## table of contents

doubleOTHERcomputational(3) | LAPACK | doubleOTHERcomputational(3) |

# NAME¶

doubleOTHERcomputational# 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

**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 vector. 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 vector. 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

**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)

**DTPMLQT**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, integer LDB, 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*

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

*N*

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

*L*

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

*MB*

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

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*N*

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

*L*

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

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 W^H 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, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integer INFO)¶

**CTPMLQT**

**Purpose:**

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

**Parameters**

*SIDE*

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

*TRANS*

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

*M*

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

*N*

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

*K*

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

*L*

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

*MB*

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*

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 DTPLQT in B. See Further Details.

*LDV*

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

*T*

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

*LDT*

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

*A*

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

*LDA*

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

*B*

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

*LDB*

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

*WORK*

WORK is COMPLEX array. The dimension of WORK is N*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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='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, double precision, 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, double precision, 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*

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

*P*

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

*Q*

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

*THETA*

THETA is 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*

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*

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:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dgghd3 (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ, 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*

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*

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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

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

## subroutine dgghrd (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ, 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. 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 DGGHRD reduces the original problem to generalized Hessenberg form.

**Parameters**

*COMPQ*

COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the 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*

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*

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*

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*

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*

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.

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*M*

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

*P*

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

*A*

A is 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*

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. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*P*

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

*N*

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

*A*

A is 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*

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. 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 INF0= -i, the i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, integer M, integer P, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision TOLA, 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, double precision, 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*

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

*P*

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

*N*

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

*A*

A is 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*

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*

K is INTEGER

*L*

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

*U*

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*

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*

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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 (character*1 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 precision SFMIN, 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*

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

*N*

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

*A*

A is 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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

**Contributors:**

**Bugs, Examples and Comments:**

## subroutine dgsvj1 (character*1 JOBV, integer M, integer N, integer N1, 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 precision SFMIN, 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 tunning 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 A^T * A. The block-entries (tiles) of the (1,2) off-diagonal block are marked by the [x]'s in the following scheme: | * * * [x] [x] [x]| | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks. | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block. |[x] [x] [x] * * * | |[x] [x] [x] * * * | |[x] [x] [x] * * * | In terms of the columns of A, the first N1 columns are rotated 'against' the remaining N-N1 columns, trying to increase the angle between the corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is tiled using quadratic tiles of side KBL. Here, KBL is a tunning parameter. The number of sweeps is given in NSWEEP and the orthogonality threshold is given in TOL.

**Parameters**

*JOBV*

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

*M*

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

*N*

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

*N1*

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

*A*

A is 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 N1, D, TOL and NSWEEP.)

*LDA*

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

*D*

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 N1, 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine dhsein (character SIDE, character EIGSRC, character INITV, logical, dimension( * ) SELECT, integer N, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double precision, 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*

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

*H*

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

WR is DOUBLE PRECISION array, dimension (N)

*WI*

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*

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*

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*

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

*M*

M is INTEGER The number of columns in the arrays VL and/or VR required to store the eigenvectors; 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, double precision, dimension( * ) WR, double precision, dimension( * ) WI, double precision, 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*

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*

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*

WR is DOUBLE PRECISION array, dimension (N)

*WI*

WI is DOUBLE PRECISION array, dimension (N) The real and imaginary parts, respectively, of the computed eigenvalues. 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**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, double precision, 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*

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

*NZ*

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

*NRHS*

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

*RES*

RES is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dla_wwaddw (integer N, double precision, dimension( * ) X, double precision, 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*

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

*X*

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

*Y*

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

*W*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

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

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

*NR*

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

*SQRE*

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

*NRHS*

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

*B*

B is 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*

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*

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

*K*

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

*C*

C is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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, double precision, 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*

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

*NRHS*

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

*B*

B is 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*

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*

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*

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*

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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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 precision RCOND, 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*

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

*NRHS*

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

*D*

D is 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*

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

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## double precision function dlansf (character NORM, character TRANSR, character UPLO, 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 = ( 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*

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

*A*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 vector.

**Purpose:**

DLARSCL2 performs a reciprocal diagonal scaling on an vector: 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*

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

*N*

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

*D*

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

*X*

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

*LDX*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dlarz (character SIDE, integer M, integer N, integer L, double precision, dimension( * ) V, integer INCV, double precision TAU, double precision, 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*

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

*N*

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

*L*

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

*V*

V is 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*

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

*LDC*

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

*WORK*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**Further Details:**

## subroutine dlarzb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( ldc, * ) C, integer LDC, 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*

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

*N*

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

*K*

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

*L*

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

*V*

V is 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*

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

*LDT*

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

*C*

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

*LDC*

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

*WORK*

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

*LDWORK*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**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*

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

*K*

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

*V*

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

*LDV*

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

*TAU*

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

*T*

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

*LDT*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**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 vector.

**Purpose:**

DLASCL2 performs a diagonal scaling on a vector: 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*

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

*N*

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

*D*

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

*X*

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

*LDX*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*L*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**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, integer INFO)¶

**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*

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

*AP*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dopmtr (character SIDE, character UPLO, character TRANS, integer M, 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*

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

*N*

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

*AP*

AP is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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, double precision, 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, double precision, 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*

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

*P*

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

*Q*

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

*X11*

X11 is 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. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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*

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

*P*

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

*Q*

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

*X11*

X11 is 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. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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*

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

*P*

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

*Q*

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

*X11*

X11 is 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. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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*

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

*P*

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

*Q*

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

*X11*

X11 is 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*

*TAUP2*

*TAUQ1*

*WORK*

WORK is DOUBLE PRECISION array, dimension (LWORK)

*LWORK*

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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*

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

*P*

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

*Q*

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

*X11*

*LDX11*

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

*X21*

*LDX21*

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

*THETA*

*PHI*

*TAUP1*

*TAUP2*

*TAUQ1*

*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*

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

**References:**

## subroutine dorbdb5 (integer M1, integer M2, integer N, double precision, dimension(*) X1, integer INCX1, double precision, dimension(*) X2, integer INCX2, double precision, dimension(ldq1,*) Q1, integer LDQ1, double precision, 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*

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

*M2*

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

*N*

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

*X1*

X1 is 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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dorbdb6 (integer M1, integer M2, integer N, double precision, dimension(*) X1, integer INCX1, double precision, dimension(*) X2, integer INCX2, double precision, dimension(ldq1,*) Q1, integer LDQ1, double precision, 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 columns of Q must be orthonormal. If the projection is zero according to Kahan's "twice is enough" criterion, then the zero vector is returned.

**Parameters**

*M1*

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

*M2*

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

*N*

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

*X1*

X1 is 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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## recursive subroutine dorcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, character SIGNS, integer M, integer P, integer Q, double precision, dimension( ldx11, * ) X11, integer LDX11, double precision, dimension( ldx12, * ) X12, integer LDX12, double precision, dimension( ldx21, * ) X21, integer LDX21, double precision, dimension( ldx22, * ) X22, integer LDX22, 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( ldv2t, * ) V2T, integer LDV2T, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

**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 C^2 + S^2 = I, in which R = MIN(P,M-P,Q,M-Q).

**Parameters**

*JOBU1*

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

*JOBU2*

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

*JOBV1T*

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

*JOBV2T*

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

*TRANS*

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

*SIGNS*

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

*M*

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

*P*

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

*Q*

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

*X11*

X11 is 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*

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*

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:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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, integer INFO)¶

**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 C^2 + S^2 = I, in which R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

**Parameters**

*JOBU1*

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

*JOBU2*

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

*JOBV1T*

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

*M*

M is INTEGER The number of rows in X.

*P*

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

*Q*

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

*X11*

X11 is 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*

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*

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*

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:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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*

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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

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

*A*

A is 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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dorgtr (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, 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*

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

*A*

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*

*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. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dorgtsqr (integer M, integer N, integer MB, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

**DORGTSQR**

## 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**

## subroutine dorm2l (character SIDE, character TRANS, integer M, integer N, integer K, double precision, dimension( lda, * ) A, integer LDA, double precision, 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*

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

*N*

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

*K*

K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0.

*A*

A is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dorm2r (character SIDE, character TRANS, integer M, integer N, integer K, double precision, dimension( lda, * ) A, integer LDA, double precision, 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*

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

*N*

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

*K*

K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0.

*A*

A is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dormbr (character VECT, character SIDE, character TRANS, integer M, integer N, integer K, 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)¶

**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*

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

*N*

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

*K*

K is INTEGER If VECT = 'Q', the number of columns in the original matrix reduced by DGEBRD. If VECT = 'P', the number of rows in the original matrix reduced by DGEBRD. K >= 0.

*A*

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*

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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dormhr (character SIDE, character TRANS, integer M, integer N, integer ILO, integer IHI, 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)¶

**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*

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

*N*

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

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER ILO and IHI must have the same values as in the previous call of 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*

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*

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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dorml2 (character SIDE, character TRANS, integer M, integer N, integer K, double precision, dimension( lda, * ) A, integer LDA, double precision, 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*

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

*N*

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

*K*

K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0.

*A*

A is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dormlq (character SIDE, character TRANS, integer M, integer N, integer K, 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)¶

**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*

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

*N*

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

*K*

*A*

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.

*LDA*

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

*TAU*

*C*

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

*LDC*

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

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dormql (character SIDE, character TRANS, integer M, integer N, integer K, 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)¶

**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*

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

*N*

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

*K*

*A*

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.

*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*

*C*

*LDC*

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

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dormqr (character SIDE, character TRANS, integer M, integer N, integer K, 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)¶

**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*

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

*N*

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

*K*

*A*

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.

*LDA*

*TAU*

*C*

*LDC*

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

*WORK*

*LWORK*

LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For good performance, LWORK should generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dormr2 (character SIDE, character TRANS, integer M, integer N, integer K, double precision, dimension( lda, * ) A, integer LDA, double precision, 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*

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

*N*

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

*K*

*A*

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*

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

*L*

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

*A*

A is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**Further Details:**

## subroutine dormrq (character SIDE, character TRANS, integer M, integer N, integer K, 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)¶

**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*

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

*N*

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

*K*

*A*

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.

*LDA*

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

*TAU*

*C*

*LDC*

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

*WORK*

*LWORK*

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*K*

*L*

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

*A*

A is 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*

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*

*LWORK*

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**Further Details:**

## subroutine dormtr (character SIDE, character UPLO, character TRANS, integer M, integer N, 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)¶

**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*

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

*N*

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

*A*

A is 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*

*LDC*

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

*WORK*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dpbcon (character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision ANORM, double precision 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*

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

*KD*

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

*AB*

AB is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*KD*

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

*AB*

AB is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*KD*

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

*NRHS*

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

*AB*

AB is 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 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*KD*

*AB*

AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*KD*

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

*AB*

AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A.

*LDAB*

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

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*KD*

*AB*

AB is DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A.

*LDAB*

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

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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*

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

*KD*

*NRHS*

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

*AB*

AB is 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.

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dpftrf (character TRANSR, character UPLO, integer N, double precision, 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*

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

*A*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 dpftri (character TRANSR, character UPLO, integer N, double precision, 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*

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

*A*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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*

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

*NRHS*

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

*A*

A is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## 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*

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

*AP*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dppequ (character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) S, double precision SCOND, double precision 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*

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

*AP*

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The upper or lower 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*NRHS*

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

*AP*

AP is 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.

*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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*AP*

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

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*AP*

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*NRHS*

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

*AP*

AP is 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.

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dpstf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, double precision 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*

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

*A*

A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n by n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dpstrf (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, double precision 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. 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 3 BLAS.

**Parameters**

*UPLO*

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

*N*

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

*A*

A is 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.

*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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dsbgst (character VECT, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, 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*

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

*KA*

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

*KB*

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

*AB*

AB is 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*KD*

*AB*

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

*LDAB*

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

*D*

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

*E*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

Modified by Linda Kaufman, Bell Labs.

## subroutine dsfrk (character TRANSR, character UPLO, character TRANS, integer N, 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, or 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*

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

*K*

K is INTEGER On entry with TRANS = 'N' or 'n', K specifies the number of columns of the matrix A, and on entry with TRANS = '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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dspcon (character UPLO, integer N, double precision, dimension( * ) AP, integer, dimension( * ) IPIV, double precision ANORM, double precision 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*

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

*AP*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dspgst (integer ITYPE, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, integer INFO)¶

**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*

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

*AP*

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

*BP*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**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*

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

*NRHS*

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

*AP*

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

*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*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*AP*

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

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

*E*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n-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*

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

*AP*

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

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

## 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*

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

*N*

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

*AP*

AP is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*NRHS*

*AP*

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*

*B*

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dstegr (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, 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*

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

*D*

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

*E*

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*

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*

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*

IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0. Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is DOUBLE PRECISION Unused. Was the absolute error tolerance for the eigenvalues/eigenvectors in previous versions.

*M*

M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.

*Z*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine dstein (integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, integer M, double precision, dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, double precision, 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*

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

*D*

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

*E*

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*

M is INTEGER The number of eigenvectors to be found. 0 <= M <= N.

*W*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dstemr (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision 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, logical TRYRAC, 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 L^T factorizations near clusters of close eigenvalues (referred to as RRRs, Relatively Robust Representations). An informal sketch of the algorithm follows. For each unreduced block (submatrix) of T, (a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general. (b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d). (c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy. (d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain. For more details, see: - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices," Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Further Details 1.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*

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

*N*

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

*D*

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

*E*

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*

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*

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*

IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0. Not referenced if RANGE = 'A' or 'V'.

*M*

M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.

*Z*

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*

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th computed eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is relevant in the case when the matrix is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

*TRYRAC*

TRYRAC is LOGICAL If TRYRAC = .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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine dtbcon (character NORM, character UPLO, character DIAG, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision 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*

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

*KD*

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

*AB*

AB is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtbrfs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, double precision, dimension( ldab, * ) AB, integer LDAB, 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)¶

**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*

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

*KD*

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

*NRHS*

*AB*

AB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1.

*LDAB*

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

*B*

B is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtbtrs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**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*

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

*KD*

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

*NRHS*

*AB*

AB is DOUBLE PRECISION array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first kd+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1.

*LDAB*

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

*B*

B is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtfsm (character TRANSR, character SIDE, character UPLO, character TRANS, character DIAG, integer M, integer N, double precision ALPHA, double precision, 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*

M is INTEGER On entry, M specifies the number of rows of B. M must be at least zero. Unchanged on exit.

*N*

N is INTEGER On entry, N specifies the number of columns of B. N must be at least zero. Unchanged on exit.

*ALPHA*

ALPHA is 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## subroutine dtftri (character TRANSR, character UPLO, character DIAG, integer N, 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*

*UPLO*

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

*DIAG*

DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular.

*N*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## subroutine dtfttp (character TRANSR, character UPLO, integer N, double precision, 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## subroutine dtfttr (character TRANSR, character UPLO, integer N, double precision, 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## 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, double precision, 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*

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

*A*

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*

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*

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*

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*

M is INTEGER The dimension of the specified pair of left and right eigen- spaces (deflating subspaces). 0 <= M <= N.

*PL*

PL is DOUBLE PRECISION

*PR*

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*

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

*IWORK*

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

*LIWORK*

LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2 or 4, LIWORK >= N+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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

**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, integer LDB, double precision TOLA, double precision TOLB, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldq, * ) Q, integer LDQ, 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*

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

*P*

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

*N*

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

*K*

K is INTEGER

*L*

L is INTEGER K and L specify the subblocks in the input matrices A and B: A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) of A and B, whose GSVD is going to be computed by DTGSJA. See Further Details.

*A*

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*

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*

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*

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*

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.

Date

December 2016

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*

N is INTEGER The order of the square matrix pair (A, B). N >= 0.

*A*

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*

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*

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*

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*

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*

MM is INTEGER The number of elements in the arrays S and DIF. MM >= M.

*M*

M is INTEGER The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected 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*

*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. 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 (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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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:**

**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 dtpcon (character NORM, character UPLO, character DIAG, integer N, double precision, dimension( * ) AP, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**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*

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

*AP*

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

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtplqt (integer M, integer N, integer L, integer MB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT, 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*

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

*N*

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

*L*

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

*MB*

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

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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*

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

*N*

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

*L*

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 W^H 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, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer INFO)¶

**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*

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

*N*

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

*K*

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

*L*

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

*MB*

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*

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. If SIDE = 'L', LDV >= max(1,M); if SIDE = 'R', LDV >= max(1,N).

*T*

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*

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', LDC >= max(1,K); If SIDE = 'R', LDC >= max(1,M).

*B*

B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer INFO)¶

**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*

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

*N*

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

*K*

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

*L*

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

*NB*

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

*V*

V is 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*

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.

*A*

A is DOUBLE PRECISION array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of 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', LDC >= max(1,K); If SIDE = 'R', LDC >= max(1,M).

*B*

B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT, 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*

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

*N*

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

*L*

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

*NB*

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

*A*

A is 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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*

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

*N*

*L*

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

*A*

A is 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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 W^H 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, integer N, 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*

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

*NRHS*

*AP*

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

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtptri (character UPLO, character DIAG, integer N, double precision, 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*

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

*AP*

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

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero. The triangular matrix is singular and its inverse can not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, integer N, 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*

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

*NRHS*

*AP*

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

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtpttf (character TRANSR, character UPLO, integer N, double precision, 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*

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

*AP*

AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ), On entry, the upper or lower 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## 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*

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

*AP*

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

*A*

A is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtrcon (character NORM, character UPLO, character DIAG, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision RCOND, 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*

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

*A*

A is DOUBLE PRECISION array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1.

*LDA*

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

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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, double precision, 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*

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

*T*

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*

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*

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*

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

*M*

M is INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, double precision, 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 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. This uses a Level 3 BLAS version of the back transformation.

**Parameters**

*SIDE*

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

*HOWMNY*

HOWMNY is CHARACTER*1 = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, backtransformed 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*

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

*T*

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*

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*

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*

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

*M*

M is INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.

*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. 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, integer LDQ, 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*

N is INTEGER The order of the matrix T. N >= 0. If N == 0 arguments ILST and IFST may be any value.

*T*

T is 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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtrrfs (character UPLO, character TRANS, character DIAG, integer N, 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*

*DIAG*

DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular.

*N*

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

*NRHS*

*A*

A is DOUBLE PRECISION array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1.

*LDA*

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

*B*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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, double precision 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. 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**

*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*

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

*T*

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

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*

WR is DOUBLE PRECISION array, dimension (N)

*WI*

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*

M is INTEGER The dimension of the specified invariant subspace. 0 < = M <= N.

*S*

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*

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

*IWORK*

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

*LIWORK*

LIWORK is INTEGER The dimension of the array IWORK. If JOB = 'N' or 'E', LIWORK >= 1; if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: reordering of 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, double precision, 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). 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**

*JOB*

JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (SEP); = 'B': for both eigenvalues and eigenvectors (S and SEP).

*HOWMNY*

HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT.

*SELECT*

SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the 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*

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

*T*

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*

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*

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*

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*

MM is INTEGER The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M.

*M*

M is INTEGER The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N.

*WORK*

WORK is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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, double precision, 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*

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

*A*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtrtri (character UPLO, character DIAG, integer N, double precision, 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*

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

*A*

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 = -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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtrtrs (character UPLO, character TRANS, character DIAG, integer N, 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*

*DIAG*

DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular.

*N*

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

*NRHS*

*A*

A is DOUBLE PRECISION array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1.

*LDA*

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

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dtrttf (character TRANSR, character UPLO, integer N, double precision, 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*

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

*A*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## 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*

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

*A*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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*

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

*N*

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

*A*

A is 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*

*LWORK*

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

*INFO*

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**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, integer LDB, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO)¶

**STPLQT**

**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*

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

*N*

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

*L*

*MB*

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

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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*

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

*N*

*L*

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 W^H 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)¶

**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*

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

*N*

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

*K*

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

*L*

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

*MB*

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*

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 DTPLQT in B. See Further Details.

*LDV*

*T*

T is REAL 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*

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*

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) WORK, integer INFO)¶

**ZTPLQT**

**Purpose:**

DTPLQT 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*

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

*N*

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

*L*

*MB*

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

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## subroutine ztplqt2 (integer M, integer N, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, 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*

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

*N*

*L*

*A*

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*

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*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

## subroutine ztpmlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer L, integer MB, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) WORK, integer INFO)¶

**ZTPMLQT**

**Purpose:**

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

**Parameters**

*SIDE*

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

*TRANS*

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

*M*

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

*N*

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

*K*

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

*L*

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

*MB*

*V*

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 DTPLQT in B. See Further Details.

*LDV*

*T*

T is COMPLEX*16 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*

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*

*B*

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 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='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¶

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