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

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

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

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

LDA

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

TAUA

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

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

LDB

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

TAUB

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N,M,P).
For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
where NB1 is the optimal blocksize for the RQ factorization
of an M-by-N matrix, NB2 is the optimal blocksize for the
QR factorization of a P-by-N matrix, and NB3 is the optimal
blocksize for a call of DORMRQ.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - taua * v * v**T

where taua is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine DORGRQ.
To use Q to update another matrix, use LAPACK subroutine DORMRQ.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(p,n).

Each H(i) has the form

H(i) = I - taub * v * v**T

where taub is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine DORGQR.
To use Z to update another matrix, use LAPACK subroutine DORMQR.

subroutine dggsvp3 (character JOBU, character JOBV, character JOBQ, integerM, integer P, integer N, double precision, dimension( lda, * ) A, integerLDA, double precision, dimension( ldb, * ) B, integer LDB, double precisionTOLA, double precision TOLB, integer K, integer L, double precision,dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * )V, integer LDV, double precision, dimension( ldq, * ) Q, integer LDQ,integer, dimension( * ) IWORK, double precision, dimension( * ) TAU, doubleprecision, dimension( * ) WORK, integer LWORK, integer INFO)

DGGSVP3

Purpose:

DGGSVP3 computes orthogonal matrices U, V and Q such that

N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )

N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )

N-K-L K L
V**T*B*Q = L ( 0 0 B13 )
P-L ( 0 0 0 )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.

This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
DGGSVD3.

Parameters

JOBU

JOBU is CHARACTER*1
= ’U’: Orthogonal matrix U is computed;
= ’N’: U is not computed.

JOBV

JOBV is CHARACTER*1
= ’V’: Orthogonal matrix V is computed;
= ’N’: V is not computed.

JOBQ

JOBQ is CHARACTER*1
= ’Q’: Orthogonal matrix Q is computed;
= ’N’: Q is not computed.

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

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

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

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

LDA

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

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

LDB

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

TOLA

TOLA is DOUBLE PRECISION

TOLB

TOLB is DOUBLE PRECISION

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

K is INTEGER

L is INTEGER

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

U is DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = ’U’, U contains the orthogonal matrix U.
If JOBU = ’N’, U is not referenced.

LDU

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

V is DOUBLE PRECISION array, dimension (LDV,P)
If JOBV = ’V’, V contains the orthogonal matrix V.
If JOBV = ’N’, V is not referenced.

LDV

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

Q is DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = ’Q’, Q contains the orthogonal matrix Q.
If JOBQ = ’N’, Q is not referenced.

LDQ

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

IWORK

IWORK is INTEGER array, dimension (N)

TAU

TAU is DOUBLE PRECISION array, dimension (N)

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The subroutine uses LAPACK subroutine DGEQP3 for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.

DGGSVP3 replaces the deprecated subroutine DGGSVP.

subroutine dgsvj0 (character1 JOBV, integer M, integer N, double precision,dimension( lda, ) A, integer LDA, double precision, dimension( n ) D,double precision, dimension( n ) SVA, integer MV, double precision,dimension( ldv, * ) V, integer LDV, double precision EPS, double precisionSFMIN, double precision TOL, integer NSWEEP, double precision, dimension(lwork ) WORK, integer LWORK, integer INFO)

DGSVJ0 pre-processor for the routine dgesvj.

Purpose:

DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
it does not check convergence (stopping criterion). Few tuning
parameters (marked by [TP]) are available for the implementer.

Parameters

JOBV

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

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

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

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

LDA

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

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

SVA

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

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

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

LDV

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

EPS

EPS is DOUBLE PRECISION
EPS = DLAMCH(’Epsilon’)

SFMIN

SFMIN is DOUBLE PRECISION
SFMIN = DLAMCH(’Safe Minimum’)

TOL

TOL is DOUBLE PRECISION
TOL is the threshold for Jacobi rotations. For a pair
A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.

NSWEEP

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

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
LWORK is the dimension of WORK. LWORK >= M.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

DGSVJ0 is used just to enable DGESVJ to call a simplified version of itself to work on a submatrix of the original matrix.

Contributors:

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

Bugs, Examples and Comments:

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

subroutine dgsvj1 (character1 JOBV, integer M, integer N, integer N1, doubleprecision, dimension( lda, ) A, integer LDA, double precision, dimension(n ) D, double precision, dimension( n ) SVA, integer MV, double precision,dimension( ldv, * ) V, integer LDV, double precision EPS, double precisionSFMIN, double precision TOL, integer NSWEEP, double precision, dimension(lwork ) WORK, integer LWORK, integer INFO)

DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots.

Purpose:

DGSVJ1 is called from DGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
it targets only particular pivots and it does not check convergence
(stopping criterion). Few tuning parameters (marked by [TP]) are
available for the implementer.

Further Details
˜˜˜˜˜˜˜˜˜˜˜˜˜˜˜
DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
off-diagonal block in the corresponding N-by-N Gram matrix AˆT * A. The
block-entries (tiles) of the (1,2) off-diagonal block are marked by the
[x]’s in the following scheme:

| * * * [x] [x] [x]|
| * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
| * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
|[x] [x] [x] * * * |
|[x] [x] [x] * * * |
|[x] [x] [x] * * * |

In terms of the columns of A, the first N1 columns are rotated ’against’
the remaining N-N1 columns, trying to increase the angle between the
corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
The number of sweeps is given in NSWEEP and the orthogonality threshold
is given in TOL.

Parameters

JOBV

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

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

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

LDA

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

SVA

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

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

LDV

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

EPS

EPS is DOUBLE PRECISION
EPS = DLAMCH(’Epsilon’)

SFMIN

SFMIN is DOUBLE PRECISION
SFMIN = DLAMCH(’Safe Minimum’)

TOL

TOL is DOUBLE PRECISION
TOL is the threshold for Jacobi rotations. For a pair
A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.

NSWEEP

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

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
LWORK is the dimension of WORK. LWORK >= M.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

subroutine dhsein (character SIDE, character EIGSRC, character INITV,logical, dimension( * ) SELECT, integer N, double precision, dimension(ldh, * ) H, integer LDH, double precision, dimension( * ) WR, doubleprecision, dimension( * ) WI, double precision, dimension( ldvl, * ) VL,integer LDVL, double precision, dimension( ldvr, * ) VR, integer LDVR,integer MM, integer M, double precision, dimension( * ) WORK, integer,dimension( * ) IFAILL, integer, dimension( * ) IFAILR, integer INFO)

DHSEIN

Purpose:

DHSEIN uses inverse iteration to find specified right and/or left
eigenvectors of a real upper Hessenberg matrix H.

The right eigenvector x and the left eigenvector y of the matrix H
corresponding to an eigenvalue w are defined by:

H * x = w * x, y**h * H = w * y**h

where y**h denotes the conjugate transpose of the vector y.

Parameters

SIDE

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

EIGSRC

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

INITV

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

SELECT

SELECT is LOGICAL array, dimension (N)
Specifies the eigenvectors to be computed. To select the
real eigenvector corresponding to a real eigenvalue WR(j),
SELECT(j) must be set to .TRUE.. To select the complex
eigenvector corresponding to a complex eigenvalue
(WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
.FALSE..

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

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

LDH

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

WR is DOUBLE PRECISION array, dimension (N)

WI is DOUBLE PRECISION array, dimension (N)

On entry, the real and imaginary parts of the eigenvalues of
H; a complex conjugate pair of eigenvalues must be stored in
consecutive elements of WR and WI.
On exit, WR may have been altered since close eigenvalues
are perturbed slightly in searching for independent
eigenvectors.

VL is DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if INITV = ’U’ and SIDE = ’L’ or ’B’, VL must
contain starting vectors for the inverse iteration for the
left eigenvectors; the starting vector for each eigenvector
must be in the same column(s) in which the eigenvector will
be stored.
On exit, if SIDE = ’L’ or ’B’, the left eigenvectors
specified by SELECT will be stored consecutively in the
columns of VL, in the same order as their eigenvalues. A
complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = ’R’, VL is not referenced.

LDVL

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

VR is DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if INITV = ’U’ and SIDE = ’R’ or ’B’, VR must
contain starting vectors for the inverse iteration for the
right eigenvectors; the starting vector for each eigenvector
must be in the same column(s) in which the eigenvector will
be stored.
On exit, if SIDE = ’R’ or ’B’, the right eigenvectors
specified by SELECT will be stored consecutively in the
columns of VR, in the same order as their eigenvalues. A
complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = ’L’, VR is not referenced.

LDVR

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

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

M is INTEGER
The number of columns in the arrays VL and/or VR required to
store the eigenvectors; each selected real eigenvector
occupies one column and each selected complex eigenvector
occupies two columns.

WORK

WORK is DOUBLE PRECISION array, dimension ((N+2)*N)

IFAILL

IFAILL is INTEGER array, dimension (MM)
If SIDE = ’L’ or ’B’, IFAILL(i) = j > 0 if the left
eigenvector in the i-th column of VL (corresponding to the
eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
eigenvector converged satisfactorily. If the i-th and (i+1)th
columns of VL hold a complex eigenvector, then IFAILL(i) and
IFAILL(i+1) are set to the same value.
If SIDE = ’R’, IFAILL is not referenced.

IFAILR

IFAILR is INTEGER array, dimension (MM)
If SIDE = ’R’ or ’B’, IFAILR(i) = j > 0 if the right
eigenvector in the i-th column of VR (corresponding to the
eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
eigenvector converged satisfactorily. If the i-th and (i+1)th
columns of VR hold a complex eigenvector, then IFAILR(i) and
IFAILR(i+1) are set to the same value.
If SIDE = ’L’, IFAILR is not referenced.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

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

subroutine dhseqr (character JOB, character COMPZ, integer N, integer ILO,integer IHI, double precision, dimension( ldh, * ) H, integer LDH, doubleprecision, dimension( * ) WR, double precision, dimension( * ) WI, doubleprecision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension(* ) WORK, integer LWORK, integer INFO)

DHSEQR

Purpose:

DHSEQR 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

JOB

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

COMPZ

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

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. ILO and IHI are normally
set by a previous call to DGEBAL, and then passed to ZGEHRD
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 is DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = ’S’, 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 JOB = ’E’, the
contents of H are unspecified on exit. (The output value of
H when INFO > 0 is given under the description of INFO
below.)

Unlike earlier versions of DHSEQR, this subroutine may
explicitly 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 is DOUBLE PRECISION array, dimension (N)

WI is DOUBLE PRECISION array, dimension (N)

The real and imaginary parts, respectively, of the computed
eigenvalues. 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 JOB = ’S’, 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).

Z is DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = ’N’, Z is not referenced.
If COMPZ = ’I’, on entry Z need not be set and on exit,
if INFO = 0, Z contains the orthogonal matrix Z of the Schur
vectors of H. If COMPZ = ’V’, on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the orthogonal matrix generated by DORGHR
after the call to DGEHRD which formed the Hessenberg matrix
H. (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 COMPZ = ’I’ or
COMPZ = ’V’, then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N)
is sufficient and delivers very good and sometimes
optimal performance. However, LWORK as large as 11*N
may be required for optimal performance. A workspace
query is recommended to determine the optimal workspace
size.

If LWORK = -1, then DHSEQR does a workspace query.
In this case, DHSEQR 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, the i-th argument had an illegal
value
> 0: if INFO = i, DHSEQR 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 JOB = ’E’, then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.

If INFO > 0 and JOB = ’S’, 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 COMPZ = ’V’, then on exit

(final value of Z) = (initial value of Z)*U

where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = ’I’, then on exit
(final value of Z) = U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)

If INFO > 0 and COMPZ = ’N’, then Z is not
accessed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

Default values supplied by
ILAENV(ISPEC,’DHSEQR’,JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.

ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
Default: 75. (Must be at least 11.)

ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS. NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15). (See ISPEC=15 below.)
The default for (IHI-ILO+1) <= 500 is NS.
The default for (IHI-ILO+1) > 500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for
details.) Default: 14% of deflation window
size.

ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.

If IHI-ILO+1 is ...

greater than ...but less ... the
or equal to ... than default is

1 30 NS = 2(+)
30 60 NS = 4(+)
60 150 NS = 10(+)
150 590 NS = **
590 3000 NS = 64
3000 6000 NS = 128
6000 infinity NS = 256

(+) By default some or all matrices of this order
are passed to the implicit double shift routine
DLAHQR and this parameter is ignored. See
ISPEC=12 above and comments in IPARMQ for
details.

(**) The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0. Otherwise the default for
ISPEC=16 is 2.

References:

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

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

subroutine dla_lin_berr (integer N, integer NZ, integer NRHS, doubleprecision, dimension( n, nrhs ) RES, double precision, dimension( n, nrhs )AYB, double precision, dimension( nrhs ) BERR)

DLA_LIN_BERR computes a component-wise relative backward error.

Purpose:

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

Parameters

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

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

NRHS

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

RES

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

AYB

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

BERR

BERR is DOUBLE PRECISION array, dimension (NRHS)
The component-wise relative backward error from the formula above.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dla_wwaddw (integer N, double precision, dimension( * ) X, doubleprecision, dimension( * ) Y, double precision, dimension( * ) W)

DLA_WWADDW adds a vector into a doubled-single vector.

Purpose:

DLA_WWADDW adds a vector W into a doubled-single vector (X, Y).

This works for all extant IBM’s hex and binary floating point
arithmetic, but not for decimal.

Parameters

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

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

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

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlals0 (integer ICOMPQ, integer NL, integer NR, integer SQRE,integer NRHS, double precision, dimension( ldb, * ) B, integer LDB, doubleprecision, dimension( ldbx, * ) BX, integer LDBX, integer, dimension( * )PERM, integer GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integerLDGCOL, double precision, dimension( ldgnum, * ) GIVNUM, integer LDGNUM,double precision, dimension( ldgnum, * ) POLES, double precision,dimension( * ) DIFL, double precision, dimension( ldgnum, * ) DIFR, doubleprecision, dimension( * ) Z, integer K, double precision C, doubleprecision S, double precision, dimension( * ) WORK, integer INFO)

DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

DLALS0 applies back the multiplying factors of either the left or the
right singular vector matrix of a diagonal matrix appended by a row
to the right hand side matrix B in solving the least squares problem
using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal
matrices are involved:

(1L) Givens rotations: the number of such rotations is GIVPTR; the
pairs of columns/rows they were applied to are stored in GIVCOL;
and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
row, and for J=2:N, PERM(J)-th row of B is to be moved to the
J-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal
matrices are involved:

(1R) The right singular vector matrix of the remaining matrix.

(2R) If SQRE = 1, one extra Givens rotation to generate the right
null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).

Parameters

ICOMPQ

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

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

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

SQRE

SQRE is INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular matrix.

The bidiagonal matrix has row dimension N = NL + NR + 1,
and column dimension M = N + SQRE.

NRHS

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

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

LDB

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

BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )

LDBX

LDBX is INTEGER
The leading dimension of BX.

PERM

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

GIVPTR

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

GIVCOL

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

LDGCOL

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

GIVNUM

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

LDGNUM

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

POLES

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

DIFL

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

DIFR

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

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

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

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

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

WORK

WORK is DOUBLE PRECISION array, dimension ( K )

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

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

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

Purpose:

DLALSA is an itermediate step in solving the least squares problem
by computing the SVD of the coefficient matrix in compact form (The
singular vectors are computed as products of simple orthorgonal
matrices.).

If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by DLALSA.

Parameters

ICOMPQ

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

SMLSIZ

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

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

NRHS

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

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

LDB

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

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

LDBX

LDBX is INTEGER
The leading dimension of BX.

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

LDU

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

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

K is INTEGER array, dimension ( N ).

DIFL

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

DIFR

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

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

POLES

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

GIVPTR

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

GIVCOL

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

LDGCOL

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

PERM

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

GIVNUM

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

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

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

WORK

WORK is DOUBLE PRECISION array, dimension (N)

IWORK

IWORK is INTEGER array, dimension (3*N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

subroutine dlalsd (character UPLO, integer SMLSIZ, integer N, integer NRHS,double precision, dimension( * ) D, double precision, dimension( * ) E,double precision, dimension( ldb, * ) B, integer LDB, double precisionRCOND, integer RANK, double precision, dimension( * ) WORK, integer,dimension( * ) IWORK, integer INFO)

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

Purpose:

DLALSD uses the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
are N-by-NRHS. The solution X overwrites B.

The singular values of A smaller than RCOND times the largest
singular value are treated as zero in solving the least squares
problem; in this case a minimum norm solution is returned.
The actual singular values are returned in D in ascending order.

This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.

Parameters

UPLO

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

SMLSIZ

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

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

NRHS

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

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

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

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

LDB

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

RCOND

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

RANK

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

WORK

WORK is DOUBLE PRECISION array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

IWORK

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

double precision function dlansf (character NORM, character TRANSR, characterUPLO, integer N, double precision, dimension( 0: * ) A, double precision,dimension( 0: * ) WORK)

DLANSF 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 in RFP format.

Purpose:

DLANSF 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 in RFP format.

Returns

DLANSF

DLANSF = ( max(abs(A(i,j))), NORM = ’M’ or ’m’
(
( norm1(A), NORM = ’1’, ’O’ or ’o’
(
( normI(A), NORM = ’I’ or ’i’
(
( normF(A), NORM = ’F’, ’f’, ’E’ or ’e’

where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a matrix norm.

Parameters

NORM

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

TRANSR

TRANSR is CHARACTER*1
Specifies whether the RFP format of A is normal or
transposed format.
= ’N’: RFP format is Normal;
= ’T’: RFP format is Transpose.

UPLO

UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
= ’U’: RFP A came from an upper triangular matrix;
= ’L’: RFP A came from a lower triangular matrix.

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

A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
On entry, the upper (if UPLO = ’U’) or lower (if UPLO = ’L’)
part of the symmetric matrix A stored in RFP format. See the
’Notes’ below for more details.
Unchanged on exit.

WORK

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = ’N’.

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

Let TRANSR = ’N’. RFP holds AP as follows:
For UPLO = ’U’ the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = ’L’ the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = ’N’.

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dlarscl2 (integer M, integer N, double precision, dimension( * )D, double precision, dimension( ldx, * ) X, integer LDX)

DLARSCL2 performs reciprocal diagonal scaling on a matrix.

Purpose:

DLARSCL2 performs a reciprocal diagonal scaling on a matrix:
x <-- inv(D) * x
where the diagonal matrix D is stored as a vector.

Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
standard.

Parameters

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

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

D is DOUBLE PRECISION array, dimension (M)
Diagonal matrix D, stored as a vector of length M.

X is DOUBLE PRECISION array, dimension (LDX,N)
On entry, the matrix X to be scaled by D.
On exit, the scaled matrix.

LDX

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

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

Purpose:

DLARZ 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.

H is a product of k elementary reflectors as returned by DTZRZF.

Parameters

SIDE

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

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

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

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

V is DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV))
The vector v in the representation of H as returned by
DTZRZF. 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 is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by the matrix H * C if SIDE = ’L’,
or C * H if SIDE = ’R’.

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’
or (M) if SIDE = ’R’

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

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

DLARZB applies a block reflector or its transpose to a general matrix.

Purpose:

DLARZB applies a real block reflector H or its transpose H**T to
a real distributed M-by-N C from the left or the right.

Currently, only STOREV = ’R’ and DIRECT = ’B’ are supported.

Parameters

SIDE

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

TRANS

TRANS is CHARACTER*1
= ’N’: apply H (No transpose)
= ’C’: 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, not supported yet)
= ’B’: H = H(k) . . . H(2) H(1) (Backward)

STOREV

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

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

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

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

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

V is DOUBLE PRECISION array, dimension (LDV,NV).
If STOREV = ’C’, NV = K; if STOREV = ’R’, NV = L.

LDV

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

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

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension (LDWORK,K)

LDWORK

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

subroutine dlarzt (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)

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

Purpose:

DLARZT 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

Currently, only STOREV = ’R’ and DIRECT = ’B’ are supported.

Parameters

DIRECT

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

STOREV

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

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

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

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

LDT

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored; the corresponding
array elements are modified but restored on exit. The rest of the
array is not used.

DIRECT = ’F’ and STOREV = ’C’: DIRECT = ’F’ and STOREV = ’R’:

______V_____
( v1 v2 v3 ) / \
( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
( v1 v2 v3 )
. . .
. . .
1 . .
1 .
1

DIRECT = ’B’ and STOREV = ’C’: DIRECT = ’B’ and STOREV = ’R’:

______V_____
1 / \
. 1 ( 1 . . . . v1 v1 v1 v1 v1 )
. . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
. . . ( . . 1 . . v3 v3 v3 v3 v3 )
. . .
( v1 v2 v3 )
( v1 v2 v3 )
V = ( v1 v2 v3 )
( v1 v2 v3 )
( v1 v2 v3 )

subroutine dlascl2 (integer M, integer N, double precision, dimension( * ) D,double precision, dimension( ldx, * ) X, integer LDX)

DLASCL2 performs diagonal scaling on a matrix.

Purpose:

DLASCL2 performs a diagonal scaling on a matrix:
x <-- D * x
where the diagonal matrix D is stored as a vector.

Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
standard.

Parameters

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

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

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

X is DOUBLE PRECISION array, dimension (LDX,N)
On entry, the matrix X to be scaled by D.
On exit, the scaled matrix.

LDX

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlatrz (integer M, integer N, integer L, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK)

DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Purpose:

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

Parameters

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

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

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

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

LDA

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

TAU

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

WORK

WORK is DOUBLE PRECISION array, dimension (M)

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

The factorization is obtained by Householder’s method. The kth
transformation matrix, Z( k ), which is used to introduce zeros into
the ( m - k + 1 )th row of A, is given in the form

Z( k ) = ( I 0 ),
( 0 T( k ) )

where

T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
( 0 )
( z( k ) )

tau is a scalar and z( k ) is an l element vector. tau and z( k )
are chosen to annihilate the elements of the kth row of A2.

The scalar tau is returned in the kth element of TAU and the vector
u( k ) in the kth row of A2, such that the elements of z( k ) are
in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
the upper triangular part of A1.

Z is given by

Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).

subroutine dopgtr (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( ) TAU, double precision, dimension(ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK, integerINFO)

DOPGTR

Purpose:

DOPGTR generates a real orthogonal matrix Q which is defined as the
product of n-1 elementary reflectors H(i) of order n, as returned by
DSPTRD using packed storage:

if UPLO = ’U’, Q = H(n-1) . . . H(2) H(1),

if UPLO = ’L’, Q = H(1) H(2) . . . H(n-1).

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangular packed storage used in previous
call to DSPTRD;
= ’L’: Lower triangular packed storage used in previous
call to DSPTRD.

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

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
The vectors which define the elementary reflectors, as
returned by DSPTRD.

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DSPTRD.

Q is DOUBLE PRECISION array, dimension (LDQ,N)
The N-by-N orthogonal matrix Q.

LDQ

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

WORK

WORK is DOUBLE PRECISION array, dimension (N-1)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dopmtr (character SIDE, character UPLO, character TRANS, integerM, integer N, double precision, dimension( * ) AP, double precision,dimension( * ) TAU, double precision, dimension( ldc, * ) C, integer LDC,double precision, dimension( * ) WORK, integer INFO)

DOPMTR

Purpose:

DOPMTR overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = ’L’ and nq = n if SIDE = ’R’. Q is defined as the product of
nq-1 elementary reflectors, as returned by DSPTRD using packed
storage:

if UPLO = ’U’, Q = H(nq-1) . . . H(2) H(1);

if UPLO = ’L’, Q = H(1) H(2) . . . H(nq-1).

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangular packed storage used in previous
call to DSPTRD;
= ’L’: Lower triangular packed storage used in previous
call to DSPTRD.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

AP is DOUBLE PRECISION array, dimension
(M*(M+1)/2) if SIDE = ’L’
(N*(N+1)/2) if SIDE = ’R’
The vectors which define the elementary reflectors, as
returned by DSPTRD. AP is modified by the routine but
restored on exit.

TAU

TAU is DOUBLE PRECISION array, dimension (M-1) if SIDE = ’L’
or (N-1) if SIDE = ’R’
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DSPTRD.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’
(M) if SIDE = ’R’

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorbdb (character TRANS, character SIGNS, integer M, integer P,integer Q, double precision, dimension( ldx11, * ) X11, integer LDX11,double precision, dimension( ldx12, * ) X12, integer LDX12, doubleprecision, dimension( ldx21, * ) X21, integer LDX21, double precision,dimension( ldx22, * ) X22, integer LDX22, double precision, dimension( * )THETA, double precision, dimension( * ) PHI, double precision, dimension( ) TAUP1, double precision, dimension( ) TAUP2, double precision,dimension( * ) TAUQ1, double precision, dimension( * ) TAUQ2, doubleprecision, dimension( * ) WORK, integer LWORK, integer INFO)

DORBDB

Purpose:

DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned orthogonal matrix X:

[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]

X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See DORCSD
for details.)

The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.

B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.

Parameters

TRANS

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

SIGNS

SIGNS is CHARACTER
= ’O’: The lower-left block is made nonpositive (the
’other’ convention);
otherwise: The upper-right block is made nonpositive (the
’default’ convention).

M is INTEGER
The number of rows and columns in X.

P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q).

X11

X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
On entry, the top-left block of the orthogonal matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = ’T’, and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.

LDX11

LDX11 is INTEGER
The leading dimension of X11. If TRANS = ’N’, then LDX11 >=
P; else LDX11 >= Q.

X12

X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
On entry, the top-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the rows of triu(X12) specify the first P reflectors for
Q2;
else TRANS = ’T’, and
the columns of tril(X12) specify the first P reflectors
for Q2.

LDX12

LDX12 is INTEGER
The leading dimension of X12. If TRANS = ’N’, then LDX12 >=
P; else LDX11 >= M-Q.

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom-left block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the columns of tril(X21) specify reflectors for P2;
else TRANS = ’T’, and
the rows of triu(X21) specify reflectors for P2.

LDX21

LDX21 is INTEGER
The leading dimension of X21. If TRANS = ’N’, then LDX21 >=
M-P; else LDX21 >= Q.

X22

X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = ’N’, then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = ’T’, and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.

LDX22

LDX22 is INTEGER
The leading dimension of X22. If TRANS = ’N’, then LDX22 >=
M-P; else LDX22 >= M-Q.

THETA

THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.

PHI

PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B12, B21, B22 can
be computed from the angles THETA and PHI. See Further
Details.

TAUP1

TAUP1 is DOUBLE PRECISION array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.

TAUP2

TAUP2 is DOUBLE PRECISION array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.

TAUQ1

TAUQ1 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.

TAUQ2

TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. Every entry in each bidiagonal band is a product
of a sine or cosine of a THETA with a sine or cosine of a PHI. See
[1] or DORCSD for details.

P1, P2, Q1, and Q2 are represented as products of elementary
reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
using DORGQR and DORGLQ.

References:

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

subroutine dorbdb1 (integer M, integer P, integer Q, double precision,dimension(ldx11,) X11, integer LDX11, double precision, dimension(ldx21,)X21, integer LDX21, double precision, dimension() THETA, double precision,dimension() PHI, double precision, dimension() TAUP1, double precision,dimension() TAUP2, double precision, dimension() TAUQ1, double precision,dimension() WORK, integer LWORK, integer INFO)

DORBDB1

Purpose:

DORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:

[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
M-P, or M-Q. Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in
which Q is not the minimum dimension.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.

B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
angles THETA, PHI.

Parameters

M is INTEGER
The number of rows X11 plus the number of rows in X21.

P is INTEGER
The number of rows in X11. 0 <= P <= M.

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q).

X11

X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
On entry, the top block of the matrix X to be reduced. On
exit, the columns of tril(X11) specify reflectors for P1 and
the rows of triu(X11,1) specify reflectors for Q1.

LDX11

LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.

LDX21

LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.

THETA

THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

PHI

PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

TAUP1

TAUP1 is DOUBLE PRECISION array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.

TAUP2

TAUP2 is DOUBLE PRECISION array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.

TAUQ1

TAUQ1 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or DORCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors.
See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
and DORGLQ.

References:

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

subroutine dorbdb2 (integer M, integer P, integer Q, double precision,dimension(ldx11,) X11, integer LDX11, double precision, dimension(ldx21,)X21, integer LDX21, double precision, dimension() THETA, double precision,dimension() PHI, double precision, dimension() TAUP1, double precision,dimension() TAUP2, double precision, dimension() TAUQ1, double precision,dimension() WORK, integer LWORK, integer INFO)

DORBDB2

Purpose:

DORBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:

[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
Q, or M-Q. Routines DORBDB1, DORBDB3, and DORBDB4 handle cases in
which P is not the minimum dimension.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.

B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
angles THETA, PHI.

Parameters

M is INTEGER
The number of rows X11 plus the number of rows in X21.

P is INTEGER
The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.

X11

LDX11

LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.

LDX21

LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.

THETA

THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

PHI

PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

TAUP1

TAUP1 is DOUBLE PRECISION array, dimension (P-1)
The scalar factors of the elementary reflectors that define
P1.

TAUP2

TAUP2 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
P2.

TAUQ1

TAUQ1 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

P1, P2, and Q1 are represented as products of elementary reflectors.
See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
and DORGLQ.

References:

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

subroutine dorbdb3 (integer M, integer P, integer Q, double precision,dimension(ldx11,) X11, integer LDX11, double precision, dimension(ldx21,)X21, integer LDX21, double precision, dimension() THETA, double precision,dimension() PHI, double precision, dimension() TAUP1, double precision,dimension() TAUP2, double precision, dimension() TAUQ1, double precision,dimension() WORK, integer LWORK, integer INFO)

DORBDB3

Purpose:

DORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:

[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in
which M-P is not the minimum dimension.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
implicitly by angles THETA, PHI.

Parameters

M is INTEGER
The number of rows X11 plus the number of rows in X21.

P is INTEGER
The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.

X11

LDX11

LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.

LDX21

LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.

THETA

THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

PHI

PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

TAUP1

TAUP1 is DOUBLE PRECISION array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.

TAUP2

TAUP2 is DOUBLE PRECISION array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.

TAUQ1

TAUQ1 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

P1, P2, and Q1 are represented as products of elementary reflectors.
See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
and DORGLQ.

References:

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

subroutine dorbdb4 (integer M, integer P, integer Q, double precision,dimension(ldx11,) X11, integer LDX11, double precision, dimension(ldx21,)X21, integer LDX21, double precision, dimension() THETA, double precision,dimension() PHI, double precision, dimension() TAUP1, double precision,dimension() TAUP2, double precision, dimension() TAUQ1, double precision,dimension() PHANTOM, double precision, dimension(*) WORK, integer LWORK,integer INFO)

DORBDB4

Purpose:

DORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:

[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
which M-Q is not the minimum dimension.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.

B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
implicitly by angles THETA, PHI.

Parameters

M is INTEGER
The number of rows X11 plus the number of rows in X21.

P is INTEGER
The number of rows in X11. 0 <= P <= M.

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M and
M-Q <= min(P,M-P,Q).

X11

LDX11

LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced. On
exit, the columns of tril(X21) specify reflectors for P2.

LDX21

LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.

THETA

THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

PHI

PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.

TAUP1

TAUP1 is DOUBLE PRECISION array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
P1.

TAUP2

TAUP2 is DOUBLE PRECISION array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
P2.

TAUQ1

TAUQ1 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.

PHANTOM

PHANTOM is DOUBLE PRECISION array, dimension (M)
The routine computes an M-by-1 column vector Y that is
orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
Y(P+1:M), respectively.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

P1, P2, and Q1 are represented as products of elementary reflectors.
See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
and DORGLQ.

References:

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

subroutine dorbdb5 (integer M1, integer M2, integer N, double precision,dimension() X1, integer INCX1, double precision, dimension() X2, integerINCX2, double precision, dimension(ldq1,) Q1, integer LDQ1, doubleprecision, dimension(ldq2,) Q2, integer LDQ2, double precision,dimension(*) WORK, integer LWORK, integer INFO)

DORBDB5

Purpose:

DORBDB5 orthogonalizes the column vector
X = [ X1 ]
[ X2 ]
with respect to the columns of
Q = [ Q1 ] .
[ Q2 ]
The columns of Q must be orthonormal.

If the projection is zero according to Kahan’s ’twice is enough’
criterion, then some other vector from the orthogonal complement
is returned. This vector is chosen in an arbitrary but deterministic
way.

Parameters

M1 is INTEGER
The dimension of X1 and the number of rows in Q1. 0 <= M1.

M2 is INTEGER
The dimension of X2 and the number of rows in Q2. 0 <= M2.

N is INTEGER
The number of columns in Q1 and Q2. 0 <= N.

X1 is DOUBLE PRECISION array, dimension (M1)
On entry, the top part of the vector to be orthogonalized.
On exit, the top part of the projected vector.

INCX1

INCX1 is INTEGER
Increment for entries of X1.

X2 is DOUBLE PRECISION array, dimension (M2)
On entry, the bottom part of the vector to be
orthogonalized. On exit, the bottom part of the projected
vector.

INCX2

INCX2 is INTEGER
Increment for entries of X2.

Q1 is DOUBLE PRECISION array, dimension (LDQ1, N)
The top part of the orthonormal basis matrix.

LDQ1

LDQ1 is INTEGER
The leading dimension of Q1. LDQ1 >= M1.

Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
The bottom part of the orthonormal basis matrix.

LDQ2

LDQ2 is INTEGER
The leading dimension of Q2. LDQ2 >= M2.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= N.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorbdb6 (integer M1, integer M2, integer N, double precision,dimension() X1, integer INCX1, double precision, dimension() X2, integerINCX2, double precision, dimension(ldq1,) Q1, integer LDQ1, doubleprecision, dimension(ldq2,) Q2, integer LDQ2, double precision,dimension(*) WORK, integer LWORK, integer INFO)

DORBDB6

Purpose:

DORBDB6 orthogonalizes the column vector
X = [ X1 ]
[ X2 ]
with respect to the columns of
Q = [ Q1 ] .
[ Q2 ]
The Euclidean norm of X must be one and the columns of Q must be
orthonormal. The orthogonalized vector will be zero if and only if it
lies entirely in the range of Q.

The projection is computed with at most two iterations of the
classical Gram-Schmidt algorithm, see
* L. Giraud, J. Langou, M. RozloÅ¾nÃk. ’On the round-off error
analysis of the Gram-Schmidt algorithm with reorthogonalization.’
2002. CERFACS Technical Report No. TR/PA/02/33. URL:
https://www.cerfacs.fr/algor/reports/2002/TR_PA_02_33.pdf

Parameters

M1 is INTEGER
The dimension of X1 and the number of rows in Q1. 0 <= M1.

M2 is INTEGER
The dimension of X2 and the number of rows in Q2. 0 <= M2.

N is INTEGER
The number of columns in Q1 and Q2. 0 <= N.

X1 is DOUBLE PRECISION array, dimension (M1)
On entry, the top part of the vector to be orthogonalized.
On exit, the top part of the projected vector.

INCX1

INCX1 is INTEGER
Increment for entries of X1.

X2 is DOUBLE PRECISION array, dimension (M2)
On entry, the bottom part of the vector to be
orthogonalized. On exit, the bottom part of the projected
vector.

INCX2

INCX2 is INTEGER
Increment for entries of X2.

Q1 is DOUBLE PRECISION array, dimension (LDQ1, N)
The top part of the orthonormal basis matrix.

LDQ1

LDQ1 is INTEGER
The leading dimension of Q1. LDQ1 >= M1.

Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
The bottom part of the orthonormal basis matrix.

LDQ2

LDQ2 is INTEGER
The leading dimension of Q2. LDQ2 >= M2.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= N.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

recursive subroutine dorcsd (character JOBU1, character JOBU2, characterJOBV1T, character JOBV2T, character TRANS, character SIGNS, integer M,integer P, integer Q, double precision, dimension( ldx11, * ) X11, integerLDX11, double precision, dimension( ldx12, * ) X12, integer LDX12, doubleprecision, dimension( ldx21, * ) X21, integer LDX21, double precision,dimension( ldx22, * ) X22, integer LDX22, doubleprecision, dimension( * ) THETA, double precision, dimension( ldu1, * ) U1,integer LDU1, double precision, dimension( ldu2, * ) U2, integer LDU2,double precision, dimension( ldv1t, * ) V1T, integer LDV1T, doubleprecision, dimension( ldv2t, * ) V2T, integer LDV2T, double precision,dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integerINFO)

DORCSD

Purpose:

DORCSD computes the CS decomposition of an M-by-M partitioned
orthogonal matrix X:

[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
X = [-----------] = [---------] [---------------------] [---------] .
[ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
[ 0 S 0 | 0 C 0 ]
[ 0 0 I | 0 0 0 ]

X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
(M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
R-by-R nonnegative diagonal matrices satisfying Cˆ2 + Sˆ2 = I, in
which R = MIN(P,M-P,Q,M-Q).

Parameters

JOBU1

JOBU1 is CHARACTER
= ’Y’: U1 is computed;
otherwise: U1 is not computed.

JOBU2

JOBU2 is CHARACTER
= ’Y’: U2 is computed;
otherwise: U2 is not computed.

JOBV1T

JOBV1T is CHARACTER
= ’Y’: V1T is computed;
otherwise: V1T is not computed.

JOBV2T

JOBV2T is CHARACTER
= ’Y’: V2T is computed;
otherwise: V2T is not computed.

TRANS

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

SIGNS

SIGNS is CHARACTER
= ’O’: The lower-left block is made nonpositive (the
’other’ convention);
otherwise: The upper-right block is made nonpositive (the
’default’ convention).

M is INTEGER
The number of rows and columns in X.

P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.

X11

X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).

X12

X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX12

LDX12 is INTEGER
The leading dimension of X12. LDX12 >= MAX(1,P).

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

LDX21 is INTEGER
The leading dimension of X11. LDX21 >= MAX(1,M-P).

X22

X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX22

LDX22 is INTEGER
The leading dimension of X11. LDX22 >= MAX(1,M-P).

THETA

THETA is DOUBLE PRECISION array, dimension (R), in which R =
MIN(P,M-P,Q,M-Q).
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1 is DOUBLE PRECISION array, dimension (LDU1,P)
If JOBU1 = ’Y’, U1 contains the P-by-P orthogonal matrix U1.

LDU1

LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = ’Y’, LDU1 >=
MAX(1,P).

U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
If JOBU2 = ’Y’, U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.

LDU2

LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = ’Y’, LDU2 >=
MAX(1,M-P).

V1T

V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
If JOBV1T = ’Y’, V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.

LDV1T

LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = ’Y’, LDV1T >=
MAX(1,Q).

V2T

V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
If JOBV2T = ’Y’, V2T contains the (M-Q)-by-(M-Q) orthogonal
matrix V2**T.

LDV2T

LDV2T is INTEGER
The leading dimension of V2T. If JOBV2T = ’Y’, LDV2T >=
MAX(1,M-Q).

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI’s.

LWORK

LWORK is INTEGER
The dimension of the array WORK.

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

IWORK

IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: DBBCSD did not converge. See the description of WORK
above for details.

References:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorcsd2by1 (character JOBU1, character JOBU2, character JOBV1T,integer M, integer P, integer Q, double precision, dimension(ldx11,) X11,integer LDX11, double precision, dimension(ldx21,) X21, integer LDX21,double precision, dimension() THETA, double precision, dimension(ldu1,)U1, integer LDU1, double precision, dimension(ldu2,) U2, integer LDU2,double precision, dimension(ldv1t,) V1T, integer LDV1T, double precision,dimension() WORK, integer LWORK, integer, dimension() IWORK, integerINFO)

DORCSD2BY1

Purpose:

DORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
orthonormal columns that has been partitioned into a 2-by-1 block
structure:

[ I1 0 0 ]
[ 0 C 0 ]
[ X11 ] [ U1 | ] [ 0 0 0 ]
X = [-----] = [---------] [----------] V1**T .
[ X21 ] [ | U2 ] [ 0 0 0 ]
[ 0 S 0 ]
[ 0 0 I2]

X11 is P-by-Q. The orthogonal matrices U1, U2, and V1 are P-by-P,
(M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R
nonnegative diagonal matrices satisfying Cˆ2 + Sˆ2 = I, in which
R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a
K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

Parameters

JOBU1

JOBU1 is CHARACTER
= ’Y’: U1 is computed;
otherwise: U1 is not computed.

JOBU2

JOBU2 is CHARACTER
= ’Y’: U2 is computed;
otherwise: U2 is not computed.

JOBV1T

JOBV1T is CHARACTER
= ’Y’: V1T is computed;
otherwise: V1T is not computed.

M is INTEGER
The number of rows in X.

P is INTEGER
The number of rows in X11. 0 <= P <= M.

Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.

X11

X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

LDX21 is INTEGER
The leading dimension of X21. LDX21 >= MAX(1,M-P).

THETA

THETA is DOUBLE PRECISION array, dimension (R), in which R =
MIN(P,M-P,Q,M-Q).
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1 is DOUBLE PRECISION array, dimension (P)
If JOBU1 = ’Y’, U1 contains the P-by-P orthogonal matrix U1.

LDU1

LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = ’Y’, LDU1 >=
MAX(1,P).

U2 is DOUBLE PRECISION array, dimension (M-P)
If JOBU2 = ’Y’, U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.

LDU2

LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = ’Y’, LDU2 >=
MAX(1,M-P).

V1T

V1T is DOUBLE PRECISION array, dimension (Q)
If JOBV1T = ’Y’, V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.

LDV1T

LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = ’Y’, LDV1T >=
MAX(1,Q).

WORK

LWORK

LWORK is INTEGER
The dimension of the array WORK.

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

IWORK

IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: DBBCSD did not converge. See the description of WORK
above for details.

References:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorg2l (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

DORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).

Purpose:

DORG2L generates an m by n real matrix Q with orthonormal columns,
which is defined as the last n columns of a product of k elementary
reflectors of order m

Q = H(k) . . . H(2) H(1)

as returned by DGEQLF.

Parameters

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

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

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the (n-k+i)-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by DGEQLF in the last k columns of its array
argument A.
On exit, the m by n matrix Q.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQLF.

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorg2r (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).

Purpose:

DORG2R generates an m by n real matrix Q with orthonormal columns,
which is defined as the first n columns of a product of k elementary
reflectors of order m

Q = H(1) H(2) . . . H(k)

as returned by DGEQRF.

Parameters

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

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

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the i-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by DGEQRF in the first k columns of its array
argument A.
On exit, the m-by-n matrix Q.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQRF.

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorghr (integer N, integer ILO, integer IHI, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DORGHR

Purpose:

DORGHR generates a real orthogonal matrix Q which is defined as the
product of IHI-ILO elementary reflectors of order N, as returned by
DGEHRD:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters

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

ILO

ILO is INTEGER

IHI

IHI is INTEGER

ILO and IHI must have the same values as in the previous call
of DGEHRD. Q is equal to the unit matrix except in the
submatrix Q(ilo+1:ihi,ilo+1:ihi).
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by DGEHRD.
On exit, the N-by-N orthogonal matrix Q.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEHRD.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= IHI-ILO.
For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
the optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgl2 (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

DORGL2

Purpose:

DORGL2 generates an m by n real matrix Q with orthonormal rows,
which is defined as the first m rows of a product of k elementary
reflectors of order n

Q = H(k) . . . H(2) H(1)

as returned by DGELQF.

Parameters

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

N is INTEGER
The number of columns of the matrix Q. N >= M.

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the i-th row must contain the vector which defines
the elementary reflector H(i), for i = 1,2,...,k, as returned
by DGELQF in the first k rows of its array argument A.
On exit, the m-by-n matrix Q.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGELQF.

WORK

WORK is DOUBLE PRECISION array, dimension (M)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorglq (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DORGLQ

Purpose:

DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the first M rows of a product of K elementary
reflectors of order N

Q = H(k) . . . H(2) H(1)

as returned by DGELQF.

Parameters

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

N is INTEGER
The number of columns of the matrix Q. N >= M.

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGELQF.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgql (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DORGQL

Purpose:

DORGQL generates an M-by-N real matrix Q with orthonormal columns,
which is defined as the last N columns of a product of K elementary
reflectors of order M

Q = H(k) . . . H(2) H(1)

as returned by DGEQLF.

Parameters

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

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

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQLF.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgqr (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DORGQR

Purpose:

DORGQR generates an M-by-N real matrix Q with orthonormal columns,
which is defined as the first N columns of a product of K elementary
reflectors of order M

Q = H(1) H(2) . . . H(k)

as returned by DGEQRF.

Parameters

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

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

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. N >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQRF.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgr2 (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).

Purpose:

DORGR2 generates an m by n real matrix Q with orthonormal rows,
which is defined as the last m rows of a product of k elementary
reflectors of order n

Q = H(1) H(2) . . . H(k)

as returned by DGERQF.

Parameters

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

N is INTEGER
The number of columns of the matrix Q. N >= M.

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the (m-k+i)-th row must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by DGERQF in the last k rows of its array argument
A.
On exit, the m by n matrix Q.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGERQF.

WORK

WORK is DOUBLE PRECISION array, dimension (M)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgrq (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DORGRQ

Purpose:

DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the last M rows of a product of K elementary
reflectors of order N

Q = H(1) H(2) . . . H(k)

as returned by DGERQF.

Parameters

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

N is INTEGER
The number of columns of the matrix Q. N >= M.

K is INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGERQF.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is the
optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgtr (character UPLO, integer N, double precision, dimension(lda, * ) A, integer LDA, double precision, dimension( * ) TAU, doubleprecision, dimension( * ) WORK, integer LWORK, integer INFO)

DORGTR

Purpose:

DORGTR generates a real orthogonal matrix Q which is defined as the
product of n-1 elementary reflectors of order N, as returned by
DSYTRD:

if UPLO = ’U’, Q = H(n-1) . . . H(2) H(1),

if UPLO = ’L’, Q = H(1) H(2) . . . H(n-1).

Parameters

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A contains elementary reflectors
from DSYTRD;
= ’L’: Lower triangle of A contains elementary reflectors
from DSYTRD.

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

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by DSYTRD.
On exit, the N-by-N orthogonal matrix Q.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DSYTRD.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N-1).
For optimum performance LWORK >= (N-1)*NB, where NB is
the optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorgtsqr (integer M, integer N, integer MB, integer NB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integerLWORK, integer INFO)

DORGTSQR

Purpose:

DORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns,
which are the first N columns of a product of real orthogonal
matrices of order M which are returned by DLATSQR

Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).

See the documentation for DLATSQR.

Parameters

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

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

MB is INTEGER
The row block size used by DLATSQR to return
arrays A and T. MB > N.
(Note that if MB > M, then M is used instead of MB
as the row block size).

NB is INTEGER
The column block size used by DLATSQR to return
arrays A and T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size).

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry:

The elements on and above the diagonal are not accessed.
The elements below the diagonal represent the unit
lower-trapezoidal blocked matrix V computed by DLATSQR
that defines the input matrices Q_in(k) (ones on the
diagonal are not stored) (same format as the output A
below the diagonal in DLATSQR).

On exit:

The array A contains an M-by-N orthonormal matrix Q_out,
i.e the columns of A are orthogonal unit vectors.

LDA

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

T is DOUBLE PRECISION array,
dimension (LDT, N * NIRB)
where NIRB = Number_of_input_row_blocks
= MAX( 1, CEIL((M-N)/(MB-N)) )
Let NICB = Number_of_input_col_blocks
= CEIL(N/NB)

The upper-triangular block reflectors used to define the
input matrices Q_in(k), k=(1:NIRB*NICB). The block
reflectors are stored in compact form in NIRB block
reflector sequences. Each of NIRB block reflector sequences
is stored in a larger NB-by-N column block of T and consists
of NICB smaller NB-by-NB upper-triangular column blocks.
(same format as the output T in DLATSQR).

LDT

LDT is INTEGER
The leading dimension of the array T.
LDT >= max(1,min(NB1,N)).

WORK

(workspace) DOUBLE PRECISION array, dimension (MAX(2,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

The dimension of the array WORK. LWORK >= (M+NB)*N.
If LWORK = -1, then a workspace query is assumed.
The routine only calculates the optimal size of the WORK
array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued
by XERBLA.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dorgtsqr_row (integer M, integer N, integer MB, integer NB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integerLWORK, integer INFO)

DORGTSQR_ROW

Purpose:

DORGTSQR_ROW generates an M-by-N real matrix Q_out with
orthonormal columns from the output of DLATSQR. These N orthonormal
columns are the first N columns of a product of complex unitary
matrices Q(k)_in of order M, which are returned by DLATSQR in
a special format.

Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).

The input matrices Q(k)_in are stored in row and column blocks in A.
See the documentation of DLATSQR for more details on the format of
Q(k)_in, where each Q(k)_in is represented by block Householder
transformations. This routine calls an auxiliary routine DLARFB_GETT,
where the computation is performed on each individual block. The
algorithm first sweeps NB-sized column blocks from the right to left
starting in the bottom row block and continues to the top row block
(hence _ROW in the routine name). This sweep is in reverse order of
the order in which DLATSQR generates the output blocks.

Parameters

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

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

MB is INTEGER
The row block size used by DLATSQR to return
arrays A and T. MB > N.
(Note that if MB > M, then M is used instead of MB
as the row block size).

NB is INTEGER
The column block size used by DLATSQR to return
arrays A and T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size).

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry:

The elements on and above the diagonal are not used as
input. The elements below the diagonal represent the unit
lower-trapezoidal blocked matrix V computed by DLATSQR
that defines the input matrices Q_in(k) (ones on the
diagonal are not stored). See DLATSQR for more details.

On exit:

The array A contains an M-by-N orthonormal matrix Q_out,
i.e the columns of A are orthogonal unit vectors.

LDA

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

T is DOUBLE PRECISION array,
dimension (LDT, N * NIRB)
where NIRB = Number_of_input_row_blocks
= MAX( 1, CEIL((M-N)/(MB-N)) )
Let NICB = Number_of_input_col_blocks
= CEIL(N/NB)

The upper-triangular block reflectors used to define the
input matrices Q_in(k), k=(1:NIRB*NICB). The block
reflectors are stored in compact form in NIRB block
reflector sequences. Each of the NIRB block reflector
sequences is stored in a larger NB-by-N column block of T
and consists of NICB smaller NB-by-NB upper-triangular
column blocks. See DLATSQR for more details on the format
of T.

LDT

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

WORK

(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

The dimension of the array WORK.
LWORK >= NBLOCAL * MAX(NBLOCAL,(N-NBLOCAL)),
where NBLOCAL=MIN(NB,N).
If LWORK = -1, then a workspace query is assumed.
The routine only calculates the optimal size of the WORK
array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued
by XERBLA.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2020, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dorhr_col (integer M, integer N, integer NB, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, double precision, dimension( * ) D, integer INFO)

DORHR_COL

Purpose:

DORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
as input, stored in A, and performs Householder Reconstruction (HR),
i.e. reconstructs Householder vectors V(i) implicitly representing
another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
where S is an N-by-N diagonal matrix with diagonal entries
equal to +1 or -1. The Householder vectors (columns V(i) of V) are
stored in A on output, and the diagonal entries of S are stored in D.
Block reflectors are also returned in T
(same output format as DGEQRT).

Parameters

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

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

NB is INTEGER
The column block size to be used in the reconstruction
of Householder column vector blocks in the array A and
corresponding block reflectors in the array T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size.)

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry:

The array A contains an M-by-N orthonormal matrix Q_in,
i.e the columns of A are orthogonal unit vectors.

On exit:

The elements below the diagonal of A represent the unit
lower-trapezoidal matrix V of Householder column vectors
V(i). The unit diagonal entries of V are not stored
(same format as the output below the diagonal in A from
DGEQRT). The matrix T and the matrix V stored on output
in A implicitly define Q_out.

The elements above the diagonal contain the factor U
of the ’modified’ LU-decomposition:
Q_in - ( S ) = V * U
( 0 )
where 0 is a (M-N)-by-(M-N) zero matrix.

LDA

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

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

Let NOCB = Number_of_output_col_blocks
= CEIL(N/NB)

On exit, T(1:NB, 1:N) contains NOCB upper-triangular
block reflectors used to define Q_out stored in compact
form as a sequence of upper-triangular NB-by-NB column
blocks (same format as the output T in DGEQRT).
The matrix T and the matrix V stored on output in A
implicitly define Q_out. NOTE: The lower triangles
below the upper-triangular blocks will be filled with
zeros. See Further Details.

LDT

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

D is DOUBLE PRECISION array, dimension min(M,N).
The elements can be only plus or minus one.

D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
1 <= i <= min(M,N), and Q_in_i is Q_in after performing
i-1 steps of âmodifiedâ Gaussian elimination.
See Further Details.

INFO

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

Further Details:

The computed M-by-M orthogonal factor Q_out is defined implicitly as
a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
the compact WY-representation format in the corresponding blocks of
matrices V (stored in A) and T.

The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
matrix A contains the column vectors V(i) in NB-size column
blocks VB(j). For example, VB(1) contains the columns
V(1), V(2), ... V(NB). NOTE: The unit entries on
the diagonal of Y are not stored in A.

The number of column blocks is

NOCB = Number_of_output_col_blocks = CEIL(N/NB)

where each block is of order NB except for the last block, which
is of order LAST_NB = N - (NOCB-1)*NB.

For example, if M=6, N=5 and NB=2, the matrix V is

V = ( VB(1), VB(2), VB(3) ) =

= ( 1 )
( v21 1 )
( v31 v32 1 )
( v41 v42 v43 1 )
( v51 v52 v53 v54 1 )
( v61 v62 v63 v54 v65 )

For each of the column blocks VB(i), an upper-triangular block
reflector TB(i) is computed. These blocks are stored as
a sequence of upper-triangular column blocks in the NB-by-N
matrix T. The size of each TB(i) block is NB-by-NB, except
for the last block, whose size is LAST_NB-by-LAST_NB.

For example, if M=6, N=5 and NB=2, the matrix T is

T = ( TB(1), TB(2), TB(3) ) =

= ( t11 t12 t13 t14 t15 )
( t22 t24 )

The M-by-M factor Q_out is given as a product of NOCB
orthogonal M-by-M matrices Q_out(i).

Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),

where each matrix Q_out(i) is given by the WY-representation
using corresponding blocks from the matrices V and T:

Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,

where I is the identity matrix. Here is the formula with matrix
dimensions:

Q(i){M-by-M} = I{M-by-M} -
VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},

where INB = NB, except for the last block NOCB
for which INB=LAST_NB.

=====
NOTE:
=====

If Q_in is the result of doing a QR factorization
B = Q_in * R_in, then:

B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.

So if one wants to interpret Q_out as the result
of the QR factorization of B, then the corresponding R_out
should be equal to R_out = S * R_in, i.e. some rows of R_in
should be multiplied by -1.

For the details of the algorithm, see [1].

[1] ’Reconstructing Householder vectors from tall-skinny QR’,
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dorm2l (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).

Purpose:

DORM2L overwrites the general real m by n matrix C with

Q * C if SIDE = ’L’ and TRANS = ’N’, or

Q**T * C if SIDE = ’L’ and TRANS = ’T’, or

C * Q if SIDE = ’R’ and TRANS = ’N’, or

C * Q**T if SIDE = ’R’ and TRANS = ’T’,

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(k) . . . H(2) H(1)

as returned by DGEQLF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.

Parameters

SIDE

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

TRANS

TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

A is DOUBLE PRECISION array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DGEQLF in the last k columns of its array argument A.
A is modified by the routine but restored on exit.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQLF.

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

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorm2r (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).

Purpose:

DORM2R overwrites the general real m by n matrix C with

Q * C if SIDE = ’L’ and TRANS = ’N’, or

Q**T* C if SIDE = ’L’ and TRANS = ’T’, or

C * Q if SIDE = ’R’ and TRANS = ’N’, or

C * Q**T if SIDE = ’R’ and TRANS = ’T’,

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(1) H(2) . . . H(k)

as returned by DGEQRF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.

Parameters

SIDE

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

TRANS

TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

A is DOUBLE PRECISION array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DGEQRF in the first k columns of its array argument A.
A is modified by the routine but restored on exit.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQRF.

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

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormbr (character VECT, character SIDE, character TRANS, integerM, integer N, integer K, double precision, dimension( lda, * ) A, integerLDA, double precision, dimension( * ) TAU, double precision, dimension(ldc, * ) C, integer LDC, double precision, dimension( * ) WORK, integerLWORK, integer INFO)

DORMBR

Purpose:

If VECT = ’Q’, DORMBR overwrites the general real M-by-N matrix C
with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

If VECT = ’P’, DORMBR overwrites the general real M-by-N matrix C
with
SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: P * C C * P
TRANS = ’T’: P**T * C C * P**T

Here Q and P**T are the orthogonal matrices determined by DGEBRD when
reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
P**T are defined as products of elementary reflectors H(i) and G(i)
respectively.

Let nq = m if SIDE = ’L’ and nq = n if SIDE = ’R’. Thus nq is the
order of the orthogonal matrix Q or P**T that is applied.

If VECT = ’Q’, A is assumed to have been an NQ-by-K matrix:
if nq >= k, Q = H(1) H(2) . . . H(k);
if nq < k, Q = H(1) H(2) . . . H(nq-1).

If VECT = ’P’, A is assumed to have been a K-by-NQ matrix:
if k < nq, P = G(1) G(2) . . . G(k);
if k >= nq, P = G(1) G(2) . . . G(nq-1).

Parameters

VECT

VECT is CHARACTER*1
= ’Q’: apply Q or Q**T;
= ’P’: apply P or P**T.

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q, Q**T, P or P**T from the Left;
= ’R’: apply Q, Q**T, P or P**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q or P;
= ’T’: Transpose, apply Q**T or P**T.

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

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

K is INTEGER
If VECT = ’Q’, the number of columns in the original
matrix reduced by DGEBRD.
If VECT = ’P’, the number of rows in the original
matrix reduced by DGEBRD.
K >= 0.

A is DOUBLE PRECISION array, dimension
(LDA,min(nq,K)) if VECT = ’Q’
(LDA,nq) if VECT = ’P’
The vectors which define the elementary reflectors H(i) and
G(i), whose products determine the matrices Q and P, as
returned by DGEBRD.

LDA

LDA is INTEGER
The leading dimension of the array A.
If VECT = ’Q’, LDA >= max(1,nq);
if VECT = ’P’, LDA >= max(1,min(nq,K)).

TAU

TAU is DOUBLE PRECISION array, dimension (min(nq,K))
TAU(i) must contain the scalar factor of the elementary
reflector H(i) or G(i) which determines Q or P, as returned
by DGEBRD in the array argument TAUQ or TAUP.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
or P*C or P**T*C or C*P or C*P**T.

LDC

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For optimum performance LWORK >= N*NB if SIDE = ’L’, and
LWORK >= M*NB if SIDE = ’R’, where NB is the optimal
blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormhr (character SIDE, character TRANS, integer M, integer N,integer ILO, integer IHI, double precision, dimension( lda, * ) A, integerLDA, double precision, dimension( * ) TAU, double precision, dimension(ldc, * ) C, integer LDC, double precision, dimension( * ) WORK, integerLWORK, integer INFO)

DORMHR

Purpose:

DORMHR overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = ’L’ and nq = n if SIDE = ’R’. Q is defined as the product of
IHI-ILO elementary reflectors, as returned by DGEHRD:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

ILO

ILO is INTEGER

IHI

IHI is INTEGER

ILO and IHI must have the same values as in the previous call
of DGEHRD. Q is equal to the unit matrix except in the
submatrix Q(ilo+1:ihi,ilo+1:ihi).
If SIDE = ’L’, then 1 <= ILO <= IHI <= M, if M > 0, and
ILO = 1 and IHI = 0, if M = 0;
if SIDE = ’R’, then 1 <= ILO <= IHI <= N, if N > 0, and
ILO = 1 and IHI = 0, if N = 0.

A is DOUBLE PRECISION array, dimension
(LDA,M) if SIDE = ’L’
(LDA,N) if SIDE = ’R’
The vectors which define the elementary reflectors, as
returned by DGEHRD.

LDA

LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,M) if SIDE = ’L’; LDA >= max(1,N) if SIDE = ’R’.

TAU

TAU is DOUBLE PRECISION array, dimension
(M-1) if SIDE = ’L’
(N-1) if SIDE = ’R’
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEHRD.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

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

LWORK

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dorml2 (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).

Purpose:

DORML2 overwrites the general real m by n matrix C with

Q * C if SIDE = ’L’ and TRANS = ’N’, or

Q**T* C if SIDE = ’L’ and TRANS = ’T’, or

C * Q if SIDE = ’R’ and TRANS = ’N’, or

C * Q**T if SIDE = ’R’ and TRANS = ’T’,

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(k) . . . H(2) H(1)

as returned by DGELQF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.

Parameters

SIDE

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

TRANS

TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

A is DOUBLE PRECISION array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DGELQF in the first k rows of its array argument A.
A is modified by the routine but restored on exit.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGELQF.

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

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormlq (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

DORMLQ

Purpose:

DORMLQ overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(k) . . . H(2) H(1)

as returned by DGELQF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGELQF.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormql (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

DORMQL

Purpose:

DORMQL overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(k) . . . H(2) H(1)

as returned by DGEQLF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQLF.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormqr (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

DORMQR

Purpose:

DORMQR overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(1) H(2) . . . H(k)

as returned by DGEQRF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGEQRF.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormr2 (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).

Purpose:

DORMR2 overwrites the general real m by n matrix C with

Q * C if SIDE = ’L’ and TRANS = ’N’, or

Q**T* C if SIDE = ’L’ and TRANS = ’T’, or

C * Q if SIDE = ’R’ and TRANS = ’N’, or

C * Q**T if SIDE = ’R’ and TRANS = ’T’,

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(1) H(2) . . . H(k)

as returned by DGERQF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.

Parameters

SIDE

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

TRANS

TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q’ (Transpose)

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

A is DOUBLE PRECISION array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DGERQF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGERQF.

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

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormr3 (character SIDE, character TRANS, integer M, integer N,integer K, integer L, double precision, dimension( lda, * ) A, integer LDA,double precision, dimension( * ) TAU, double precision, dimension( ldc, * )C, integer LDC, double precision, dimension( * ) WORK, integer INFO)

DORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).

Purpose:

DORMR3 overwrites the general real m by n matrix C with

Q * C if SIDE = ’L’ and TRANS = ’N’, or

Q**T* C if SIDE = ’L’ and TRANS = ’C’, or

C * Q if SIDE = ’R’ and TRANS = ’N’, or

C * Q**T if SIDE = ’R’ and TRANS = ’C’,

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(1) H(2) . . . H(k)

as returned by DTZRZF. Q is of order m if SIDE = ’L’ and of order n
if SIDE = ’R’.

Parameters

SIDE

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

TRANS

TRANS is CHARACTER*1
= ’N’: apply Q (No transpose)
= ’T’: apply Q**T (Transpose)

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

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

A is DOUBLE PRECISION array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DTZRZF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DTZRZF.

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

LDC

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

WORK

WORK is DOUBLE PRECISION array, dimension
(N) if SIDE = ’L’,
(M) if SIDE = ’R’

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

subroutine dormrq (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

DORMRQ

Purpose:

DORMRQ overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(1) H(2) . . . H(k)

as returned by DGERQF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DGERQF.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dormrz (character SIDE, character TRANS, integer M, integer N,integer K, integer L, double precision, dimension( lda, * ) A, integer LDA,double precision, dimension( * ) TAU, double precision, dimension( ldc, * )C, integer LDC, double precision, dimension( * ) WORK, integer LWORK,integer INFO)

DORMRZ

Purpose:

DORMRZ overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix defined as the product of k
elementary reflectors

Q = H(1) H(2) . . . H(k)

as returned by DTZRZF. Q is of order M if SIDE = ’L’ and of order N
if SIDE = ’R’.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = ’L’, M >= K >= 0;
if SIDE = ’R’, N >= K >= 0.

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

A is DOUBLE PRECISION array, dimension
(LDA,M) if SIDE = ’L’,
(LDA,N) if SIDE = ’R’
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DTZRZF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.

LDA

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

TAU

TAU is DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DTZRZF.

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

LDC

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If SIDE = ’L’, LWORK >= max(1,N);
if SIDE = ’R’, LWORK >= max(1,M).
For good performance, LWORK should generally be larger.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

subroutine dormtr (character SIDE, character UPLO, character TRANS, integerM, integer N, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

DORMTR

Purpose:

DORMTR overwrites the general real M-by-N matrix C with

SIDE = ’L’ SIDE = ’R’
TRANS = ’N’: Q * C C * Q
TRANS = ’T’: Q**T * C C * Q**T

where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = ’L’ and nq = n if SIDE = ’R’. Q is defined as the product of
nq-1 elementary reflectors, as returned by DSYTRD:

if UPLO = ’U’, Q = H(nq-1) . . . H(2) H(1);

if UPLO = ’L’, Q = H(1) H(2) . . . H(nq-1).

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A contains elementary reflectors
from DSYTRD;
= ’L’: Lower triangle of A contains elementary reflectors
from DSYTRD.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

A is DOUBLE PRECISION array, dimension
(LDA,M) if SIDE = ’L’
(LDA,N) if SIDE = ’R’
The vectors which define the elementary reflectors, as
returned by DSYTRD.

LDA

LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,M) if SIDE = ’L’; LDA >= max(1,N) if SIDE = ’R’.

TAU

TAU is DOUBLE PRECISION array, dimension
(M-1) if SIDE = ’L’
(N-1) if SIDE = ’R’
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by DSYTRD.

C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

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

WORK

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

LWORK

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpbcon (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, double precision ANORM, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

DPBCON

Purpose:

DPBCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite band matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters

UPLO

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

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

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

AB is DOUBLE PRECISION array, dimension (LDAB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A, stored in the
first KD+1 rows of the array. The j-th column of U or L is
stored in the j-th column of the array AB as follows:
if UPLO =’U’, AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
if UPLO =’L’, AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).

LDAB

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

ANORM

ANORM is DOUBLE PRECISION
The 1-norm (or infinity-norm) of the symmetric band matrix A.

RCOND

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

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

DPBEQU

Purpose:

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

Parameters

UPLO

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

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

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

AB is DOUBLE PRECISION array, dimension (LDAB,N)
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).

LDAB

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

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

SCOND

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

AMAX

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpbrfs (character UPLO, integer N, integer KD, integer NRHS,double precision, dimension( ldab, * ) AB, integer LDAB, double precision,dimension( ldafb, * ) AFB, integer LDAFB, double precision, dimension( ldb,* ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX,double precision, dimension( * ) FERR, double precision, dimension( * )BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,integer INFO)

DPBRFS

Purpose:

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

Parameters

UPLO

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

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

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

NRHS

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

LDAB

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

AFB

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

LDAFB

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

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

LDB

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

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

LDX

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

FERR

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

BERR

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

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpbstf (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, integer INFO)

DPBSTF

Purpose:

DPBSTF computes a split Cholesky factorization of a real
symmetric positive definite band matrix A.

This routine is designed to be used in conjunction with DSBGST.

The factorization has the form A = S**T*S where S is a band matrix
of the same bandwidth as A and the following structure:

S = ( U )
( M L )

where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.

Parameters

UPLO

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

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

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

AB 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 factor S from the split Cholesky
factorization A = S**T*S. See Further Details.

LDAB

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when
N = 7, KD = 2:

S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )

If UPLO = ’U’, the array AB holds:

on entry: on exit:

* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77

If UPLO = ’L’, the array AB holds:

on entry: on exit:

a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *

Array elements marked * are not used by the routine.

subroutine dpbtf2 (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, integer INFO)

DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

Purpose:

DPBTF2 computes the Cholesky factorization of a real symmetric
positive definite band matrix A.

The factorization has the form
A = U**T * U , if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix, U**T is the transpose of U, and
L is lower triangular.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

UPLO

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

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

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.

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.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = ’U’:

On entry: On exit:

* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66

Similarly, if UPLO = ’L’ the format of A is as follows:

On entry: On exit:

a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *

Array elements marked * are not used by the routine.

subroutine dpbtrf (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, integer INFO)

DPBTRF

Purpose:

DPBTRF computes the Cholesky factorization of a real symmetric
positive definite band matrix A.

The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

Parameters

UPLO

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

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

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

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.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = ’U’:

On entry: On exit:

* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66

Similarly, if UPLO = ’L’ the format of A is as follows:

On entry: On exit:

a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *

Array elements marked * are not used by the routine.

Contributors:

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

subroutine dpbtrs (character UPLO, integer N, integer KD, integer NRHS,double precision, dimension( ldab, * ) AB, integer LDAB, double precision,dimension( ldb, * ) B, integer LDB, integer INFO)

DPBTRS

Purpose:

DPBTRS solves a system of linear equations A*X = B with a symmetric
positive definite band matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by DPBTRF.

Parameters

UPLO

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

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

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

NRHS

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

LDAB

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

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

LDB

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpftrf (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) A, integer INFO)

DPFTRF

Purpose:

DPFTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A.

The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.

UPLO

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

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

A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
On entry, the symmetric matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = ’N’
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = ’T’ then RFP is
the transpose of RFP A as defined when
TRANSR = ’N’. The contents of RFP A are defined by UPLO as
follows: If UPLO = ’U’ the RFP A contains the NT elements of
upper packed A. If UPLO = ’L’ the RFP A contains the elements
of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
’T’. When TRANSR is ’N’ the LDA is N+1 when N is even and N
is odd. See the Note below for more details.

On exit, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**T*U or RFP A = L*L**T.

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dpftri (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) A, integer INFO)

DPFTRI

Purpose:

DPFTRI computes the inverse of a (real) symmetric positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by DPFTRF.

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.

UPLO

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

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

A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
On entry, the symmetric matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = ’N’
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = ’T’ then RFP is
the transpose of RFP A as defined when
TRANSR = ’N’. The contents of RFP A are defined by UPLO as
follows: If UPLO = ’U’ the RFP A contains the nt elements of
upper packed A. If UPLO = ’L’ the RFP A contains the elements
of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
’T’. When TRANSR is ’N’ the LDA is N+1 when N is even and N
is odd. See the Note below for more details.

On exit, the symmetric inverse of the original matrix, in the
same storage format.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dpftrs (character TRANSR, character UPLO, integer N, integer NRHS,double precision, dimension( 0: * ) A, double precision, dimension( ldb, *) B, integer LDB, integer INFO)

DPFTRS

Purpose:

DPFTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by DPFTRF.

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.

UPLO

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

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

NRHS

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

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

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

LDB

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dppcon (character UPLO, integer N, double precision, dimension( ) AP, double precision ANORM, double precision RCOND, double precision,dimension( ) WORK, integer, dimension( * ) IWORK, integer INFO)

DPPCON

Purpose:

DPPCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite packed matrix using
the Cholesky factorization A = U**T*U or A = L*L**T computed by
DPPTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters

UPLO

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

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

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, packed columnwise in a linear
array. The j-th column of U or L is stored in the array AP
as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

ANORM

ANORM is DOUBLE PRECISION
The 1-norm (or infinity-norm) of the symmetric matrix A.

RCOND

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

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dppequ (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( ) S, double precision SCOND, doubleprecision AMAX, integer INFO)

DPPEQU

Purpose:

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

Parameters

UPLO

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

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

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.

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

SCOND

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

AMAX

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpprfs (character UPLO, integer N, integer NRHS, double precision,dimension( * ) AP, double precision, dimension( * ) AFP, double precision,dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * )X, integer LDX, double precision, dimension( * ) FERR, double precision,dimension( * ) BERR, double precision, dimension( * ) WORK, integer,dimension( * ) IWORK, integer INFO)

DPPRFS

Purpose:

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

Parameters

UPLO

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

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

NRHS

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

AFP

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

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

LDB

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

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

LDX

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

FERR

BERR

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpptrf (character UPLO, integer N, double precision, dimension( *) AP, integer INFO)

DPPTRF

Purpose:

DPPTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A stored in packed format.

The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

Parameters

UPLO

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

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

On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T, in the same
storage format as A.

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The packed storage scheme is illustrated by the following example
when N = 4, UPLO = ’U’:

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

subroutine dpptri (character UPLO, integer N, double precision, dimension( *) AP, integer INFO)

DPPTRI

Purpose:

DPPTRI computes the inverse of a real symmetric positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by DPPTRF.

Parameters

UPLO

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

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

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T, packed columnwise as
a linear array. The j-th column of U or L is stored in the
array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

On exit, the upper or lower triangle of the (symmetric)
inverse of A, overwriting the input factor U or L.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpptrs (character UPLO, integer N, integer NRHS, double precision,dimension( * ) AP, double precision, dimension( ldb, * ) B, integer LDB,integer INFO)

DPPTRS

Purpose:

DPPTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A in packed storage using the Cholesky
factorization A = U**T*U or A = L*L**T computed by DPPTRF.

Parameters

UPLO

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

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

NRHS

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

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

LDB

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpstf2 (character UPLO, integer N, double precision, dimension(lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, doubleprecision TOL, double precision, dimension( 2*n ) WORK, integer INFO)

DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

Purpose:

DPSTF2 computes the Cholesky factorization with complete
pivoting of a real symmetric positive semidefinite matrix A.

The factorization has the form
P**T * A * P = U**T * U , if UPLO = ’U’,
P**T * A * P = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular, and
P is stored as vector PIV.

This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 2 BLAS.

Parameters

UPLO

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

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

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 INFO = 0, the factor U or L from the Cholesky
factorization as above.

PIV

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

RANK

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

TOL

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

LDA

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

WORK

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpstrf (character UPLO, integer N, double precision, dimension(lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, doubleprecision TOL, double precision, dimension( 2*n ) WORK, integer INFO)

DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

Purpose:

DPSTRF computes the Cholesky factorization with complete
pivoting of a real symmetric positive semidefinite matrix A.

This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 3 BLAS.

Parameters

UPLO

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

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

On exit, if INFO = 0, the factor U or L from the Cholesky
factorization as above.

LDA

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

PIV

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

RANK

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

TOL

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

WORK

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

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsbgst (character VECT, character UPLO, integer N, integer KA,integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, doubleprecision, dimension( ldbb, * ) BB, integer LDBB, double precision,dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) WORK,integer INFO)

DSBGST

Purpose:

DSBGST reduces a real symmetric-definite banded generalized
eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y,
such that C has the same bandwidth as A.

B must have been previously factorized as S**T*S by DPBSTF, using a
split Cholesky factorization. A is overwritten by C = X**T*A*X, where
X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
bandwidth of A.

Parameters

VECT

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

UPLO

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

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

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

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

AB is DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first ka+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = ’U’, AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).

On exit, the transformed matrix X**T*A*X, stored in the same
format as A.

LDAB

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

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

LDBB

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

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

LDX

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

WORK

WORK is DOUBLE PRECISION array, dimension (2*N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsbtrd (character VECT, character UPLO, integer N, integer KD,double precision, dimension( ldab, * ) AB, integer LDAB, double precision,dimension( * ) D, double precision, dimension( * ) E, double precision,dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK,integer INFO)

DSBTRD

Purpose:

DSBTRD reduces a real symmetric band matrix A to symmetric
tridiagonal form T by an orthogonal similarity transformation:
Q**T * A * Q = T.

Parameters

VECT

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

UPLO

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

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

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

AB 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, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = ’U’) or the
first subdiagonal (if UPLO = ’L’) are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.

LDAB

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

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

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

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if VECT = ’U’, then Q must contain an N-by-N
matrix X; if VECT = ’N’ or ’V’, then Q need not be set.

On exit:
if VECT = ’V’, Q contains the N-by-N orthogonal matrix Q;
if VECT = ’U’, Q contains the product X*Q;
if VECT = ’N’, the array Q is not referenced.

LDQ

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

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

Modified by Linda Kaufman, Bell Labs.

subroutine dsfrk (character TRANSR, character UPLO, character TRANS, integerN, integer K, double precision ALPHA, double precision, dimension( lda, * )A, integer LDA, double precision BETA, double precision, dimension( * ) C)

DSFRK performs a symmetric rank-k operation for matrix in RFP format.

Purpose:

Level 3 BLAS like routine for C in RFP Format.

DSFRK performs one of the symmetric rank--k operations

C := alpha*A*A**T + beta*C,

C := alpha*A**T*A + beta*C,

where alpha and beta are real scalars, C is an n--by--n symmetric
matrix and A is an n--by--k matrix in the first case and a k--by--n
matrix in the second case.

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal Form of RFP A is stored;
= ’T’: The Transpose Form of RFP A is stored.

UPLO

UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the array C is to be referenced as
follows:

UPLO = ’U’ or ’u’ Only the upper triangular part of C
is to be referenced.

UPLO = ’L’ or ’l’ Only the lower triangular part of C
is to be referenced.

Unchanged on exit.

TRANS

TRANS is CHARACTER*1
On entry, TRANS specifies the operation to be performed as
follows:

TRANS = ’N’ or ’n’ C := alpha*A*A**T + beta*C.

TRANS = ’T’ or ’t’ C := alpha*A**T*A + beta*C.

Unchanged on exit.

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

K is INTEGER
On entry with TRANS = ’N’ or ’n’, K specifies the number
of columns of the matrix A, and on entry with TRANS = ’T’
or ’t’, K specifies the number of rows of the matrix A. K
must be at least zero.
Unchanged on exit.

ALPHA

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

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

LDA

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

BETA

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

C is DOUBLE PRECISION array, dimension (NT)
NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
Format. RFP Format is described by TRANSR, UPLO and N.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dspcon (character UPLO, integer N, double precision, dimension( ) AP, integer, dimension( ) IPIV, double precision ANORM, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

DSPCON

Purpose:

DSPCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric packed matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by DSPTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters

UPLO

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

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

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

IPIV

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

ANORM

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

RCOND

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

WORK

WORK is DOUBLE PRECISION array, dimension (2*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dspgst (integer ITYPE, character UPLO, integer N, doubleprecision, dimension( * ) AP, double precision, dimension( * ) BP, integerINFO)

DSPGST

Purpose:

DSPGST reduces a real symmetric-definite generalized eigenproblem
to standard form, using packed storage.

If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

B must have been previously factorized as U**T*U or L*L**T by DPPTRF.

Parameters

ITYPE

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

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored and B is factored as
U**T*U;
= ’L’: Lower triangle of A is stored and B is factored as
L*L**T.

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

On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsprfs (character UPLO, integer N, integer NRHS, double precision,dimension( * ) AP, double precision, dimension( * ) AFP, integer,dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB,double precision, dimension( ldx, * ) X, integer LDX, double precision,dimension( * ) FERR, double precision, dimension( * ) BERR, doubleprecision, dimension( * ) WORK, integer, dimension( * ) IWORK, integerINFO)

DSPRFS

Purpose:

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

Parameters

UPLO

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

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

NRHS

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

AFP

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

IPIV

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

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

LDB

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

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

LDX

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

FERR

BERR

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsptrd (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( ) D, double precision, dimension( * )E, double precision, dimension( * ) TAU, integer INFO)

DSPTRD

Purpose:

DSPTRD reduces a real symmetric matrix A stored in packed form to
symmetric tridiagonal form T by an orthogonal similarity
transformation: Q**T * A * Q = T.

Parameters

UPLO

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

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

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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = ’U’, the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= ’L’, the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.

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

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

TAU

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

If UPLO = ’U’, the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = ’L’, the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**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+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).

subroutine dsptrf (character UPLO, integer N, double precision, dimension( ) AP, integer, dimension( ) IPIV, integer INFO)

DSPTRF

Purpose:

DSPTRF computes the factorization of a real symmetric matrix A stored
in packed format using the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.

Parameters

UPLO

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

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

On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L, stored as a packed triangular
matrix overwriting A (see below for further details).

IPIV

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

If UPLO = ’U’, then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = ’L’, then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:

J. Lewis, Boeing Computer Services Company

subroutine dsptri (character UPLO, integer N, double precision, dimension( ) AP, integer, dimension( ) IPIV, double precision, dimension( * ) WORK,integer INFO)

DSPTRI

Purpose:

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

Parameters

UPLO

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

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

On exit, if INFO = 0, the (symmetric) inverse of the original
matrix, stored as a packed triangular matrix. The j-th column
of inv(A) is stored in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
if UPLO = ’L’,
AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

IPIV

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

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsptrs (character UPLO, integer N, integer NRHS, double precision,dimension( * ) AP, integer, dimension( * ) IPIV, double precision,dimension( ldb, * ) B, integer LDB, integer INFO)

DSPTRS

Purpose:

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

Parameters

UPLO

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

NRHS

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

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

IPIV

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

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

LDB

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dstegr (character JOBZ, character RANGE, integer N, doubleprecision, dimension( * ) D, double precision, dimension( * ) E, doubleprecision VL, double precision VU, integer IL, integer IU, double precisionABSTOL, integer M, double precision, dimension( * ) W, double precision,dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, doubleprecision, dimension( * ) WORK, integer LWORK, integer, dimension( * )IWORK, integer LIWORK, integer INFO)

DSTEGR

Purpose:

DSTEGR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
a well defined set of pairwise different real eigenvalues, the corresponding
real eigenvectors are pairwise orthogonal.

The spectrum may be computed either completely or partially by specifying
either an interval (VL,VU] or a range of indices IL:IU for the desired
eigenvalues.

DSTEGR is a compatibility wrapper around the improved DSTEMR routine.
See DSTEMR for further details.

One important change is that the ABSTOL parameter no longer provides any
benefit and hence is no longer used.

Note : DSTEGR and DSTEMR work only on machines which follow
IEEE-754 floating-point standard in their handling of infinities and
NaNs. Normal execution may create these exceptiona values and hence
may abort due to a floating point exception in environments which
do not conform to the IEEE-754 standard.

Parameters

JOBZ

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

RANGE

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

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

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

E is DOUBLE PRECISION array, dimension (N)
On entry, the (N-1) subdiagonal elements of the tridiagonal
matrix T in elements 1 to N-1 of E. E(N) need not be set on
input, but is used internally as workspace.
On exit, E is overwritten.

VL is DOUBLE PRECISION

If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU is DOUBLE PRECISION

If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL is INTEGER

If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.

IU is INTEGER

If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.

ABSTOL

ABSTOL is DOUBLE PRECISION
Unused. Was the absolute error tolerance for the
eigenvalues/eigenvectors in previous versions.

M is INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = ’A’, M = N, and if RANGE = ’I’, M = IU-IL+1.

W is DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
If JOBZ = ’V’, and if INFO = 0, then the first M columns of Z
contain the orthonormal eigenvectors of the matrix T
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and an upper bound must be used.
Supplying N columns is always safe.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = ’V’, then LDZ >= max(1,N).

ISUPPZ

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The i-th computed eigenvector
is nonzero only in elements ISUPPZ( 2*i-1 ) through
ISUPPZ( 2*i ). This is relevant in the case when the matrix
is split. ISUPPZ is only accessed when JOBZ is ’V’ and N > 0.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
(and minimal) LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,18*N)
if JOBZ = ’V’, and LWORK >= max(1,12*N) if JOBZ = ’N’.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

IWORK

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

LIWORK

LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= max(1,10*N)
if the eigenvectors are desired, and LIWORK >= max(1,8*N)
if only the eigenvalues are to be computed.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.

INFO

INFO is INTEGER
On exit, INFO
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = 1X, internal error in DLARRE,
if INFO = 2X, internal error in DLARRV.
Here, the digit X = ABS( IINFO ) < 10, where IINFO is
the nonzero error code returned by DLARRE or
DLARRV, respectively.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, LBNL/NERSC, USA

subroutine dstein (integer N, double precision, dimension( * ) D, doubleprecision, dimension( * ) E, integer M, double precision, dimension( * ) W,integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, doubleprecision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension(* ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,integer INFO)

DSTEIN

Purpose:

DSTEIN computes the eigenvectors of a real symmetric tridiagonal
matrix T corresponding to specified eigenvalues, using inverse
iteration.

The maximum number of iterations allowed for each eigenvector is
specified by an internal parameter MAXITS (currently set to 5).

Parameters

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

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

E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix
T, in elements 1 to N-1.

M is INTEGER
The number of eigenvectors to be found. 0 <= M <= N.

W is DOUBLE PRECISION array, dimension (N)
The first M elements of W contain the eigenvalues for
which eigenvectors are to be computed. The eigenvalues
should be grouped by split-off block and ordered from
smallest to largest within the block. ( The output array
W from DSTEBZ with ORDER = ’B’ is expected here. )

IBLOCK

IBLOCK is INTEGER array, dimension (N)
The submatrix indices associated with the corresponding
eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
the first submatrix from the top, =2 if W(i) belongs to
the second submatrix, etc. ( The output array IBLOCK
from DSTEBZ is expected here. )

ISPLIT

ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to
ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
through ISPLIT( 2 ), etc.
( The output array ISPLIT from DSTEBZ is expected here. )

Z is DOUBLE PRECISION array, dimension (LDZ, M)
The computed eigenvectors. The eigenvector associated
with the eigenvalue W(i) is stored in the i-th column of
Z. Any vector which fails to converge is set to its current
iterate after MAXITS iterations.

LDZ

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

WORK

WORK is DOUBLE PRECISION array, dimension (5*N)

IWORK

IWORK is INTEGER array, dimension (N)

IFAIL

IFAIL is INTEGER array, dimension (M)
On normal exit, all elements of IFAIL are zero.
If one or more eigenvectors fail to converge after
MAXITS iterations, then their indices are stored in
array IFAIL.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, then i eigenvectors failed to converge
in MAXITS iterations. Their indices are stored in
array IFAIL.

Internal Parameters:

MAXITS INTEGER, default = 5
The maximum number of iterations performed.

EXTRA INTEGER, default = 2
The number of iterations performed after norm growth
criterion is satisfied, should be at least 1.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dstemr (character JOBZ, character RANGE, integer N, doubleprecision, dimension( * ) D, double precision, dimension( * ) E, doubleprecision VL, double precision VU, integer IL, integer IU, integer M,double precision, dimension( * ) W, double precision, dimension( ldz, * )Z, integer LDZ, integer NZC, integer, dimension( * ) ISUPPZ, logicalTRYRAC, double precision, dimension( * ) WORK, integer LWORK, integer,dimension( * ) IWORK, integer LIWORK, integer INFO)

DSTEMR

Purpose:

DSTEMR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
a well defined set of pairwise different real eigenvalues, the corresponding
real eigenvectors are pairwise orthogonal.

The spectrum may be computed either completely or partially by specifying
either an interval (VL,VU] or a range of indices IL:IU for the desired
eigenvalues.

Depending on the number of desired eigenvalues, these are computed either
by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
computed by the use of various suitable L D LˆT factorizations near clusters
of close eigenvalues (referred to as RRRs, Relatively Robust
Representations). An informal sketch of the algorithm follows.

For each unreduced block (submatrix) of T,
(a) Compute T - sigma I = L D LˆT, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
the corresponding eigenvector by forming a rank revealing twisted
factorization. Go back to (c) for any clusters that remain.

For more details, see:
- Inderjit S. Dhillon and Beresford N. Parlett: ’Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,’
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: ’Orthogonal Eigenvectors and
Relative Gaps,’ SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004. Also LAPACK Working Note 154.
- Inderjit Dhillon: ’A new O(nˆ2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem’,
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.

Further Details
1.DSTEMR works only on machines which follow IEEE-754
floating-point standard in their handling of infinities and NaNs.
This permits the use of efficient inner loops avoiding a check for
zero divisors.

Parameters

JOBZ

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

RANGE

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

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

VL is DOUBLE PRECISION

If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU is DOUBLE PRECISION

If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL is INTEGER

If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.

IU is INTEGER

If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0.
Not referenced if RANGE = ’A’ or ’V’.

M is INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = ’A’, M = N, and if RANGE = ’I’, M = IU-IL+1.

W is DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.

Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
If JOBZ = ’V’, and if INFO = 0, then the first M columns of Z
contain the orthonormal eigenvectors of the matrix T
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If JOBZ = ’N’, then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = ’V’, the exact value of M
is not known in advance and can be computed with a workspace
query by setting NZC = -1, see below.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = ’V’, then LDZ >= max(1,N).

NZC

NZC is INTEGER
The number of eigenvectors to be held in the array Z.
If RANGE = ’A’, then NZC >= max(1,N).
If RANGE = ’V’, then NZC >= the number of eigenvalues in (VL,VU].
If RANGE = ’I’, then NZC >= IU-IL+1.
If NZC = -1, then a workspace query is assumed; the
routine calculates the number of columns of the array Z that
are needed to hold the eigenvectors.
This value is returned as the first entry of the Z array, and
no error message related to NZC is issued by XERBLA.

ISUPPZ

TRYRAC

TRYRAC is LOGICAL
If TRYRAC = .TRUE., indicates that the code should check whether
the tridiagonal matrix defines its eigenvalues to high relative
accuracy. If so, the code uses relative-accuracy preserving
algorithms that might be (a bit) slower depending on the matrix.
If the matrix does not define its eigenvalues to high relative
accuracy, the code can uses possibly faster algorithms.
If TRYRAC = .FALSE., the code is not required to guarantee
relatively accurate eigenvalues and can use the fastest possible
techniques.
On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
does not define its eigenvalues to high relative accuracy.

WORK

WORK is DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
(and minimal) LWORK.

LWORK

IWORK

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

LIWORK

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

subroutine dtbcon (character NORM, character UPLO, character DIAG, integer N,integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

DTBCON

Purpose:

DTBCON estimates the reciprocal of the condition number of a
triangular band matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for
norm(inv(A)), then the reciprocal of the condition number is
computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters

NORM

NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.

UPLO

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

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

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

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).
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.

LDAB

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

RCOND

RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtbrfs (character UPLO, character TRANS, character DIAG, integerN, integer KD, integer NRHS, double precision, dimension( ldab, * ) AB,integer LDAB, double precision, dimension( ldb, * ) B, integer LDB, doubleprecision, dimension( ldx, * ) X, integer LDX, double precision, dimension(* ) FERR, double precision, dimension( * ) BERR, double precision,dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DTBRFS

Purpose:

DTBRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular band
coefficient matrix.

The solution matrix X must be computed by DTBTRS or some other
means before entering this routine. DTBRFS does not do iterative
refinement because doing so cannot improve the backward error.

Parameters

UPLO

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

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

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

NRHS

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

LDAB

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

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

LDB

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

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

LDX

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

FERR

BERR

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtbtrs (character UPLO, character TRANS, character DIAG, integerN, integer KD, integer NRHS, double precision, dimension( ldab, * ) AB,integer LDAB, double precision, dimension( ldb, * ) B, integer LDB, integerINFO)

DTBTRS

Purpose:

DTBTRS solves a triangular system of the form

A * X = B or A**T * X = B,

where A is a triangular band matrix of order N, and B is an
N-by NRHS matrix. A check is made to verify that A is nonsingular.

Parameters

UPLO

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

TRANS

TRANS is CHARACTER*1
Specifies the form the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

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

NRHS

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

AB is DOUBLE PRECISION array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first kd+1 rows of AB. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = ’L’, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.

LDAB

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

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.

LDB

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

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the
solutions X have not been computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtfsm (character TRANSR, character SIDE, character UPLO, characterTRANS, character DIAG, integer M, integer N, double precision ALPHA, doubleprecision, dimension( 0: * ) A, double precision, dimension( 0: ldb-1, 0: *) B, integer LDB)

DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

Level 3 BLAS like routine for A in RFP Format.

DTFSM solves the matrix equation

op( A )*X = alpha*B or X*op( A ) = alpha*B

where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit, upper or lower triangular matrix and op( A ) is one of

op( A ) = A or op( A ) = A**T.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B.

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal Form of RFP A is stored;
= ’T’: The Transpose Form of RFP A is stored.

SIDE

SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:

SIDE = ’L’ or ’l’ op( A )*X = alpha*B.

SIDE = ’R’ or ’r’ X*op( A ) = alpha*B.

Unchanged on exit.

UPLO

UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = ’U’ or ’u’ RFP A came from an upper triangular matrix
UPLO = ’L’ or ’l’ RFP A came from a lower triangular matrix

Unchanged on exit.

TRANS

TRANS is CHARACTER*1
On entry, TRANS specifies the form of op( A ) to be used
in the matrix multiplication as follows:

TRANS = ’N’ or ’n’ op( A ) = A.

TRANS = ’T’ or ’t’ op( A ) = A’.

Unchanged on exit.

DIAG

DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:

DIAG = ’U’ or ’u’ A is assumed to be unit triangular.

DIAG = ’N’ or ’n’ A is not assumed to be unit
triangular.

Unchanged on exit.

M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.

N is INTEGER
On entry, N specifies the number of columns of B. N must be
at least zero.
Unchanged on exit.

ALPHA

ALPHA is DOUBLE PRECISION
On entry, ALPHA specifies the scalar alpha. When alpha is
zero then A is not referenced and B need not be set before
entry.
Unchanged on exit.

A is DOUBLE PRECISION array, dimension (NT)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR=’N’ then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = ’T’ then RFP is the transpose of RFP A as
defined when TRANSR = ’N’. The contents of RFP A are defined
by UPLO as follows: If UPLO = ’U’ the RFP A contains the NT
elements of upper packed A either in normal or
transpose Format. If UPLO = ’L’ the RFP A contains
the NT elements of lower packed A either in normal or
transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = ’T’. When TRANSR is ’N’ the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.

B is DOUBLE PRECISION array, dimension (LDB,N)
Before entry, the leading m by n part of the array B must
contain the right-hand side matrix B, and on exit is
overwritten by the solution matrix X.

LDB

LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in the calling (sub) program. LDB must be at least
max( 1, m ).
Unchanged on exit.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dtftri (character TRANSR, character UPLO, character DIAG, integerN, double precision, dimension( 0: * ) A, integer INFO)

DTFTRI

Purpose:

DTFTRI computes the inverse of a triangular matrix A stored in RFP
format.

This is a Level 3 BLAS version of the algorithm.

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: The Normal TRANSR of RFP A is stored;
= ’T’: The Transpose TRANSR of RFP A is stored.

UPLO

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

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

A is DOUBLE PRECISION array, dimension (0:nt-1);
nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
Positive Definite matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = ’N’
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = ’T’ then RFP is
the transpose of RFP A as defined when
TRANSR = ’N’. The contents of RFP A are defined by UPLO as
follows: If UPLO = ’U’ the RFP A contains the nt elements of
upper packed A; If UPLO = ’L’ the RFP A contains the nt
elements of lower packed A. The LDA of RFP A is (N+1)/2 when
TRANSR = ’T’. When TRANSR is ’N’ the LDA is N+1 when N is
even and N is odd. See the Note below for more details.

On exit, the (triangular) inverse of the original matrix, in
the same storage format.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, A(i,i) is exactly zero. The triangular
matrix is singular and its inverse can not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dtfttp (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) ARF, double precision, dimension( 0: * ) AP,integer INFO)

DTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).

Purpose:

DTFTTP copies a triangular matrix A from rectangular full packed
format (TF) to standard packed format (TP).

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: ARF is in Normal format;
= ’T’: ARF is in Transpose format;

UPLO

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

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

ARF

ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
On entry, the upper or lower triangular matrix A stored in
RFP format. For a further discussion see Notes below.

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dtfttr (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) ARF, double precision, dimension( 0: lda-1, 0:* ) A, integer LDA, integer INFO)

DTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).

Purpose:

DTFTTR copies a triangular matrix A from rectangular full packed
format (TF) to standard full format (TR).

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: ARF is in Normal format;
= ’T’: ARF is in Transpose format.

UPLO

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

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

ARF

ARF is DOUBLE PRECISION array, dimension (N*(N+1)/2).
On entry, the upper (if UPLO = ’U’) or lower (if UPLO = ’L’)
matrix A in RFP format. See the ’Notes’ below for more
details.

A is DOUBLE PRECISION array, dimension (LDA,N)
On exit, the triangular matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced.

LDA

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dtgsen (integer IJOB, logical WANTQ, logical WANTZ, logical,dimension( * ) SELECT, integer N, double precision, dimension( lda, * ) A,integer LDA, double precision, dimension( ldb, * ) B, integer LDB, doubleprecision, dimension( * ) ALPHAR, double precision, dimension( * ) ALPHAI,double precision, dimension( * ) BETA, double precision, dimension( ldq, ) Q, integer LDQ, double precision, dimension( ldz, ) Z, integer LDZ,integer M, double precision PL, double precision PR, double precision,dimension( * ) DIF, double precision, dimension( * ) WORK, integer LWORK,integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DTGSEN

Purpose:

DTGSEN reorders the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans-
formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
appears in the leading diagonal blocks of the upper quasi-triangular
matrix A and the upper triangular B. The leading columns of Q and
Z form orthonormal bases of the corresponding left and right eigen-
spaces (deflating subspaces). (A, B) must be in generalized real
Schur canonical form (as returned by DGGES), i.e. A is block upper
triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
triangular.

DTGSEN also computes the generalized eigenvalues

w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

of the reordered matrix pair (A, B).

Optionally, DTGSEN computes the estimates of reciprocal condition
numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
(A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
between the matrix pairs (A11, B11) and (A22,B22) that correspond to
the selected cluster and the eigenvalues outside the cluster, resp.,
and norms of ’projections’ onto left and right eigenspaces w.r.t.
the selected cluster in the (1,1)-block.

Parameters

IJOB

IJOB is INTEGER
Specifies whether condition numbers are required for the
cluster of eigenvalues (PL and PR) or the deflating subspaces
(Difu and Difl):
=0: Only reorder w.r.t. SELECT. No extras.
=1: Reciprocal of norms of ’projections’ onto left and right
eigenspaces w.r.t. the selected cluster (PL and PR).
=2: Upper bounds on Difu and Difl. F-norm-based estimate
(DIF(1:2)).
=3: Estimate of Difu and Difl. 1-norm-based estimate
(DIF(1:2)).
About 5 times as expensive as IJOB = 2.
=4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
version to get it all.
=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

WANTQ

WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.

WANTZ

WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.

SELECT

SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.

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

A is DOUBLE PRECISION array, dimension(LDA,N)
On entry, the upper quasi-triangular matrix A, with (A, B) in
generalized real Schur canonical form.
On exit, A is overwritten by the reordered matrix A.

LDA

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

B is DOUBLE PRECISION array, dimension(LDB,N)
On entry, the upper triangular matrix B, with (A, B) in
generalized real Schur canonical form.
On exit, B is overwritten by the reordered matrix B.

LDB

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

ALPHAR

ALPHAR is DOUBLE PRECISION array, dimension (N)

ALPHAI

ALPHAI is DOUBLE PRECISION array, dimension (N)

BETA

BETA is DOUBLE PRECISION array, dimension (N)

On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
and BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real generalized Schur form of (A,B) were further reduced
to triangular form using complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
On exit, Q has been postmultiplied by the left orthogonal
transformation matrix which reorder (A, B); The leading M
columns of Q form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTQ = .FALSE., Q is not referenced.

LDQ

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

Z is DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
On exit, Z has been postmultiplied by the left orthogonal
transformation matrix which reorder (A, B); The leading M
columns of Z form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTZ = .FALSE., Z is not referenced.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1;
If WANTZ = .TRUE., LDZ >= N.

M is INTEGER
The dimension of the specified pair of left and right eigen-
spaces (deflating subspaces). 0 <= M <= N.

PL is DOUBLE PRECISION

PR is DOUBLE PRECISION

If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
reciprocal of the norm of ’projections’ onto left and right
eigenspaces with respect to the selected cluster.
0 < PL, PR <= 1.
If M = 0 or M = N, PL = PR = 1.
If IJOB = 0, 2 or 3, PL and PR are not referenced.

DIF

DIF is DOUBLE PRECISION array, dimension (2).
If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
estimates of Difu and Difl.
If M = 0 or N, DIF(1:2) = F-norm([A, B]).
If IJOB = 0 or 1, DIF is not referenced.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= 4*N+16.
If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).

IWORK

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

LIWORK

LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= 1.
If IJOB = 1, 2 or 4, LIWORK >= N+6.
If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

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

INFO

INFO is INTEGER
=0: Successful exit.
<0: If INFO = -i, the i-th argument had an illegal value.
=1: Reordering of (A, B) failed because the transformed
matrix pair (A, B) would be too far from generalized
Schur form; the problem is very ill-conditioned.
(A, B) may have been partially reordered.
If requested, 0 is returned in DIF(*), PL and PR.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

DTGSEN first collects the selected eigenvalues by computing
orthogonal U and W that move them to the top left corner of (A, B).
In other words, the selected eigenvalues are the eigenvalues of
(A11, B11) in:

U**T*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) n2
n1 n2 n1 n2

where N = n1+n2 and U**T means the transpose of U. The first n1 columns
of U and W span the specified pair of left and right eigenspaces
(deflating subspaces) of (A, B).

If (A, B) has been obtained from the generalized real Schur
decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
reordered generalized real Schur form of (C, D) is given by

(C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,

and the first n1 columns of Q*U and Z*W span the corresponding
deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

Note that if the selected eigenvalue is sufficiently ill-conditioned,
then its value may differ significantly from its value before
reordering.

The reciprocal condition numbers of the left and right eigenspaces
spanned by the first n1 columns of U and W (or Q*U and Z*W) may
be returned in DIF(1:2), corresponding to Difu and Difl, resp.

The Difu and Difl are defined as:

Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
and
Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the
(2*n1*n2)-by-(2*n1*n2) matrix

Zu = [ kron(In2, A11) -kron(A22**T, In1) ]
[ kron(In2, B11) -kron(B22**T, In1) ].

Here, Inx is the identity matrix of size nx and A22**T is the
transpose of A22. kron(X, Y) is the Kronecker product between
the matrices X and Y.

When DIF(2) is small, small changes in (A, B) can cause large changes
in the deflating subspace. An approximate (asymptotic) bound on the
maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / DIF(2),

where EPS is the machine precision.

The reciprocal norm of the projectors on the left and right
eigenspaces associated with (A11, B11) may be returned in PL and PR.
They are computed as follows. First we compute L and R so that
P*(A, B)*Q is block diagonal, where

P = ( I -L ) n1 Q = ( I R ) n1
( 0 I ) n2 and ( 0 I ) n2
n1 n2 n1 n2

and (L, R) is the solution to the generalized Sylvester equation

A11*R - L*A22 = -A12
B11*R - L*B22 = -B12

Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is

EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up
to a certain restriction: A lower bound (x) on the smallest
F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
(i.e. (A + E, B + F), is

x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR.

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
(L’, R’) and unperturbed (L, R) left and right deflating subspaces
associated with the selected cluster in the (1,1)-blocks can be
bounded as

max-angle(L, L’) <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
max-angle(R, R’) <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User’s Guide section 4.11 or the following references
for more information.

Note that if the default method for computing the Frobenius-norm-
based estimate DIF is not wanted (see DLATDF), then the parameter
IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
(IJOB = 2 will be used)). See DTGSYL for more details.

Contributors:

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

References:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea
University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.

[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
1996.

subroutine dtgsja (character JOBU, character JOBV, character JOBQ, integer M,integer P, integer N, integer K, integer L, double precision, dimension(lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integerLDB, double precision TOLA, double precision TOLB, double precision,dimension( * ) ALPHA, double precision, dimension( * ) BETA, doubleprecision, dimension( ldu, * ) U, integer LDU, double precision, dimension(ldv, * ) V, integer LDV, double precision, dimension( ldq, * ) Q, integerLDQ, double precision, dimension( * ) WORK, integer NCYCLE, integer INFO)

DTGSJA

Purpose:

DTGSJA computes the generalized singular value decomposition (GSVD)
of two real upper triangular (or trapezoidal) matrices A and B.

On entry, it is assumed that matrices A and B have the following
forms, which may be obtained by the preprocessing subroutine DGGSVP
from a general M-by-N matrix A and P-by-N matrix B:

N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )

N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )

N-K-L K L
B = L ( 0 0 B13 )
P-L ( 0 0 0 )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal.

On exit,

U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),

where U, V and Q are orthogonal matrices.
R is a nonsingular upper triangular matrix, and D1 and D2 are
‘‘diagonal’’ matrices, which are of the following structures:

If M-K-L >= 0,

K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )

K L
D2 = L ( 0 S )
P-L ( 0 0 )

N-K-L K L
( 0 R ) = K ( 0 R11 R12 ) K
L ( 0 0 R22 ) L

where

C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.

R is stored in A(1:K+L,N-K-L+1:N) on exit.

If M-K-L < 0,

K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )

K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )

N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )

where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.

R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.

The computation of the orthogonal transformation matrices U, V or Q
is optional. These matrices may either be formed explicitly, or they
may be postmultiplied into input matrices U1, V1, or Q1.

Parameters

JOBU

JOBU is CHARACTER*1
= ’U’: U must contain an orthogonal matrix U1 on entry, and
the product U1*U is returned;
= ’I’: U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= ’N’: U is not computed.

JOBV

JOBV is CHARACTER*1
= ’V’: V must contain an orthogonal matrix V1 on entry, and
the product V1*V is returned;
= ’I’: V is initialized to the unit matrix, and the
orthogonal matrix V is returned;
= ’N’: V is not computed.

JOBQ

JOBQ is CHARACTER*1
= ’Q’: Q must contain an orthogonal matrix Q1 on entry, and
the product Q1*Q is returned;
= ’I’: Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= ’N’: Q is not computed.

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

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

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

K is INTEGER

L is INTEGER

K and L specify the subblocks in the input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
of A and B, whose GSVD is going to be computed by DTGSJA.
See Further Details.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
matrix R or part of R. See Purpose for details.

LDA

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

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
a part of R. See Purpose for details.

LDB

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

TOLA

TOLA is DOUBLE PRECISION

TOLB

TOLB is DOUBLE PRECISION

TOLA and TOLB are the convergence criteria for the Jacobi-
Kogbetliantz iteration procedure. Generally, they are the
same as used in the preprocessing step, say
TOLA = max(M,N)*norm(A)*MAZHEPS,
TOLB = max(P,N)*norm(B)*MAZHEPS.

ALPHA

ALPHA is DOUBLE PRECISION array, dimension (N)

BETA

BETA is DOUBLE PRECISION array, dimension (N)

On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
BETA(1:K) = 0,
and if M-K-L >= 0,
ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(S),
or if M-K-L < 0,
ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0 and
BETA(K+L+1:N) = 0.

U is DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = ’U’, U must contain a matrix U1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBU = ’I’, U contains the orthogonal matrix U;
if JOBU = ’U’, U contains the product U1*U.
If JOBU = ’N’, U is not referenced.

LDU

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

V is DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = ’V’, V must contain a matrix V1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBV = ’I’, V contains the orthogonal matrix V;
if JOBV = ’V’, V contains the product V1*V.
If JOBV = ’N’, V is not referenced.

LDV

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

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = ’Q’, Q must contain a matrix Q1 (usually
the orthogonal matrix returned by DGGSVP).
On exit,
if JOBQ = ’I’, Q contains the orthogonal matrix Q;
if JOBQ = ’Q’, Q contains the product Q1*Q.
If JOBQ = ’N’, Q is not referenced.

LDQ

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

WORK

WORK is DOUBLE PRECISION array, dimension (2*N)

NCYCLE

NCYCLE is INTEGER
The number of cycles required for convergence.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.

Internal Parameters
===================

MAXIT INTEGER
MAXIT specifies the total loops that the iterative procedure
may take. If after MAXIT cycles, the routine fails to
converge, we return INFO = 1..fi

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
matrix B13 to the form:

U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
of Z. C1 and S1 are diagonal matrices satisfying

C1**2 + S1**2 = I,

and R1 is an L-by-L nonsingular upper triangular matrix.

subroutine dtgsna (character JOB, character HOWMNY, logical, dimension( * )SELECT, integer N, double precision, dimension( lda, * ) A, integer LDA,double precision, dimension( ldb, * ) B, integer LDB, double precision,dimension( ldvl, * ) VL, integer LDVL, double precision, dimension( ldvr, ) VR, integer LDVR, double precision, dimension( ) S, double precision,dimension( * ) DIF, integer MM, integer M, double precision, dimension( * )WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

DTGSNA

Purpose:

DTGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B) in
generalized real Schur canonical form (or of any matrix pair
(Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
Z**T denotes the transpose of Z.

(A, B) must be in generalized real Schur form (as returned by DGGES),
i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
blocks. B is upper triangular.

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (DIF):
= ’E’: for eigenvalues only (S);
= ’V’: for eigenvectors only (DIF);
= ’B’: for both eigenvalues and eigenvectors (S and DIF).

HOWMNY

HOWMNY is CHARACTER*1
= ’A’: compute condition numbers for all eigenpairs;
= ’S’: compute condition numbers for selected eigenpairs
specified by the array SELECT.

SELECT

SELECT is LOGICAL array, dimension (N)
If HOWMNY = ’S’, SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the eigenpair corresponding to a real eigenvalue w(j),
SELECT(j) must be set to .TRUE.. To select condition numbers
corresponding to a complex conjugate pair of eigenvalues w(j)
and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
set to .TRUE..
If HOWMNY = ’A’, SELECT is not referenced.

N is INTEGER
The order of the square matrix pair (A, B). N >= 0.

A is DOUBLE PRECISION array, dimension (LDA,N)
The upper quasi-triangular matrix A in the pair (A,B).

LDA

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

B is DOUBLE PRECISION array, dimension (LDB,N)
The upper triangular matrix B in the pair (A,B).

LDB

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

VL is DOUBLE PRECISION array, dimension (LDVL,M)
If JOB = ’E’ or ’B’, VL must contain left eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns of VL, as returned by DTGEVC.
If JOB = ’V’, VL is not referenced.

LDVL

LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1.
If JOB = ’E’ or ’B’, LDVL >= N.

VR is DOUBLE PRECISION array, dimension (LDVR,M)
If JOB = ’E’ or ’B’, VR must contain right eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns ov VR, as returned by DTGEVC.
If JOB = ’V’, VR is not referenced.

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1.
If JOB = ’E’ or ’B’, LDVR >= N.

S is DOUBLE PRECISION array, dimension (MM)
If JOB = ’E’ or ’B’, the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus
S(j), DIF(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = ’V’, S is not referenced.

DIF

DIF is DOUBLE PRECISION array, dimension (MM)
If JOB = ’V’ or ’B’, the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of DIF are set to the same value. If
the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
is set to 0; this can only occur when the true value would be
very small anyway.
If JOB = ’E’, DIF is not referenced.

MM is INTEGER
The number of elements in the arrays S and DIF. MM >= M.

M is INTEGER
The number of elements of the arrays S and DIF used to store
the specified condition numbers; for each selected real
eigenvalue one element is used, and for each selected complex
conjugate pair of eigenvalues, two elements are used.
If HOWMNY = ’A’, M is set to N.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If JOB = ’V’ or ’B’ LWORK >= 2*N*(N+2)+16.

IWORK

IWORK is INTEGER array, dimension (N + 6)
If JOB = ’E’, IWORK is not referenced.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The reciprocal of the condition number of a generalized eigenvalue
w = (a, b) is defined as

S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))

where u and v are the left and right eigenvectors of (A, B)
corresponding to w; |z| denotes the absolute value of the complex
number, and norm(u) denotes the 2-norm of the vector u.
The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
of the matrix pair (A, B). If both a and b equal zero, then (A B) is
singular and S(I) = -1 is returned.

An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact
eigenvalue lambda is

chord(w, lambda) <= EPS * norm(A, B) / S(I)

where EPS is the machine precision.

The reciprocal of the condition number DIF(i) of right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows:

a) If the i-th eigenvalue w = (a,b) is real

Suppose U and V are orthogonal transformations such that

U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
( 0 S22 ),( 0 T22 ) n-1
1 n-1 1 n-1

Then the reciprocal condition number DIF(i) is

Difl((a, b), (S22, T22)) = sigma-min( Zl ),

where sigma-min(Zl) denotes the smallest singular value of the
2(n-1)-by-2(n-1) matrix

Zl = [ kron(a, In-1) -kron(1, S22) ]
[ kron(b, In-1) -kron(1, T22) ] .

Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
Kronecker product between the matrices X and Y.

Note that if the default method for computing DIF(i) is wanted
(see DLATDF), then the parameter DIFDRI (see below) should be
changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
See DTGSYL for more details.

b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

Suppose U and V are orthogonal transformations such that

U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
( 0 S22 ),( 0 T22) n-2
2 n-2 2 n-2

and (S11, T11) corresponds to the complex conjugate eigenvalue
pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
that

U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
( 0 s22 ) ( 0 t22 )

where the generalized eigenvalues w = s11/t11 and
conjg(w) = s22/t22.

Then the reciprocal condition number DIF(i) is bounded by

min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
Z1 is the complex 2-by-2 matrix

Z1 = [ s11 -s22 ]
[ t11 -t22 ],

This is done by computing (using real arithmetic) the
roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
where Z1**T denotes the transpose of Z1 and det(X) denotes
the determinant of X.

and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ]
[ kron(T11**T, In-2) -kron(I2, T22) ]

Note that if the default method for computing DIF is wanted (see
DLATDF), then the parameter DIFDRI (see below) should be changed
from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
for more details.

For each eigenvalue/vector specified by SELECT, DIF stores a
Frobenius norm-based estimate of Difl.

An approximate error bound for the i-th computed eigenvector VL(i) or
VR(i) is given by

EPS * norm(A, B) / DIF(i).

See ref. [2-3] for more details and further references.

Contributors:

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

References:

subroutine dtpcon (character NORM, character UPLO, character DIAG, integer N,double precision, dimension( * ) AP, double precision RCOND, doubleprecision, dimension( * ) WORK, integer, dimension( * ) IWORK, integerINFO)

DTPCON

Purpose:

DTPCON estimates the reciprocal of the condition number of a packed
triangular matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for
norm(inv(A)), then the reciprocal of the condition number is
computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters

NORM

NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.

UPLO

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

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

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.
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.

RCOND

RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtplqt (integer M, integer N, integer L, integer MB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integerLDT, double precision, dimension( * ) WORK, integer INFO)

DTPLQT

Purpose:

DTPLQT computes a blocked LQ factorization of a real
’triangular-pentagonal’ matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.

Parameters

M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.

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

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

MB is INTEGER
The block size to be used in the blocked QR. M >= MB >= 1.

A is DOUBLE PRECISION array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

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

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.

LDB

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

T is DOUBLE PRECISION array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.

LDT

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

WORK

WORK is DOUBLE PRECISION array, dimension (MB*M)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

The rows of V represent the vectors which define the H(i)’s.

T = [T1 T2 ... TB].

subroutine dtplqt2 (integer M, integer N, integer L, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * )B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT,integer INFO)

DTPLQT2 computes a LQ factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

DTPLQT2 computes a LQ a factorization of a real ’triangular-pentagonal’
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.

Parameters

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

N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

A is DOUBLE PRECISION array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

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

LDB

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

T is DOUBLE PRECISION array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.

LDT

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ][ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:

B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C

C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,

W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)’s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W**T * T * W

where WˆH is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.

subroutine dtpmlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer L, integer MB, double precision, dimension( ldv, * ) V,integer LDV, double precision, dimension( ldt, * ) T, integer LDT, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integerINFO)

DTPMLQT

Purpose:

DTPMQRT applies a real orthogonal matrix Q obtained from a
’triangular-pentagonal’ real block reflector H to a general
real matrix C, which consists of two blocks A and B.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.

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

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in DTPLQT.

V is DOUBLE PRECISION array, dimension (LDV,K)
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
DTPLQT in B. See Further Details.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= K.

T is DOUBLE PRECISION array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by DTPLQT, stored as a MB-by-K matrix.

LDT

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

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
Q*C or Q**T*C or C*Q or C*Q**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 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
Q*C or Q**T*C or C*Q or C*Q**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. The dimension of WORK is
N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:

V = [V1] [V2].

If SIDE = ’L’: C = [A] where A is K-by-N, B is M-by-N and V is K-by-M.
[B]

If SIDE = ’R’: C = [A B] where A is M-by-K, B is M-by-N and V is K-by-N.

The real orthogonal matrix Q is formed from V and T.

If TRANS=’N’ and SIDE=’L’, C is on exit replaced with Q * C.

If TRANS=’T’ and SIDE=’L’, C is on exit replaced with Q**T * C.

If TRANS=’N’ and SIDE=’R’, C is on exit replaced with C * Q.

If TRANS=’T’ and SIDE=’R’, C is on exit replaced with C * Q**T.

subroutine dtpmqrt (character SIDE, character TRANS, integer M, integer N,integer K, integer L, integer NB, double precision, dimension( ldv, * ) V,integer LDV, double precision, dimension( ldt, * ) T, integer LDT, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integerINFO)

DTPMQRT

Purpose:

DTPMQRT applies a real orthogonal matrix Q obtained from a
’triangular-pentagonal’ real block reflector H to a general
real matrix C, which consists of two blocks A and B.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

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

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

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.

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

NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in CTPQRT.

V is DOUBLE PRECISION array, dimension (LDV,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
CTPQRT in B. See Further Details.

LDV

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

T is DOUBLE PRECISION array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by CTPQRT, stored as a NB-by-K matrix.

LDT

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

LDA

LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDC >= max(1,K);
If SIDE = ’R’, LDC >= max(1,M).

B 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
Q*C or Q**T*C or C*Q or C*Q**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. The dimension of WORK is
N*NB if SIDE = ’L’, or M*NB if SIDE = ’R’.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:

V = [V1]
[V2].

The size of the trapezoidal block V2 is determined by the parameter L,
where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular;
if L=0, there is no trapezoidal block, hence V = V1 is rectangular.

If SIDE = ’L’: C = [A] where A is K-by-N, B is M-by-N and V is M-by-K.
[B]

If SIDE = ’R’: C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K.

The real orthogonal matrix Q is formed from V and T.

If TRANS=’N’ and SIDE=’L’, C is on exit replaced with Q * C.

If TRANS=’T’ and SIDE=’L’, C is on exit replaced with Q**T * C.

If TRANS=’N’ and SIDE=’R’, C is on exit replaced with C * Q.

If TRANS=’T’ and SIDE=’R’, C is on exit replaced with C * Q**T.

subroutine dtpqrt (integer M, integer N, integer L, integer NB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integerLDT, double precision, dimension( * ) WORK, integer INFO)

DTPQRT

Purpose:

DTPQRT computes a blocked QR factorization of a real
’triangular-pentagonal’ matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.

Parameters

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

N is INTEGER
The number of columns of the matrix B, and the order of the
triangular matrix A.
N >= 0.

L is INTEGER
The number of rows of the upper trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

NB is INTEGER
The block size to be used in the blocked QR. N >= NB >= 1.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.

LDA

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

B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first M-L rows
are rectangular, and the last L rows are upper trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.

LDB

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

T is DOUBLE PRECISION array, dimension (LDT,N)
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.

LDT

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

WORK

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a (N+M)-by-N matrix

C = [ A ]
[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:

B = [ B1 ] <- (M-L)-by-N rectangular
[ B2 ] <- L-by-N upper trapezoidal.

The upper trapezoidal matrix B2 consists of the first L rows of a
N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is upper triangular.

The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ] <- upper triangular N-by-N
[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ] <- identity, N-by-N
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,

V = [ V1 ] <- (M-L)-by-N rectangular
[ V2 ] <- L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)’s.

The number of blocks is B = ceiling(N/NB), where each
block is of order NB except for the last block, which is of order
IB = N - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T’s are stored in the NB-by-N matrix T as

T = [T1 T2 ... TB].

subroutine dtpqrt2 (integer M, integer N, integer L, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * )B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT,integer INFO)

DTPQRT2 computes a QR factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

DTPQRT2 computes a QR factorization of a real ’triangular-pentagonal’
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.

Parameters

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

N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.

L is INTEGER
The number of rows of the upper trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.

LDA

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

LDB

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

T is DOUBLE PRECISION array, dimension (LDT,N)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.

LDT

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a (N+M)-by-N matrix

C = [ A ]
[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:

B = [ B1 ] <- (M-L)-by-N rectangular
[ B2 ] <- L-by-N upper trapezoidal.

The upper trapezoidal matrix B2 consists of the first L rows of a
N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is upper triangular.

The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ] <- upper triangular N-by-N
[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ] <- identity, N-by-N
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,

V = [ V1 ] <- (M-L)-by-N rectangular
[ V2 ] <- L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)’s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W * T * W**T

where WˆH is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.

subroutine dtprfs (character UPLO, character TRANS, character DIAG, integerN, integer NRHS, double precision, dimension( * ) AP, double precision,dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * )X, integer LDX, double precision, dimension( * ) FERR, double precision,dimension( * ) BERR, double precision, dimension( * ) WORK, integer,dimension( * ) IWORK, integer INFO)

DTPRFS

Purpose:

DTPRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular packed
coefficient matrix.

The solution matrix X must be computed by DTPTRS or some other
means before entering this routine. DTPRFS does not do iterative
refinement because doing so cannot improve the backward error.

Parameters

UPLO

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

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

NRHS

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

AP 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
If DIAG = ’U’, the diagonal elements of A are not referenced
and are assumed to be 1.

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

LDB

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

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

LDX

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

FERR

BERR

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtptri (character UPLO, character DIAG, integer N, doubleprecision, dimension( * ) AP, integer INFO)

DTPTRI

Purpose:

DTPTRI computes the inverse of a real upper or lower triangular
matrix A stored in packed format.

Parameters

UPLO

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

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the upper or lower triangular matrix A, stored
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = ’U’, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ’L’, AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.
On exit, the (triangular) inverse of the original matrix, in
the same packed storage format.

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

A triangular matrix A can be transferred to packed storage using one
of the following program segments:

UPLO = ’U’: UPLO = ’L’:

JC = 1 JC = 1
DO 2 J = 1, N DO 2 J = 1, N
DO 1 I = 1, J DO 1 I = J, N
AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)
1 CONTINUE 1 CONTINUE
JC = JC + J JC = JC + N - J + 1
2 CONTINUE 2 CONTINUE

subroutine dtptrs (character UPLO, character TRANS, character DIAG, integerN, integer NRHS, double precision, dimension( * ) AP, double precision,dimension( ldb, * ) B, integer LDB, integer INFO)

DTPTRS

Purpose:

DTPTRS solves a triangular system of the form

A * X = B or A**T * X = B,

where A is a triangular matrix of order N stored in packed format,
and B is an N-by-NRHS matrix. A check is made to verify that A is
nonsingular.

Parameters

UPLO

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

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

NRHS

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

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.

LDB

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

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtpttf (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) AP, double precision, dimension( 0: * ) ARF,integer INFO)

DTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).

Purpose:

DTPTTF copies a triangular matrix A from standard packed format (TP)
to rectangular full packed format (TF).

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: ARF in Normal format is wanted;
= ’T’: ARF in Conjugate-transpose format is wanted.

UPLO

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

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

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

ARF

ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
On exit, the upper or lower triangular matrix A stored in
RFP format. For a further discussion see Notes below.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

subroutine dtpttr (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( lda, ) A, integer LDA, integer INFO)

DTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).

Purpose:

DTPTTR copies a triangular matrix A from standard packed format (TP)
to standard full format (TR).

Parameters

UPLO

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

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

A is DOUBLE PRECISION array, dimension ( LDA, N )
On exit, the triangular matrix A. If UPLO = ’U’, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

LDA

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrcon (character NORM, character UPLO, character DIAG, integer N,double precision, dimension( lda, * ) A, integer LDA, double precisionRCOND, double precision, dimension( * ) WORK, integer, dimension( * )IWORK, integer INFO)

DTRCON

Purpose:

DTRCON estimates the reciprocal of the condition number of a
triangular matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for
norm(inv(A)), then the reciprocal of the condition number is
computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters

NORM

NORM is CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.

UPLO

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

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

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).

RCOND

RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrevc (character SIDE, character HOWMNY, logical, dimension( * )SELECT, integer N, double precision, dimension( ldt, * ) T, integer LDT,double precision, dimension( ldvl, * ) VL, integer LDVL, double precision,dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, doubleprecision, dimension( * ) WORK, integer INFO)

DTREVC

Purpose:

DTREVC computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x, (y**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.

This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.

Parameters

SIDE

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

HOWMNY

HOWMNY is CHARACTER*1
= ’A’: compute all right and/or left eigenvectors;
= ’B’: compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= ’S’: compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.

SELECT

SELECT is LOGICAL array, dimension (N)
If HOWMNY = ’S’, SELECT specifies the eigenvectors to be
computed.
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
.FALSE..
Not referenced if HOWMNY = ’A’ or ’B’.

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

T is DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.

LDT

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

VL is DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = ’L’ or ’B’ and HOWMNY = ’B’, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by DHSEQR).
On exit, if SIDE = ’L’ or ’B’, VL contains:
if HOWMNY = ’A’, the matrix Y of left eigenvectors of T;
if HOWMNY = ’B’, the matrix Q*Y;
if HOWMNY = ’S’, the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = ’R’.

LDVL

LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1, and if
SIDE = ’L’ or ’B’, LDVL >= N.

VR is DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = ’R’ or ’B’ and HOWMNY = ’B’, VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by DHSEQR).
On exit, if SIDE = ’R’ or ’B’, VR contains:
if HOWMNY = ’A’, the matrix X of right eigenvectors of T;
if HOWMNY = ’B’, the matrix Q*X;
if HOWMNY = ’S’, the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = ’L’.

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = ’R’ or ’B’, LDVR >= N.

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

M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = ’A’ or ’B’, M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

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

subroutine dtrevc3 (character SIDE, character HOWMNY, logical, dimension( * )SELECT, integer N, double precision, dimension( ldt, * ) T, integer LDT,double precision, dimension( ldvl, * ) VL, integer LDVL, double precision,dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, doubleprecision, dimension( * ) WORK, integer LWORK, integer INFO)

DTREVC3

Purpose:

DTREVC3 computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x, (y**T)*T = w*(y**T)

where y**T denotes the transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.

This uses a Level 3 BLAS version of the back transformation.

Parameters

SIDE

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

HOWMNY

SELECT

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

T is DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.

LDT

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

LDVL

LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = ’L’ or ’B’, LDVL >= N.

LDVR

LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = ’R’ or ’B’, LDVR >= N.

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,3*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

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

subroutine dtrexc (character COMPQ, integer N, double precision, dimension(ldt, * ) T, integer LDT, double precision, dimension( ldq, * ) Q, integerLDQ, integer IFST, integer ILST, double precision, dimension( * ) WORK,integer INFO)

DTREXC

Purpose:

DTREXC reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
moved to row ILST.

The real Schur form T is reordered by an orthogonal similarity
transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
is updated by postmultiplying it with Z.

T must be in Schur canonical form (as returned by DHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.

Parameters

COMPQ

COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.

N is INTEGER
The order of the matrix T. N >= 0.
If N == 0 arguments ILST and IFST may be any value.

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

LDT

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

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
orthogonal transformation matrix Z which reorders T.
If COMPQ = ’N’, Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1, and if
COMPQ = ’V’, LDQ >= max(1,N).

IFST

IFST is INTEGER

ILST

ILST is INTEGER

Specify the reordering of the diagonal blocks of T.
The block with row index IFST is moved to row ILST, by a
sequence of transpositions between adjacent blocks.
On exit, if IFST pointed on entry to the second row of a
2-by-2 block, it is changed to point to the first row; ILST
always points to the first row of the block in its final
position (which may differ from its input value by +1 or -1).
1 <= IFST <= N; 1 <= ILST <= N.

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: two adjacent blocks were too close to swap (the problem
is very ill-conditioned); T may have been partially
reordered, and ILST points to the first row of the
current position of the block being moved.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrrfs (character UPLO, character TRANS, character DIAG, integerN, integer NRHS, double precision, dimension( lda, * ) A, integer LDA,double precision, dimension( ldb, * ) B, integer LDB, double precision,dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR,double precision, dimension( * ) BERR, double precision, dimension( * )WORK, integer, dimension( * ) IWORK, integer INFO)

DTRRFS

Purpose:

DTRRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular
coefficient matrix.

The solution matrix X must be computed by DTRTRS or some other
means before entering this routine. DTRRFS does not do iterative
refinement because doing so cannot improve the backward error.

Parameters

UPLO

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

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

NRHS

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

LDA

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

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

LDB

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

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

LDX

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

FERR

BERR

WORK

WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

IWORK is INTEGER array, dimension (N)

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrsen (character JOB, character COMPQ, logical, dimension( * )SELECT, integer N, double precision, dimension( ldt, * ) T, integer LDT,double precision, dimension( ldq, * ) Q, integer LDQ, double precision,dimension( * ) WR, double precision, dimension( * ) WI, integer M, doubleprecision S, double precision SEP, double precision, dimension( * ) WORK,integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DTRSEN

Purpose:

DTRSEN reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
the leading diagonal blocks of the upper quasi-triangular matrix T,
and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= ’N’: none;
= ’E’: for eigenvalues only (S);
= ’V’: for invariant subspace only (SEP);
= ’B’: for both eigenvalues and invariant subspace (S and
SEP).

COMPQ

COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.

SELECT

SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.

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

T is DOUBLE PRECISION array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, T is overwritten by the reordered matrix T, again in
Schur canonical form, with the selected eigenvalues in the
leading diagonal blocks.

LDT

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

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
orthogonal transformation matrix which reorders T; the
leading M columns of Q form an orthonormal basis for the
specified invariant subspace.
If COMPQ = ’N’, Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.

WR is DOUBLE PRECISION array, dimension (N)

WI is DOUBLE PRECISION array, dimension (N)

The real and imaginary parts, respectively, of the reordered
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T, with WR(i) = T(i,i) and, if
T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
WI(i+1) = -WI(i). Note that if a complex eigenvalue is
sufficiently ill-conditioned, then its value may differ
significantly from its value before reordering.

M is INTEGER
The dimension of the specified invariant subspace.
0 < = M <= N.

S is DOUBLE PRECISION
If JOB = ’E’ or ’B’, S is a lower bound on the reciprocal
condition number for the selected cluster of eigenvalues.
S cannot underestimate the true reciprocal condition number
by more than a factor of sqrt(N). If M = 0 or N, S = 1.
If JOB = ’N’ or ’V’, S is not referenced.

SEP

SEP is DOUBLE PRECISION
If JOB = ’V’ or ’B’, SEP is the estimated reciprocal
condition number of the specified invariant subspace. If
M = 0 or N, SEP = norm(T).
If JOB = ’N’ or ’E’, SEP is not referenced.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK.
If JOB = ’N’, LWORK >= max(1,N);
if JOB = ’E’, LWORK >= max(1,M*(N-M));
if JOB = ’V’ or ’B’, LWORK >= max(1,2*M*(N-M)).

IWORK

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

LIWORK

LIWORK is INTEGER
The dimension of the array IWORK.
If JOB = ’N’ or ’E’, LIWORK >= 1;
if JOB = ’V’ or ’B’, LIWORK >= max(1,M*(N-M)).

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very ill-conditioned);
T may have been partially reordered, and WR and WI
contain the eigenvalues in the same order as in T; S and
SEP (if requested) are set to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

DTRSEN first collects the selected eigenvalues by computing an
orthogonal transformation Z to move them to the top left corner of T.
In other words, the selected eigenvalues are the eigenvalues of T11
in:

Z**T * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2

where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.

If T has been obtained from the real Schur factorization of a matrix
A = Q*T*Q**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
the corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that

P = ( I R ) n1
( 0 0 ) n2
n1 n2

is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:

T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).

An approximate error bound for the computed average of the
eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix

C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is

EPS * norm(T) / SEP

subroutine dtrsna (character JOB, character HOWMNY, logical, dimension( * )SELECT, integer N, double precision, dimension( ldt, * ) T, integer LDT,double precision, dimension( ldvl, * ) VL, integer LDVL, double precision,dimension( ldvr, * ) VR, integer LDVR, double precision, dimension( * ) S,double precision, dimension( * ) SEP, integer MM, integer M, doubleprecision, dimension( ldwork, * ) WORK, integer LDWORK, integer, dimension(* ) IWORK, integer INFO)

DTRSNA

Purpose:

DTRSNA estimates reciprocal condition numbers for specified
eigenvalues and/or right eigenvectors of a real upper
quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
orthogonal).

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (SEP):
= ’E’: for eigenvalues only (S);
= ’V’: for eigenvectors only (SEP);
= ’B’: for both eigenvalues and eigenvectors (S and SEP).

HOWMNY

HOWMNY is CHARACTER*1
= ’A’: compute condition numbers for all eigenpairs;
= ’S’: compute condition numbers for selected eigenpairs
specified by the array SELECT.

SELECT

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

T is DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canonical form.

LDT

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

VL is DOUBLE PRECISION array, dimension (LDVL,M)
If JOB = ’E’ or ’B’, VL must contain left eigenvectors of T
(or of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VL, as returned by
DHSEIN or DTREVC.
If JOB = ’V’, VL is not referenced.

LDVL

LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = ’E’ or ’B’, LDVL >= N.

VR is DOUBLE PRECISION array, dimension (LDVR,M)
If JOB = ’E’ or ’B’, VR must contain right eigenvectors of T
(or of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VR, as returned by
DHSEIN or DTREVC.
If JOB = ’V’, VR is not referenced.

LDVR

LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = ’E’ or ’B’, LDVR >= N.

S is DOUBLE PRECISION array, dimension (MM)
If JOB = ’E’ or ’B’, the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus
S(j), SEP(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = ’V’, S is not referenced.

SEP

SEP is DOUBLE PRECISION array, dimension (MM)
If JOB = ’V’ or ’B’, the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of SEP are set to the same value. If
the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
is set to 0; this can only occur when the true value would be
very small anyway.
If JOB = ’E’, SEP is not referenced.

MM is INTEGER
The number of elements in the arrays S (if JOB = ’E’ or ’B’)
and/or SEP (if JOB = ’V’ or ’B’). MM >= M.

M is INTEGER
The number of elements of the arrays S and/or SEP actually
used to store the estimated condition numbers.
If HOWMNY = ’A’, M is set to N.

WORK

WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
If JOB = ’E’, WORK is not referenced.

LDWORK

LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = ’V’ or ’B’, LDWORK >= N.

IWORK

IWORK is INTEGER array, dimension (2*(N-1))
If JOB = ’E’, IWORK is not referenced.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The reciprocal of the condition number of an eigenvalue lambda is
defined as

S(lambda) = |v**T*u| / (norm(u)*norm(v))

where u and v are the right and left eigenvectors of T corresponding
to lambda; v**T denotes the transpose of v, and norm(u)
denotes the Euclidean norm. These reciprocal condition numbers always
lie between zero (very badly conditioned) and one (very well
conditioned). If n = 1, S(lambda) is defined to be 1.

An approximate error bound for a computed eigenvalue W(i) is given by

EPS * norm(T) / S(i)

where EPS is the machine precision.

The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose

T = ( lambda c )
( 0 T22 )

Then the reciprocal condition number is

SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )

where sigma-min denotes the smallest singular value. We approximate
the smallest singular value by the reciprocal of an estimate of the
one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
defined to be abs(T(1,1)).

An approximate error bound for a computed right eigenvector VR(i)
is given by

EPS * norm(T) / SEP(i)

subroutine dtrti2 (character UPLO, character DIAG, integer N, doubleprecision, dimension( lda, * ) A, integer LDA, integer INFO)

DTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).

Purpose:

DTRTI2 computes the inverse of a real upper or lower triangular
matrix.

This is the Level 2 BLAS version of the algorithm.

Parameters

UPLO

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

DIAG

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

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

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = ’U’, the
leading n by n upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading n by n lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If DIAG = ’U’, the
diagonal elements of A are also not referenced and are
assumed to be 1.

On exit, the (triangular) inverse of the original matrix, in
the same storage format.

LDA

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrtri (character UPLO, character DIAG, integer N, doubleprecision, dimension( lda, * ) A, integer LDA, integer INFO)

DTRTRI

Purpose:

DTRTRI computes the inverse of a real upper or lower triangular
matrix A.

This is the Level 3 BLAS version of the algorithm.

Parameters

UPLO

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

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If DIAG = ’U’, the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.

LDA

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

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrtrs (character UPLO, character TRANS, character DIAG, integerN, integer NRHS, double precision, dimension( lda, * ) A, integer LDA,double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DTRTRS

Purpose:

DTRTRS solves a triangular system of the form

A * X = B or A**T * X = B,

where A is a triangular matrix of order N, and B is an N-by-NRHS
matrix. A check is made to verify that A is nonsingular.

Parameters

UPLO

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

TRANS

TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)

DIAG

DIAG is CHARACTER*1
= ’N’: A is non-unit triangular;
= ’U’: A is unit triangular.

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

NRHS

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

LDA

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

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.

LDB

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

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtrttf (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: lda-1, 0: * ) A, integer LDA, double precision,dimension( 0: * ) ARF, integer INFO)

DTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).

Purpose:

DTRTTF copies a triangular matrix A from standard full format (TR)
to rectangular full packed format (TF) .

Parameters

TRANSR

TRANSR is CHARACTER*1
= ’N’: ARF in Normal form is wanted;
= ’T’: ARF in Transpose form is wanted.

UPLO

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

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

A is DOUBLE PRECISION array, dimension (LDA,N).
On entry, the triangular matrix A. If UPLO = ’U’, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced.

LDA

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

ARF

ARF is DOUBLE PRECISION array, dimension (NT).
NT=N*(N+1)/2. On exit, the triangular matrix A in RFP format.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper AP is Lower

00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55

RFP A RFP A

03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper AP is Lower

00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44

RFP A RFP A

02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42

Now let TRANSR = ’T’. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A RFP A

02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52

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

DTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).

Purpose:

DTRTTP copies a triangular matrix A from full format (TR) to standard
packed format (TP).

Parameters

UPLO

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

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

A is DOUBLE PRECISION array, dimension (LDA,N)
On exit, the triangular matrix A. If UPLO = ’U’, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = ’L’, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

LDA

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

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

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dtzrzf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

DTZRZF

Purpose:

DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
to upper triangular form by means of orthogonal transformations.

The upper trapezoidal matrix A is factored as

A = ( R 0 ) * Z,

where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
triangular matrix.

Parameters

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

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

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

LDA

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

TAU

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

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Further Details:

The N-by-N matrix Z can be computed by

Z = Z(1)*Z(2)* ... *Z(M)

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**T

with v(k) is the kth row vector of the M-by-N matrix

V = ( I A(:,M+1:N) )

I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from DTZRZF,
and tau(k) is the kth element of the array TAU.

subroutine stplqt (integer M, integer N, integer L, integer MB, real,dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integerLDB, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK,integer INFO)

STPLQT

Purpose:

STPLQT computes a blocked LQ factorization of a real
’triangular-pentagonal’ matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.

Parameters

M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.

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

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

MB is INTEGER
The block size to be used in the blocked QR. M >= MB >= 1.

A is REAL array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

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

B is REAL array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).

T is REAL array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

WORK

WORK is REAL array, dimension (MB*M)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

The rows of V represent the vectors which define the H(i)’s.

T = [T1 T2 ... TB].

subroutine stplqt2 (integer M, integer N, integer L, real, dimension( lda, ) A, integer LDA, real, dimension( ldb, ) B, integer LDB, real,dimension( ldt, * ) T, integer LDT, integer INFO)

STPLQT2 computes a LQ factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

STPLQT2 computes a LQ a factorization of a real ’triangular-pentagonal’
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.

Parameters

M is INTEGER
The total number of rows of the matrix B.
M >= 0.

N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

A is REAL array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).

T is REAL array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,M)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ][ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:

B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C

C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,

W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)’s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W**T * T * W

where WˆH is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.

subroutine stpmlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer L, integer MB, real, dimension( ldv, * ) V, integer LDV,real, dimension( ldt, * ) T, integer LDT, real, dimension( lda, * ) A,integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * )WORK, integer INFO)

STPMLQT

Purpose:

STPMLQT applies a real orthogonal matrix Q obtained from a
’triangular-pentagonal’ real block reflector H to a general
real matrix C, which consists of two blocks A and B.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**T from the Left;
= ’R’: apply Q or Q**T from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’T’: Transpose, apply Q**T.

M is INTEGER
The number of rows of the matrix B. M >= 0.

N is INTEGER
The number of columns of the matrix B. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.

L is INTEGER
The order of the trapezoidal part of V.
K >= L >= 0. See Further Details.

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in STPLQT.

V is REAL array, dimension (LDV,K)
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
STPLQT in B. See Further Details.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= K.

T is REAL array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by STPLQT, stored as a MB-by-K matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

A is REAL array, dimension
(LDA,N) if SIDE = ’L’ or
(LDA,K) if SIDE = ’R’
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
Q*C or Q**T*C or C*Q or C*Q**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 is REAL array, dimension (LDB,N)
On entry, the M-by-N matrix B.
On exit, B is overwritten by the corresponding block of
Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.

LDB

LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).

WORK

WORK is REAL array. The dimension of WORK is
N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:

V = [V1] [V2].

If SIDE = ’L’: C = [A] where A is K-by-N, B is M-by-N and V is K-by-M.
[B]

If SIDE = ’R’: C = [A B] where A is M-by-K, B is M-by-N and V is K-by-N.

The real orthogonal matrix Q is formed from V and T.

If TRANS=’N’ and SIDE=’L’, C is on exit replaced with Q * C.

If TRANS=’T’ and SIDE=’L’, C is on exit replaced with Q**T * C.

If TRANS=’N’ and SIDE=’R’, C is on exit replaced with C * Q.

If TRANS=’T’ and SIDE=’R’, C is on exit replaced with C * Q**T.

subroutine ztplqt (integer M, integer N, integer L, integer MB, complex16,dimension( lda, ) A, integer LDA, complex16, dimension( ldb, ) B,integer LDB, complex16, dimension( ldt, ) T, integer LDT, complex16,dimension( ) WORK, integer INFO)

ZTPLQT

Purpose:

ZTPLQT computes a blocked LQ factorization of a complex
’triangular-pentagonal’ matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.

Parameters

M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.

N is INTEGER
The number of columns of the matrix B.
N >= 0.

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

MB is INTEGER
The block size to be used in the blocked QR. M >= MB >= 1.

A is COMPLEX*16 array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B is COMPLEX*16 array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).

T is COMPLEX*16 array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See Further Details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

WORK

WORK is COMPLEX*16 array, dimension (MB*M)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

The rows of V represent the vectors which define the H(i)’s.

T = [T1 T2 ... TB].

subroutine ztplqt2 (integer M, integer N, integer L, complex16, dimension(lda, ) A, integer LDA, complex16, dimension( ldb, ) B, integer LDB,complex16, dimension( ldt, ) T, integer LDT, integer INFO)

ZTPLQT2 computes a LQ factorization of a real or complex ’triangular-pentagonal’ matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

ZTPLQT2 computes a LQ a factorization of a complex ’triangular-pentagonal’
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.

Parameters

M is INTEGER
The total number of rows of the matrix B.
M >= 0.

N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.

L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.

A is COMPLEX*16 array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

LDB

LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).

T is COMPLEX*16 array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,M)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The input matrix C is a M-by-(M+N) matrix

C = [ A ][ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:

B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C

C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,

W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)’s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W**T * T * W

where WˆH is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.

subroutine ztpmlqt (character SIDE, character TRANS, integer M, integer N,integer K, integer L, integer MB, complex16, dimension( ldv, ) V,integer LDV, complex16, dimension( ldt, ) T, integer LDT, complex16,dimension( lda, ) A, integer LDA, complex16, dimension( ldb, ) B,integer LDB, complex16, dimension( ) WORK, integer INFO)

ZTPMLQT

Purpose:

ZTPMLQT applies a complex unitary matrix Q obtained from a
’triangular-pentagonal’ complex block reflector H to a general
complex matrix C, which consists of two blocks A and B.

Parameters

SIDE

SIDE is CHARACTER*1
= ’L’: apply Q or Q**H from the Left;
= ’R’: apply Q or Q**H from the Right.

TRANS

TRANS is CHARACTER*1
= ’N’: No transpose, apply Q;
= ’C’: Conjugate transpose, apply Q**H.

M is INTEGER
The number of rows of the matrix B. M >= 0.

N is INTEGER
The number of columns of the matrix B. N >= 0.

K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.

L is INTEGER
The order of the trapezoidal part of V.
K >= L >= 0. See Further Details.

MB is INTEGER
The block size used for the storage of T. K >= MB >= 1.
This must be the same value of MB used to generate T
in ZTPLQT.

V is COMPLEX*16 array, dimension (LDV,K)
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
ZTPLQT in B. See Further Details.

LDV

LDV is INTEGER
The leading dimension of the array V. LDV >= K.

T is COMPLEX*16 array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by ZTPLQT, stored as a MB-by-K matrix.

LDT

LDT is INTEGER
The leading dimension of the array T. LDT >= MB.

A is COMPLEX*16 array, dimension
(LDA,N) if SIDE = ’L’ or
(LDA,K) if SIDE = ’R’
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A.
If SIDE = ’L’, LDA >= max(1,K);
If SIDE = ’R’, LDA >= max(1,M).

B is COMPLEX*16 array, dimension (LDB,N)
On entry, the M-by-N matrix B.
On exit, B is overwritten by the corresponding block of
Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

LDB

LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).

WORK

WORK is COMPLEX*16 array. The dimension of WORK is
N*MB if SIDE = ’L’, or M*MB if SIDE = ’R’.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:

V = [V1] [V2].

If SIDE = ’L’: C = [A] where A is K-by-N, B is M-by-N and V is K-by-M.
[B]

If SIDE = ’R’: C = [A B] where A is M-by-K, B is M-by-N and V is K-by-N.

The complex unitary matrix Q is formed from V and T.

If TRANS=’N’ and SIDE=’L’, C is on exit replaced with Q * C.

If TRANS=’C’ and SIDE=’L’, C is on exit replaced with Q**H * C.

If TRANS=’N’ and SIDE=’R’, C is on exit replaced with C * Q.

If TRANS=’C’ and SIDE=’R’, C is on exit replaced with C * Q**H.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Description

doubleOTHERcomputational

NAME

SYNOPSIS

Functions

Detailed Description

Function Documentation

subroutine ctplqt (integer M, integer N, integer L, integer MB, complex,dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integerLDB, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( * )WORK, integer INFO)

subroutine ctplqt2 (integer M, integer N, integer L, complex, dimension( lda,* ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex,dimension( ldt, * ) T, integer LDT, integer INFO)

subroutine dgetsqrhrt (integer M, integer N, integer MB1, integer NB1,integer NB2, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( ldt, * ) T, integer LDT, double precision, dimension(* ) WORK, integer LWORK, integer INFO)

subroutine dla_lin_berr (integer N, integer NZ, integer NRHS, doubleprecision, dimension( n, nrhs ) RES, double precision, dimension( n, nrhs )AYB, double precision, dimension( nrhs ) BERR)

subroutine dla_wwaddw (integer N, double precision, dimension( * ) X, doubleprecision, dimension( * ) Y, double precision, dimension( * ) W)

double precision function dlansf (character NORM, character TRANSR, characterUPLO, integer N, double precision, dimension( 0: * ) A, double precision,dimension( 0: * ) WORK)

subroutine dlarscl2 (integer M, integer N, double precision, dimension( * )D, double precision, dimension( ldx, * ) X, integer LDX)

subroutine dlarz (character SIDE, integer M, integer N, integer L, doubleprecision, dimension( * ) V, integer INCV, double precision TAU, doubleprecision, dimension( ldc, * ) C, integer LDC, double precision, dimension(* ) WORK)

subroutine dlarzt (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)

subroutine dlascl2 (integer M, integer N, double precision, dimension( * ) D,double precision, dimension( ldx, * ) X, integer LDX)

subroutine dlatrz (integer M, integer N, integer L, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK)

subroutine dopgtr (character UPLO, integer N, double precision, dimension( *) AP, double precision, dimension( * ) TAU, double precision, dimension(ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK, integerINFO)

subroutine dopmtr (character SIDE, character UPLO, character TRANS, integerM, integer N, double precision, dimension( * ) AP, double precision,dimension( * ) TAU, double precision, dimension( ldc, * ) C, integer LDC,double precision, dimension( * ) WORK, integer INFO)

subroutine dorg2l (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

subroutine dorg2r (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

subroutine dorghr (integer N, integer ILO, integer IHI, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dorgl2 (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

subroutine dorglq (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dorgql (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dorgqr (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dorgr2 (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer INFO)

subroutine dorgrq (integer M, integer N, integer K, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU,double precision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dorgtr (character UPLO, integer N, double precision, dimension(lda, * ) A, integer LDA, double precision, dimension( * ) TAU, doubleprecision, dimension( * ) WORK, integer LWORK, integer INFO)

subroutine dorgtsqr (integer M, integer N, integer MB, integer NB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integerLWORK, integer INFO)

subroutine dorgtsqr_row (integer M, integer N, integer MB, integer NB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integerLWORK, integer INFO)

subroutine dorhr_col (integer M, integer N, integer NB, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * )T, integer LDT, double precision, dimension( * ) D, integer INFO)

subroutine dorm2l (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

subroutine dorm2r (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

subroutine dorml2 (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

subroutine dormlq (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

subroutine dormql (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

subroutine dormqr (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

subroutine dormr2 (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer INFO)

subroutine dormr3 (character SIDE, character TRANS, integer M, integer N,integer K, integer L, double precision, dimension( lda, * ) A, integer LDA,double precision, dimension( * ) TAU, double precision, dimension( ldc, * )C, integer LDC, double precision, dimension( * ) WORK, integer INFO)

subroutine dormrq (character SIDE, character TRANS, integer M, integer N,integer K, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

subroutine dormtr (character SIDE, character UPLO, character TRANS, integerM, integer N, double precision, dimension( lda, * ) A, integer LDA, doubleprecision, dimension( * ) TAU, double precision, dimension( ldc, * ) C,integer LDC, double precision, dimension( * ) WORK, integer LWORK, integerINFO)

subroutine dpbcon (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, double precision ANORM, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

subroutine dpbequ (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S,double precision SCOND, double precision AMAX, integer INFO)

subroutine dpbstf (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, integer INFO)

subroutine dpbtf2 (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, integer INFO)

subroutine dpbtrf (character UPLO, integer N, integer KD, double precision,dimension( ldab, * ) AB, integer LDAB, integer INFO)

subroutine dpbtrs (character UPLO, integer N, integer KD, integer NRHS,double precision, dimension( ldab, * ) AB, integer LDAB, double precision,dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine dpftrf (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) A, integer INFO)

subroutine dpftri (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) A, integer INFO)

subroutine dpftrs (character TRANSR, character UPLO, integer N, integer NRHS,double precision, dimension( 0: * ) A, double precision, dimension( ldb, *) B, integer LDB, integer INFO)

subroutine dppcon (character UPLO, integer N, double precision, dimension( *) AP, double precision ANORM, double precision RCOND, double precision,dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

subroutine dppequ (character UPLO, integer N, double precision, dimension( *) AP, double precision, dimension( * ) S, double precision SCOND, doubleprecision AMAX, integer INFO)

subroutine dpptrf (character UPLO, integer N, double precision, dimension( *) AP, integer INFO)

subroutine dpptri (character UPLO, integer N, double precision, dimension( *) AP, integer INFO)

subroutine dpptrs (character UPLO, integer N, integer NRHS, double precision,dimension( * ) AP, double precision, dimension( ldb, * ) B, integer LDB,integer INFO)

subroutine dpstf2 (character UPLO, integer N, double precision, dimension(lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, doubleprecision TOL, double precision, dimension( 2*n ) WORK, integer INFO)

subroutine dpstrf (character UPLO, integer N, double precision, dimension(lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, doubleprecision TOL, double precision, dimension( 2*n ) WORK, integer INFO)

subroutine dsfrk (character TRANSR, character UPLO, character TRANS, integerN, integer K, double precision ALPHA, double precision, dimension( lda, * )A, integer LDA, double precision BETA, double precision, dimension( * ) C)

subroutine dspcon (character UPLO, integer N, double precision, dimension( *) AP, integer, dimension( * ) IPIV, double precision ANORM, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

subroutine dspgst (integer ITYPE, character UPLO, integer N, doubleprecision, dimension( * ) AP, double precision, dimension( * ) BP, integerINFO)

subroutine dsptrd (character UPLO, integer N, double precision, dimension( *) AP, double precision, dimension( * ) D, double precision, dimension( * )E, double precision, dimension( * ) TAU, integer INFO)

subroutine dsptrf (character UPLO, integer N, double precision, dimension( *) AP, integer, dimension( * ) IPIV, integer INFO)

subroutine dsptri (character UPLO, integer N, double precision, dimension( *) AP, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK,integer INFO)

subroutine dsptrs (character UPLO, integer N, integer NRHS, double precision,dimension( * ) AP, integer, dimension( * ) IPIV, double precision,dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine dtbcon (character NORM, character UPLO, character DIAG, integer N,integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

subroutine dtbtrs (character UPLO, character TRANS, character DIAG, integerN, integer KD, integer NRHS, double precision, dimension( ldab, * ) AB,integer LDAB, double precision, dimension( ldb, * ) B, integer LDB, integerINFO)

subroutine dtfsm (character TRANSR, character SIDE, character UPLO, characterTRANS, character DIAG, integer M, integer N, double precision ALPHA, doubleprecision, dimension( 0: * ) A, double precision, dimension( 0: ldb-1, 0: *) B, integer LDB)

subroutine dtftri (character TRANSR, character UPLO, character DIAG, integerN, double precision, dimension( 0: * ) A, integer INFO)

subroutine dtfttp (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) ARF, double precision, dimension( 0: * ) AP,integer INFO)

subroutine dtfttr (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) ARF, double precision, dimension( 0: lda-1, 0:* ) A, integer LDA, integer INFO)

subroutine dtpcon (character NORM, character UPLO, character DIAG, integer N,double precision, dimension( * ) AP, double precision RCOND, doubleprecision, dimension( * ) WORK, integer, dimension( * ) IWORK, integerINFO)

subroutine dtplqt (integer M, integer N, integer L, integer MB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integerLDT, double precision, dimension( * ) WORK, integer INFO)

subroutine dtplqt2 (integer M, integer N, integer L, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * )B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT,integer INFO)

subroutine dtpqrt (integer M, integer N, integer L, integer NB, doubleprecision, dimension( lda, * ) A, integer LDA, double precision, dimension(ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integerLDT, double precision, dimension( * ) WORK, integer INFO)

subroutine dtpqrt2 (integer M, integer N, integer L, double precision,dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * )B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT,integer INFO)

subroutine dtptri (character UPLO, character DIAG, integer N, doubleprecision, dimension( * ) AP, integer INFO)

subroutine dtptrs (character UPLO, character TRANS, character DIAG, integerN, integer NRHS, double precision, dimension( * ) AP, double precision,dimension( ldb, * ) B, integer LDB, integer INFO)

subroutine dtpttf (character TRANSR, character UPLO, integer N, doubleprecision, dimension( 0: * ) AP, double precision, dimension( 0: * ) ARF,integer INFO)

subroutine dopgtr (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( ) TAU, double precision, dimension(ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK, integerINFO)

subroutine dppcon (character UPLO, integer N, double precision, dimension( ) AP, double precision ANORM, double precision RCOND, double precision,dimension( ) WORK, integer, dimension( * ) IWORK, integer INFO)

subroutine dppequ (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( ) S, double precision SCOND, doubleprecision AMAX, integer INFO)

subroutine dspcon (character UPLO, integer N, double precision, dimension( ) AP, integer, dimension( ) IPIV, double precision ANORM, doubleprecision RCOND, double precision, dimension( * ) WORK, integer, dimension(* ) IWORK, integer INFO)

subroutine dsptrd (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( ) D, double precision, dimension( * )E, double precision, dimension( * ) TAU, integer INFO)

subroutine dsptrf (character UPLO, integer N, double precision, dimension( ) AP, integer, dimension( ) IPIV, integer INFO)

subroutine dsptri (character UPLO, integer N, double precision, dimension( ) AP, integer, dimension( ) IPIV, double precision, dimension( * ) WORK,integer INFO)

subroutine dtpttr (character UPLO, integer N, double precision, dimension( ) AP, double precision, dimension( lda, ) A, integer LDA, integer INFO)

subroutine dtzrzf (integer M, integer N, double precision, dimension( lda, ) A, integer LDA, double precision, dimension( ) TAU, double precision,dimension( * ) WORK, integer LWORK, integer INFO)

subroutine stplqt2 (integer M, integer N, integer L, real, dimension( lda, ) A, integer LDA, real, dimension( ldb, ) B, integer LDB, real,dimension( ldt, * ) T, integer LDT, integer INFO)

subroutine ztplqt (integer M, integer N, integer L, integer MB, complex16,dimension( lda, ) A, integer LDA, complex16, dimension( ldb, ) B,integer LDB, complex16, dimension( ldt, ) T, integer LDT, complex16,dimension( ) WORK, integer INFO)

subroutine ztplqt2 (integer M, integer N, integer L, complex16, dimension(lda, ) A, integer LDA, complex16, dimension( ldb, ) B, integer LDB,complex16, dimension( ldt, ) T, integer LDT, integer INFO)