
NAMEcomplex16OTHERcomputational  complex16SYNOPSISFunctionssubroutine zbbcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, RWORK, LRWORK, INFO) ZBBCSD subroutine zbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO) ZBDSQR subroutine zgghd3 (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO) ZGGHD3 subroutine zgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO) ZGGHRD subroutine zggqrf (N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO) ZGGQRF subroutine zggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO) ZGGRQF subroutine zggsvp3 (JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, LWORK, INFO) ZGGSVP3 subroutine zgsvj0 (JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO) ZGSVJ0 preprocessor for the routine zgesvj. subroutine zgsvj1 (JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO) ZGSVJ1 preprocessor for the routine zgesvj, applies Jacobi rotations targeting only particular pivots. subroutine zhbgst (VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO) ZHBGST subroutine zhbtrd (VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO) ZHBTRD subroutine zhetrd_hb2st (STAGE1, VECT, UPLO, N, KD, AB, LDAB, D, E, HOUS, LHOUS, WORK, LWORK, INFO) ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T subroutine zhfrk (TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C) ZHFRK performs a Hermitian rankk operation for matrix in RFP format. subroutine zhpcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO) ZHPCON subroutine zhpgst (ITYPE, UPLO, N, AP, BP, INFO) ZHPGST subroutine zhprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZHPRFS subroutine zhptrd (UPLO, N, AP, D, E, TAU, INFO) ZHPTRD subroutine zhptrf (UPLO, N, AP, IPIV, INFO) ZHPTRF subroutine zhptri (UPLO, N, AP, IPIV, WORK, INFO) ZHPTRI subroutine zhptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO) ZHPTRS subroutine zhsein (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO) ZHSEIN subroutine zhseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO) ZHSEQR subroutine zla_lin_berr (N, NZ, NRHS, RES, AYB, BERR) ZLA_LIN_BERR computes a componentwise relative backward error. subroutine zla_wwaddw (N, X, Y, W) ZLA_WWADDW adds a vector into a doubledsingle vector. subroutine zlaed0 (QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS, RWORK, IWORK, INFO) ZLAED0 used by ZSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. subroutine zlaed7 (N, CUTPNT, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, LDQ, RHO, INDXQ, QSTORE, QPTR, PRMPTR, PERM, GIVPTR, GIVCOL, GIVNUM, WORK, RWORK, IWORK, INFO) ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rankone symmetric matrix. Used when the original matrix is dense. subroutine zlaed8 (K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO) ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. subroutine zlals0 (ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, RWORK, INFO) ZLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd. subroutine zlalsa (ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, RWORK, IWORK, INFO) ZLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd. subroutine zlalsd (UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, RWORK, IWORK, INFO) ZLALSD uses the singular value decomposition of A to solve the least squares problem. double precision function zlanhf (NORM, TRANSR, UPLO, N, A, WORK) ZLANHF returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format. subroutine zlarscl2 (M, N, D, X, LDX) ZLARSCL2 performs reciprocal diagonal scaling on a vector. subroutine zlarz (SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK) ZLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix. subroutine zlarzb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, C, LDC, WORK, LDWORK) ZLARZB applies a block reflector or its conjugatetranspose to a general matrix. subroutine zlarzt (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT) ZLARZT forms the triangular factor T of a block reflector H = I  vtvH. subroutine zlascl2 (M, N, D, X, LDX) ZLASCL2 performs diagonal scaling on a vector. subroutine zlatrz (M, N, L, A, LDA, TAU, WORK) ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations. subroutine zpbcon (UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO) ZPBCON subroutine zpbequ (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO) ZPBEQU subroutine zpbrfs (UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZPBRFS subroutine zpbstf (UPLO, N, KD, AB, LDAB, INFO) ZPBSTF subroutine zpbtf2 (UPLO, N, KD, AB, LDAB, INFO) ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm). subroutine zpbtrf (UPLO, N, KD, AB, LDAB, INFO) ZPBTRF subroutine zpbtrs (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO) ZPBTRS subroutine zpftrf (TRANSR, UPLO, N, A, INFO) ZPFTRF subroutine zpftri (TRANSR, UPLO, N, A, INFO) ZPFTRI subroutine zpftrs (TRANSR, UPLO, N, NRHS, A, B, LDB, INFO) ZPFTRS subroutine zppcon (UPLO, N, AP, ANORM, RCOND, WORK, RWORK, INFO) ZPPCON subroutine zppequ (UPLO, N, AP, S, SCOND, AMAX, INFO) ZPPEQU subroutine zpprfs (UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZPPRFS subroutine zpptrf (UPLO, N, AP, INFO) ZPPTRF subroutine zpptri (UPLO, N, AP, INFO) ZPPTRI subroutine zpptrs (UPLO, N, NRHS, AP, B, LDB, INFO) ZPPTRS subroutine zpstf2 (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO) ZPSTF2 computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix. subroutine zpstrf (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO) ZPSTRF computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix. subroutine zspcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO) ZSPCON subroutine zsprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZSPRFS subroutine zsptrf (UPLO, N, AP, IPIV, INFO) ZSPTRF subroutine zsptri (UPLO, N, AP, IPIV, WORK, INFO) ZSPTRI subroutine zsptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO) ZSPTRS subroutine zstedc (COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) ZSTEDC subroutine zstegr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) ZSTEGR subroutine zstein (N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO) ZSTEIN subroutine zstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO) ZSTEMR subroutine zsteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO) ZSTEQR subroutine ztbcon (NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, RWORK, INFO) ZTBCON subroutine ztbrfs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZTBRFS subroutine ztbtrs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO) ZTBTRS subroutine ztfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB) ZTFSM solves a matrix equation (one operand is a triangular matrix in RFP format). subroutine ztftri (TRANSR, UPLO, DIAG, N, A, INFO) ZTFTRI subroutine ztfttp (TRANSR, UPLO, N, ARF, AP, INFO) ZTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP). subroutine ztfttr (TRANSR, UPLO, N, ARF, A, LDA, INFO) ZTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR). subroutine ztgsen (IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO) ZTGSEN subroutine ztgsja (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) ZTGSJA subroutine ztgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) ZTGSNA subroutine ztpcon (NORM, UPLO, DIAG, N, AP, RCOND, WORK, RWORK, INFO) ZTPCON subroutine ztpmqrt (SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT, A, LDA, B, LDB, WORK, INFO) ZTPMQRT subroutine ztpqrt (M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO) ZTPQRT subroutine ztpqrt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO) ZTPQRT2 computes a QR factorization of a real or complex 'triangularpentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. subroutine ztprfs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZTPRFS subroutine ztptri (UPLO, DIAG, N, AP, INFO) ZTPTRI subroutine ztptrs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO) ZTPTRS subroutine ztpttf (TRANSR, UPLO, N, AP, ARF, INFO) ZTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF). subroutine ztpttr (UPLO, N, AP, A, LDA, INFO) ZTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR). subroutine ztrcon (NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, RWORK, INFO) ZTRCON subroutine ztrevc (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO) ZTREVC subroutine ztrevc3 (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, RWORK, LRWORK, INFO) ZTREVC3 subroutine ztrexc (COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO) ZTREXC subroutine ztrrfs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZTRRFS subroutine ztrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO) ZTRSEN subroutine ztrsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO) ZTRSNA subroutine ztrti2 (UPLO, DIAG, N, A, LDA, INFO) ZTRTI2 computes the inverse of a triangular matrix (unblocked algorithm). subroutine ztrtri (UPLO, DIAG, N, A, LDA, INFO) ZTRTRI subroutine ztrtrs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO) ZTRTRS subroutine ztrttf (TRANSR, UPLO, N, A, LDA, ARF, INFO) ZTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF). subroutine ztrttp (UPLO, N, A, LDA, AP, INFO) ZTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP). subroutine ztzrzf (M, N, A, LDA, TAU, WORK, LWORK, INFO) ZTZRZF subroutine zunbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO) ZUNBDB subroutine zunbdb1 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) ZUNBDB1 subroutine zunbdb2 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) ZUNBDB2 subroutine zunbdb3 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) ZUNBDB3 subroutine zunbdb4 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) ZUNBDB4 subroutine zunbdb5 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO) ZUNBDB5 subroutine zunbdb6 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO) ZUNBDB6 recursive subroutine zuncsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK, RWORK, LRWORK, IWORK, INFO) ZUNCSD subroutine zuncsd2by1 (JOBU1, JOBU2, JOBV1T, M, P, Q, X11, LDX11, X21, LDX21, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, WORK, LWORK, RWORK, LRWORK, IWORK, INFO) ZUNCSD2BY1 subroutine zung2l (M, N, K, A, LDA, TAU, WORK, INFO) ZUNG2L generates all or part of the unitary matrix Q from a QL factorization determined by cgeqlf (unblocked algorithm). subroutine zung2r (M, N, K, A, LDA, TAU, WORK, INFO) ZUNG2R subroutine zunghr (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO) ZUNGHR subroutine zungl2 (M, N, K, A, LDA, TAU, WORK, INFO) ZUNGL2 generates all or part of the unitary matrix Q from an LQ factorization determined by cgelqf (unblocked algorithm). subroutine zunglq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO) ZUNGLQ subroutine zungql (M, N, K, A, LDA, TAU, WORK, LWORK, INFO) ZUNGQL subroutine zungqr (M, N, K, A, LDA, TAU, WORK, LWORK, INFO) ZUNGQR subroutine zungr2 (M, N, K, A, LDA, TAU, WORK, INFO) ZUNGR2 generates all or part of the unitary matrix Q from an RQ factorization determined by cgerqf (unblocked algorithm). subroutine zungrq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO) ZUNGRQ subroutine zungtr (UPLO, N, A, LDA, TAU, WORK, LWORK, INFO) ZUNGTR subroutine zungtsqr (M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO) ZUNGTSQR subroutine zungtsqr_row (M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO) ZUNGTSQR_ROW subroutine zunhr_col (M, N, NB, A, LDA, T, LDT, D, INFO) ZUNHR_COL subroutine zunm2l (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO) ZUNM2L multiplies a general matrix by the unitary matrix from a QL factorization determined by cgeqlf (unblocked algorithm). subroutine zunm2r (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO) ZUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf (unblocked algorithm). subroutine zunmbr (VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMBR subroutine zunmhr (SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMHR subroutine zunml2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO) ZUNML2 multiplies a general matrix by the unitary matrix from a LQ factorization determined by cgelqf (unblocked algorithm). subroutine zunmlq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMLQ subroutine zunmql (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMQL subroutine zunmqr (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMQR subroutine zunmr2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO) ZUNMR2 multiplies a general matrix by the unitary matrix from a RQ factorization determined by cgerqf (unblocked algorithm). subroutine zunmr3 (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, INFO) ZUNMR3 multiplies a general matrix by the unitary matrix from a RZ factorization determined by ctzrzf (unblocked algorithm). subroutine zunmrq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMRQ subroutine zunmrz (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMRZ subroutine zunmtr (SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) ZUNMTR subroutine zupgtr (UPLO, N, AP, TAU, Q, LDQ, WORK, INFO) ZUPGTR subroutine zupmtr (SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO) ZUPMTR Detailed DescriptionThis is the group of complex16 other Computational routinesFunction Documentationsubroutine zbbcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, integer M, integer P, integer Q, double precision, dimension( * ) THETA, double precision, dimension( * ) PHI, complex*16, dimension( ldu1, * ) U1, integer LDU1, complex*16, dimension( ldu2, * ) U2, integer LDU2, complex*16, dimension( ldv1t, * ) V1T, integer LDV1T, complex*16, 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( * ) RWORK, integer LRWORK, integer INFO)ZBBCSDPurpose: ZBBCSD computes the CS decomposition of a unitary matrix in bidiagonalblock form, [ B11  B12 0 0 ] [ 0  0 I 0 ] X = [] [ B21  B22 0 0 ] [ 0  0 0 I ] [ C  S 0 0 ] [ U1  ] [ 0  0 I 0 ] [ V1  ]**H = [] [] [] . [  U2 ] [ S  C 0 0 ] [  V2 ] [ 0  0 0 I ] X is MbyM, its topleft block is PbyQ, and Q must be no larger than P, MP, or MQ. (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 ZUNCSD for details.) The bidiagonal matrices B11, B12, B21, and B22 are represented implicitly by angles THETA(1:Q) and PHI(1:Q1). The unitary matrices U1, U2, V1T, and V2T are input/output. The input matrices are pre or postmultiplied 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 rowmajor order; otherwise: X, U1, U2, V1T, and V2T are stored in column major order. M M is INTEGER The number of rows and columns in X, the unitary matrix in bidiagonalblock form. P P is INTEGER The number of rows in the topleft block of X. 0 <= P <= M. Q Q is INTEGER The number of columns in the topleft block of X. 0 <= Q <= MIN(P,MP,MQ). THETA THETA is DOUBLE PRECISION array, dimension (Q) On entry, the angles THETA(1),...,THETA(Q) that, along with PHI(1), ...,PHI(Q1), define the matrix in bidiagonalblock form. On exit, the angles whose cosines and sines define the diagonal blocks in the CS decomposition. PHI PHI is DOUBLE PRECISION array, dimension (Q1) The angles PHI(1),...,PHI(Q1) that, along with THETA(1),..., THETA(Q), define the matrix in bidiagonalblock form. U1 U1 is COMPLEX*16 array, dimension (LDU1,P) On entry, a PbyP matrix. On exit, U1 is postmultiplied by the left singular vector matrix common to [ B11 ; 0 ] and [ B12 0 0 ; 0 I 0 0 ]. LDU1 LDU1 is INTEGER The leading dimension of the array U1, LDU1 >= MAX(1,P). U2 U2 is COMPLEX*16 array, dimension (LDU2,MP) On entry, an (MP)by(MP) 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,MP). V1T V1T is COMPLEX*16 array, dimension (LDV1T,Q) On entry, a QbyQ matrix. On exit, V1T is premultiplied by the conjugate transpose of the right singular vector matrix common to [ B11 ; 0 ] and [ B21 ; 0 ]. LDV1T LDV1T is INTEGER The leading dimension of the array V1T, LDV1T >= MAX(1,Q). V2T V2T is COMPLEX*16 array, dimension (LDV2T,MQ) On entry, an (MQ)by(MQ) matrix. On exit, V2T is premultiplied by the conjugate transpose of the right singular vector matrix common to [ B12 0 0 ; 0 I 0 ] and [ B22 0 0 ; 0 0 I ]. LDV2T LDV2T is INTEGER The leading dimension of the array V2T, LDV2T >= MAX(1,MQ). B11D B11D is DOUBLE PRECISION array, dimension (Q) When ZBBCSD converges, B11D contains the cosines of THETA(1), ..., THETA(Q). If ZBBCSD fails to converge, then B11D contains the diagonal of the partially reduced topleft block. B11E B11E is DOUBLE PRECISION array, dimension (Q1) When ZBBCSD converges, B11E contains zeros. If ZBBCSD fails to converge, then B11E contains the superdiagonal of the partially reduced topleft block. B12D B12D is DOUBLE PRECISION array, dimension (Q) When ZBBCSD converges, B12D contains the negative sines of THETA(1), ..., THETA(Q). If ZBBCSD fails to converge, then B12D contains the diagonal of the partially reduced topright block. B12E B12E is DOUBLE PRECISION array, dimension (Q1) When ZBBCSD converges, B12E contains zeros. If ZBBCSD fails to converge, then B12E contains the subdiagonal of the partially reduced topright block. B21D B21D is DOUBLE PRECISION array, dimension (Q) When ZBBCSD converges, B21D contains the negative sines of THETA(1), ..., THETA(Q). If ZBBCSD fails to converge, then B21D contains the diagonal of the partially reduced bottomleft block. B21E B21E is DOUBLE PRECISION array, dimension (Q1) When ZBBCSD converges, B21E contains zeros. If ZBBCSD fails to converge, then B21E contains the subdiagonal of the partially reduced bottomleft block. B22D B22D is DOUBLE PRECISION array, dimension (Q) When ZBBCSD converges, B22D contains the negative sines of THETA(1), ..., THETA(Q). If ZBBCSD fails to converge, then B22D contains the diagonal of the partially reduced bottomright block. B22E B22E is DOUBLE PRECISION array, dimension (Q1) When ZBBCSD converges, B22E contains zeros. If ZBBCSD fails to converge, then B22E contains the subdiagonal of the partially reduced bottomright block. RWORK RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. LRWORK LRWORK is INTEGER The dimension of the array RWORK. LRWORK >= MAX(1,8*Q). If LRWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the work array, and no error message related to LRWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if ZBBCSD did not converge, INFO specifies the number of nonzero entries in PHI, and B11D, B11E, etc., contain the partially reduced matrix. Internal Parameters: TOLMUL DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(1/8))) TOLMUL controls the convergence criterion of the QR loop. Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they are within TOLMUL*EPS of either bound. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 328 of file zbbcsd.f. subroutine zbdsqr (character UPLO, integer N, integer NCVT, integer NRU, integer NCC, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldvt, * ) VT, integer LDVT, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) RWORK, integer INFO)ZBDSQRPurpose: ZBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real NbyN (upper or lower) bidiagonal matrix B using the implicit zeroshift QR algorithm. The SVD of B has the form B = Q * S * P**H where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**H*VT instead of P**H, for given complex input matrices U and VT. When U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by ZGEBRD, then A = (U*Q) * S * (P**H*VT) is the SVD of A. Optionally, the subroutine may also compute Q**H*C for a given complex input matrix C. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873912, Sept 1990) and "Accurate singular values and differential qd algorithms," by B. Parlett and V. Fernando, Technical Report CPAM554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm. Parameters UPLO
UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal. N N is INTEGER The order of the matrix B. N >= 0. NCVT NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0. NRU NRU is INTEGER The number of rows of the matrix U. NRU >= 0. NCC NCC is INTEGER The number of columns of the matrix C. NCC >= 0. D D is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order. E E is DOUBLE PRECISION array, dimension (N1) On entry, the N1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input. VT VT is COMPLEX*16 array, dimension (LDVT, NCVT) On entry, an NbyNCVT matrix VT. On exit, VT is overwritten by P**H * VT. Not referenced if NCVT = 0. LDVT LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. U U is COMPLEX*16 array, dimension (LDU, N) On entry, an NRUbyN matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU). C C is COMPLEX*16 array, dimension (LDC, NCC) On entry, an NbyNCC matrix C. On exit, C is overwritten by Q**H * C. Not referenced if NCC = 0. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. RWORK RWORK is DOUBLE PRECISION array, dimension (4*N) INFO INFO is INTEGER = 0: successful exit < 0: If INFO = i, the ith argument had an illegal value > 0: the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero. Internal Parameters: TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(1/8))) TOLMUL controls the convergence criterion of the QR loop. If it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller). MAXITR INTEGER, default = 6 MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithms stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 220 of file zbdsqr.f. subroutine zgghd3 (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZGGHD3Purpose: ZGGHD3 reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = lambda*B*x, and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**H*A*Z = H and transforms B to another upper triangular matrix T: Q**H*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**H*x. The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H If Q1 is the unitary matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then ZGGHD3 reduces the original problem to generalized Hessenberg form. This is a blocked variant of CGGHRD, using matrixmatrix multiplications for parts of the computation to enhance performance. Parameters COMPQ
COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned. COMPZ COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned; = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned. N N is INTEGER The order of the matrices A and B. N >= 0. ILO ILO is INTEGER IHI IHI is INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. A A is COMPLEX*16 array, dimension (LDA, N) On entry, the NbyN general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB, N) On entry, the NbyN upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Q Q is COMPLEX*16 array, dimension (LDQ, N) On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. Z Z is COMPLEX*16 array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. WORK WORK is COMPLEX*16 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: This routine reduces A to Hessenberg form and maintains B in triangular form using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, QuintanaOrti, and QuintanaOrti (BIT 2008). Definition at line 225 of file zgghd3.f. subroutine zgghrd (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer INFO)ZGGHRDPurpose: ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = lambda*B*x, and B is typically made upper triangular by computing its QR factorization and moving the unitary matrix Q to the left side of the equation. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**H*A*Z = H and transforms B to another upper triangular matrix T: Q**H*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**H*x. The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H If Q1 is the unitary matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then ZGGHRD reduces the original problem to generalized Hessenberg form. Parameters COMPQ
COMPQ is CHARACTER*1 = 'N': do not compute Q; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'V': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned. COMPZ COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned; = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned. N N is INTEGER The order of the matrices A and B. N >= 0. ILO ILO is INTEGER IHI IHI is INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. A A is COMPLEX*16 array, dimension (LDA, N) On entry, the NbyN general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB, N) On entry, the NbyN upper triangular matrix B. On exit, the upper triangular matrix T = Q**H B Z. The elements below the diagonal are set to zero. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). Q Q is COMPLEX*16 array, dimension (LDQ, N) On entry, if COMPQ = 'V', the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the unitary matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. Z Z is COMPLEX*16 array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix Z1. On exit, if COMPZ='I', the unitary matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and van Loan (Johns Hopkins Press). Definition at line 202 of file zgghrd.f. subroutine zggqrf (integer N, integer M, integer P, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAUA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) TAUB, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZGGQRFPurpose: ZGGQRF computes a generalized QR factorization of an NbyM matrix A and an NbyP matrix B: A = Q*R, B = Q*T*Z, where Q is an NbyN unitary matrix, Z is a PbyP unitary matrix, and R and T assume one of the forms: if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) NM N MN M where R11 is upper triangular, and if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) NP, PN N ( T21 ) P P where T12 or T21 is upper triangular. In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)*A: inv(B)*A = Z**H * (inv(T)*R) where inv(B) denotes the inverse of the matrix B, and Z**H denotes the conjugate transpose of matrix Z. Parameters N
N is INTEGER The number of rows of the matrices A and B. N >= 0. M M is INTEGER The number of columns of the matrix A. M >= 0. P P is INTEGER The number of columns of the matrix B. P >= 0. A A is COMPLEX*16 array, dimension (LDA,M) On entry, the NbyM matrix A. On exit, the elements on and above the diagonal of the array contain the min(N,M)byM upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (see Further Details). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). TAUA TAUA is COMPLEX*16 array, dimension (min(N,M)) The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details). B B is COMPLEX*16 array, dimension (LDB,P) On entry, the NbyP matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,PN+1:P) contains the NbyN upper triangular matrix T; if N > P, the elements on and above the (NP)th subdiagonal contain the NbyP upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). TAUB TAUB is COMPLEX*16 array, dimension (min(N,P)) The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 NbyM matrix, NB2 is the optimal blocksize for the RQ factorization of an NbyP matrix, and NB3 is the optimal blocksize for a call of ZUNMQR. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = min(n,m). Each H(i) has the form H(i) = I  taua * v * v**H where taua is a complex scalar, and v is a complex vector with v(1:i1) = 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 ZUNGQR. To use Q to update another matrix, use LAPACK subroutine ZUNMQR. The matrix Z is represented as a product of elementary reflectors Z = H(1) H(2) . . . H(k), where k = min(n,p). Each H(i) has the form H(i) = I  taub * v * v**H where taub is a complex scalar, and v is a complex vector with v(pk+i+1:p) = 0 and v(pk+i) = 1; v(1:pk+i1) is stored on exit in B(nk+i,1:pk+i1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine ZUNGRQ. To use Z to update another matrix, use LAPACK subroutine ZUNMRQ. Definition at line 213 of file zggqrf.f. subroutine zggrqf (integer M, integer P, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAUA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) TAUB, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZGGRQFPurpose: ZGGRQF computes a generalized RQ factorization of an MbyN matrix A and a PbyN matrix B: A = R*Q, B = Z*T*Q, where Q is an NbyN unitary matrix, Z is a PbyP unitary matrix, and R and T assume one of the forms: if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) MN, NM 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 ) PN P NP N where T11 is upper triangular. In particular, if B is square and nonsingular, the GRQ factorization of A and B implicitly gives the RQ factorization of A*inv(B): A*inv(B) = (R*inv(T))*Z**H where inv(B) denotes the inverse of the matrix B, and Z**H denotes the conjugate transpose of the matrix Z. Parameters M
M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the MbyN matrix A. On exit, if M <= N, the upper triangle of the subarray A(1:M,NM+1:N) contains the MbyM upper triangular matrix R; if M > N, the elements on and above the (MN)th subdiagonal contain the MbyN upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the unitary matrix Q as a product of elementary reflectors (see Further Details). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). TAUA TAUA is COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details). B B is COMPLEX*16 array, dimension (LDB,N) On entry, the PbyN matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)byN upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). TAUB TAUB is COMPLEX*16 array, dimension (min(P,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 MbyN matrix, NB2 is the optimal blocksize for the QR factorization of a PbyN matrix, and NB3 is the optimal blocksize for a call of ZUNMRQ. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = min(m,n). Each H(i) has the form H(i) = I  taua * v * v**H where taua is a complex scalar, and v is a complex vector with v(nk+i+1:n) = 0 and v(nk+i) = 1; v(1:nk+i1) is stored on exit in A(mk+i,1:nk+i1), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine ZUNGRQ. To use Q to update another matrix, use LAPACK subroutine ZUNMRQ. The matrix Z is represented as a product of elementary reflectors Z = H(1) H(2) . . . H(k), where k = min(p,n). Each H(i) has the form H(i) = I  taub * v * v**H where taub is a complex scalar, and v is a complex vector with v(1:i1) = 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 ZUNGQR. To use Z to update another matrix, use LAPACK subroutine ZUNMQR. Definition at line 212 of file zggrqf.f. subroutine zggsvp3 (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, double precision TOLA, double precision TOLB, integer K, integer L, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldq, * ) Q, integer LDQ, integer, dimension( * ) IWORK, double precision, dimension( * ) RWORK, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZGGSVP3Purpose: ZGGSVP3 computes unitary matrices U, V and Q such that NKL K L U**H*A*Q = K ( 0 A12 A13 ) if MKL >= 0; L ( 0 0 A23 ) MKL ( 0 0 0 ) NKL K L = K ( 0 A12 A13 ) if MKL < 0; MK ( 0 0 A23 ) NKL K L V**H*B*Q = L ( 0 0 B13 ) PL ( 0 0 0 ) where the KbyK matrix A12 and LbyL matrix B13 are nonsingular upper triangular; A23 is LbyL upper triangular if MKL >= 0, otherwise A23 is (MK)byL upper trapezoidal. K+L = the effective numerical rank of the (M+P)byN matrix (A**H,B**H)**H. This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine ZGGSVD3. Parameters JOBU
JOBU is CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed. JOBV JOBV is CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed. JOBQ JOBQ is CHARACTER*1 = 'Q': Unitary matrix Q is computed; = 'N': Q is not computed. M M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the MbyN matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is COMPLEX*16 array, dimension (LDB,N) On entry, the PbyN 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)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition. K K is INTEGER L L is INTEGER On exit, K and L specify the dimension of the subblocks described in Purpose section. K + L = effective numerical rank of (A**H,B**H)**H. U U is COMPLEX*16 array, dimension (LDU,M) If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N', U is not referenced. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V V is COMPLEX*16 array, dimension (LDV,P) If JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N', V is not referenced. LDV LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q Q is COMPLEX*16 array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. IWORK IWORK is INTEGER array, dimension (N) RWORK RWORK is DOUBLE PRECISION array, dimension (2*N) TAU TAU is COMPLEX*16 array, dimension (N) WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The subroutine uses LAPACK subroutine ZGEQP3 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. ZGGSVP3 replaces the deprecated subroutine ZGGSVP. Definition at line 275 of file zggsvp3.f. subroutine zgsvj0 (character*1 JOBV, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( n ) D, double precision, dimension( n ) SVA, integer MV, complex*16, dimension( ldv, * ) V, integer LDV, double precision EPS, double precision SFMIN, double precision TOL, integer NSWEEP, complex*16, dimension( lwork ) WORK, integer LWORK, integer INFO)ZGSVJ0 preprocessor for the routine zgesvj.Purpose: ZGSVJ0 is called from ZGESVJ as a preprocessor and that is its main purpose. It applies Jacobi rotations in the same way as ZGESVJ 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 NbyN array V. (See the description of V.) = 'A': the product of the Jacobi rotations is accumulated by postmulyiplying the MVbyN array V. (See the descriptions of MV and V.) = 'N': the Jacobi rotations are not accumulated. M M is INTEGER The number of rows of the input matrix A. M >= 0. N N is INTEGER The number of columns of the input matrix A. M >= N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, MbyN matrix A, such that A*diag(D) represents the input matrix. On exit, A_onexit * diag(D_onexit) represents the input matrix A*diag(D) postmultiplied by a sequence of Jacobi rotations, where the rotation threshold and the total number of sweeps are given in TOL and NSWEEP, respectively. (See the descriptions of D, TOL and NSWEEP.) LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). D D is COMPLEX*16 array, dimension (N) The array D accumulates the scaling factors from the complex scaled Jacobi rotations. On entry, A*diag(D) represents the input matrix. On exit, A_onexit*diag(D_onexit) represents the input matrix postmultiplied 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 A_onexit*diag(D_onexit). MV MV is INTEGER If JOBV = 'A', then MV rows of V are postmultipled by a sequence of Jacobi rotations. If JOBV = 'N', then MV is not referenced. V V is COMPLEX*16 array, dimension (LDV,N) If JOBV = 'V' then N rows of V are postmultipled by a sequence of Jacobi rotations. If JOBV = 'A' then MV rows of V are postmultipled 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 ABS(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 COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: ZGSVJ0 is used just to enable ZGESVJ to call a simplified
version of itself to work on a submatrix of the original matrix.
Contributor: Zlatko Drmac (Zagreb, Croatia) Bugs, Examples and Comments: Please report all bugs and send interesting test examples
and comments to drmac@math.hr. Thank you.
Definition at line 216 of file zgsvj0.f. subroutine zgsvj1 (character*1 JOBV, integer M, integer N, integer N1, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( n ) D, double precision, dimension( n ) SVA, integer MV, complex*16, dimension( ldv, * ) V, integer LDV, double precision EPS, double precision SFMIN, double precision TOL, integer NSWEEP, complex*16, dimension( lwork ) WORK, integer LWORK, integer INFO)ZGSVJ1 preprocessor for the routine zgesvj, applies Jacobi rotations targeting only particular pivots.Purpose: ZGSVJ1 is called from ZGESVJ as a preprocessor and that is its main purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but it targets only particular pivots and it does not check convergence (stopping criterion). Few tuning parameters (marked by [TP]) are available for the implementer. Further Details ~~~~~~~~~~~~~~~ ZGSVJ1 applies few sweeps of Jacobi rotations in the column space of the input MbyN matrix A. The pivot pairs are taken from the (1,2) offdiagonal block in the corresponding NbyN Gram matrix A^T * A. The blockentries (tiles) of the (1,2) offdiagonal block are marked by the [x]'s in the following scheme:  * * * [x] [x] [x]  * * * [x] [x] [x] Rowcycling in the nblrbynblc [x] blocks.  * * * [x] [x] [x] Rowcyclic 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 NN1 columns, trying to increase the angle between the corresponding subspaces. The offdiagonal block is N1by(NN1) and it is tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter. The number of sweeps is given in NSWEEP and the orthogonality threshold is given in TOL. Parameters JOBV
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 NbyN array V. (See the description of V.) = 'A': the product of the Jacobi rotations is accumulated by postmulyiplying the MVbyN 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 NN1 columns of A. A A is COMPLEX*16 array, dimension (LDA,N) On entry, MbyN matrix A, such that A*diag(D) represents the input matrix. On exit, A_onexit * D_onexit represents the input matrix A*diag(D) postmultiplied by a sequence of Jacobi rotations, where the rotation threshold and the total number of sweeps are given in TOL and NSWEEP, respectively. (See the descriptions of N1, D, TOL and NSWEEP.) LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). D D is COMPLEX*16 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 postmultiplied 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 postmultipled by a sequence of Jacobi rotations. If JOBV = 'N', then MV is not referenced. V V is COMPLEX*16 array, dimension (LDV,N) If JOBV = 'V' then N rows of V are postmultipled by a sequence of Jacobi rotations. If JOBV = 'A' then MV rows of V are postmultipled 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 ABS(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 COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributor: Zlatko Drmac (Zagreb, Croatia)
Definition at line 234 of file zgsvj1.f. subroutine zhbgst (character VECT, character UPLO, integer N, integer KA, integer KB, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB, integer LDBB, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZHBGSTPurpose: ZHBGST reduces a complex Hermitiandefinite banded generalized eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, such that C has the same bandwidth as A. B must have been previously factorized as S**H*S by ZPBSTF, using a split Cholesky factorization. A is overwritten by C = X**H*A*X, where X = S**(1)*Q and Q is a unitary matrix chosen to preserve the bandwidth of A. Parameters VECT
VECT is CHARACTER*1 = 'N': do not form the transformation matrix X; = 'V': form X. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrices A and B. N >= 0. KA KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0. KB KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(ka+1+ij,j) = A(i,j) for max(1,jka)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the transformed matrix X**H*A*X, stored in the same format as A. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1. BB BB is COMPLEX*16 array, dimension (LDBB,N) The banded factor S from the split Cholesky factorization of B, as returned by ZPBSTF, stored in the first kb+1 rows of the array. LDBB LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1. X X is COMPLEX*16 array, dimension (LDX,N) If VECT = 'V', the nbyn matrix X. If VECT = 'N', the array X is not referenced. LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise. WORK WORK is COMPLEX*16 array, dimension (N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 163 of file zhbgst.f. subroutine zhbtrd (character VECT, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( * ) WORK, integer INFO)ZHBTRDPurpose: ZHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. Parameters VECT
VECT is CHARACTER*1 = 'N': do not form Q; = 'V': form Q; = 'U': update a matrix X, by forming X*Q. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,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 offdiagonal 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 (N1) The offdiagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if VECT = 'U', then Q must contain an NbyN matrix X; if VECT = 'N' or 'V', then Q need not be set. On exit: if VECT = 'V', Q contains the NbyN unitary matrix Q; if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. WORK WORK is COMPLEX*16 array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: Modified by Linda Kaufman, Bell Labs. Definition at line 161 of file zhbtrd.f. subroutine zhetrd_hb2st (character STAGE1, character VECT, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( * ) HOUS, integer LHOUS, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form TPurpose: ZHETRD_HB2ST reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. Parameters STAGE1
STAGE1 is CHARACTER*1 = 'N': "No": to mention that the stage 1 of the reduction from dense to band using the zhetrd_he2hb routine was not called before this routine to reproduce AB. In other term this routine is called as standalone. = 'Y': "Yes": to mention that the stage 1 of the reduction from dense to band using the zhetrd_he2hb routine has been called to produce AB (e.g., AB is the output of zhetrd_he2hb. VECT VECT is CHARACTER*1 = 'N': No need for the Housholder representation, and thus LHOUS is of size max(1, 4*N); = 'V': the Householder representation is needed to either generate or to apply Q later on, then LHOUS is to be queried and computed. (NOT AVAILABLE IN THIS RELEASE). UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,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 offdiagonal 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 (N1) The offdiagonal elements of the tridiagonal matrix T: E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. HOUS HOUS is COMPLEX*16 array, dimension LHOUS, that store the Householder representation. LHOUS LHOUS is INTEGER The dimension of the array HOUS. LHOUS = MAX(1, dimension) If LWORK = 1, or LHOUS=1, then a query is assumed; the routine only calculates the optimal size of the HOUS array, returns this value as the first entry of the HOUS array, and no error message related to LHOUS is issued by XERBLA. LHOUS = MAX(1, dimension) where dimension = 4*N if VECT='N' not available now if VECT='H' WORK WORK is COMPLEX*16 array, dimension LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK = MAX(1, dimension) If LWORK = 1, or LHOUS=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. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: Implemented by Azzam Haidar. All details are available on technical report, SC11, SC13 papers. Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated finegrained and memoryaware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPUGPU generalized eigensolver for electronic structure calculations based on finegrained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196209, May 2014. http://hpc.sagepub.com/content/28/2/196 Definition at line 228 of file zhetrd_hb2st.F. subroutine zhfrk (character TRANSR, character UPLO, character TRANS, integer N, integer K, double precision ALPHA, complex*16, dimension( lda, * ) A, integer LDA, double precision BETA, complex*16, dimension( * ) C)ZHFRK performs a Hermitian rankk operation for matrix in RFP format.Purpose: Level 3 BLAS like routine for C in RFP Format. ZHFRK performs one of the Hermitian rankk operations C := alpha*A*A**H + beta*C, or C := alpha*A**H*A + beta*C, where alpha and beta are real scalars, C is an nbyn Hermitian matrix and A is an nbyk matrix in the first case and a kbyn matrix in the second case. Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': The Normal Form of RFP A is stored; = 'C': The Conjugatetranspose Form of RFP A is stored. UPLO UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the array C is to be referenced as follows: UPLO = 'U' or 'u' Only the upper triangular part of C is to be referenced. UPLO = 'L' or 'l' Only the lower triangular part of C is to be referenced. Unchanged on exit. TRANS TRANS is CHARACTER*1 On entry, TRANS specifies the operation to be performed as follows: TRANS = 'N' or 'n' C := alpha*A*A**H + beta*C. TRANS = 'C' or 'c' C := alpha*A**H*A + beta*C. Unchanged on exit. N N is INTEGER On entry, N specifies the order of the matrix C. N must be at least zero. Unchanged on exit. K K is INTEGER On entry with TRANS = 'N' or 'n', K specifies the number of columns of the matrix A, and on entry with TRANS = 'C' or 'c', K specifies the number of rows of the matrix A. K must be at least zero. Unchanged on exit. ALPHA ALPHA is DOUBLE PRECISION On entry, ALPHA specifies the scalar alpha. Unchanged on exit. A A is COMPLEX*16 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 NbyK part of the array A must contain the matrix A, otherwise the leading KbyN 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 COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the matrix A in RFP Format. RFP Format is described by TRANSR, UPLO and N. Note that the imaginary parts of the diagonal elements need not be set, they are assumed to be zero, and on exit they are set to zero. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 166 of file zhfrk.f. subroutine zhpcon (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)ZHPCONPurpose: ZHPCON estimates the reciprocal of the condition number of a complex Hermitian packed matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHPTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 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 ZHPTRF, 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 ZHPTRF. ANORM ANORM is DOUBLE PRECISION The 1norm 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 1norm of inv(A) computed in this routine. WORK WORK is COMPLEX*16 array, dimension (2*N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 117 of file zhpcon.f. subroutine zhpgst (integer ITYPE, character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, integer INFO)ZHPGSTPurpose: ZHPGST reduces a complex Hermitiandefinite generalized eigenproblem to standard form, using packed storage. If ITYPE = 1, the problem is A*x = lambda*B*x, and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H) If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H*A*L. B must have been previously factorized as U**H*U or L*L**H by ZPPTRF. Parameters ITYPE
ITYPE is INTEGER = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H); = 2 or 3: compute U*A*U**H or L**H*A*L. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored and B is factored as U**H*U; = 'L': Lower triangle of A is stored and B is factored as L*L**H. N N is INTEGER The order of the matrices A and B. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. On exit, if INFO = 0, the transformed matrix, stored in the same format as A. BP BP is COMPLEX*16 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 ZPPTRF. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 112 of file zhpgst.f. subroutine zhprfs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( * ) AFP, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZHPRFSPurpose: ZHPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite and packed, and provides error bounds and backward error estimates for the solution. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2*nj)/2) = A(i,j) for j<=i<=n. AFP AFP is COMPLEX*16 array, dimension (N*(N+1)/2) The factored form of the matrix A. AFP contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, 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 ZHPTRF. B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZHPTRS. 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 jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Internal Parameters: ITMAX is the maximum number of steps of iterative refinement. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 178 of file zhprfs.f. subroutine zhptrd (character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( * ) TAU, integer INFO)ZHPTRDPurpose: ZHPTRD reduces a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2*nj)/2) = A(i,j) for j<=i<=n. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. D D is 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 (N1) The offdiagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. TAU TAU is COMPLEX*16 array, dimension (N1) The scalar factors of the elementary reflectors (see Further Details). INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n1) . . . H(2) H(1). Each H(i) has the form H(i) = I  tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i1) is stored on exit in AP, overwriting A(1:i1,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(n1). Each H(i) has the form H(i) = I  tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, overwriting A(i+2:n,i), and tau is stored in TAU(i). Definition at line 150 of file zhptrd.f. subroutine zhptrf (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)ZHPTRFPurpose: ZHPTRF computes the factorization of a complex Hermitian packed matrix A using the BunchKaufman diagonal pivoting method: A = U*D*U**H or A = L*D*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1by1 and 2by2 diagonal blocks. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/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 1by1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 diagonal block. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: If UPLO = 'U', then A = U*D*U**H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1by1 and 2by2 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 ) ks U(k) = ( 0 I 0 ) s ( 0 0 I ) nk ks s nk If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper triangle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1:k). If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1by1 and 2by2 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 ) k1 L(k) = ( 0 I 0 ) s ( 0 v I ) nks+1 k1 s nks+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). Contributors: J. Lewis, Boeing Computer Services Company
Definition at line 158 of file zhptrf.f. subroutine zhptri (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)ZHPTRIPurpose: ZHPTRI computes the inverse of a complex Hermitian indefinite matrix A in packed storage using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHPTRF. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 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 ZHPTRF, stored as a packed triangular matrix. On exit, if INFO = 0, the (Hermitian) inverse of the original matrix, stored as a packed triangular matrix. The jth column of inv(A) is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = inv(A)(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/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 ZHPTRF. WORK WORK is COMPLEX*16 array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 108 of file zhptri.f. subroutine zhptrs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZHPTRSPurpose: ZHPTRS solves a system of linear equations A*X = B with a complex Hermitian matrix A stored in packed format using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHPTRF. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**H; = 'L': Lower triangular, form is A = L*D*L**H. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. AP AP is COMPLEX*16 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 ZHPTRF, 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 ZHPTRF. B B is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 114 of file zhptrs.f. subroutine zhsein (character SIDE, character EIGSRC, character INITV, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IFAILL, integer, dimension( * ) IFAILR, integer INFO)ZHSEINPurpose: ZHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H. The right eigenvector x and the left eigenvector y of the matrix H corresponding to an eigenvalue w are defined by: H * x = w * x, y**h * H = w * y**h where y**h denotes the conjugate transpose of the vector y. Parameters SIDE
SIDE is CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. EIGSRC EIGSRC is CHARACTER*1 Specifies the source of eigenvalues supplied in W: = 'Q': the eigenvalues were found using ZHSEQR; thus, if H has zero subdiagonal elements, and so is blocktriangular, then the jth eigenvalue can be assumed to be an eigenvalue of the block containing the jth row/column. This property allows ZHSEIN 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, ZHSEIN must always perform inverse iteration using the whole matrix H. INITV INITV is CHARACTER*1 = 'N': no initial vectors are supplied; = 'U': usersupplied initial vectors are stored in the arrays VL and/or VR. SELECT SELECT is LOGICAL array, dimension (N) Specifies the eigenvectors to be computed. To select the eigenvector corresponding to the eigenvalue W(j), SELECT(j) must be set to .TRUE.. N N is INTEGER The order of the matrix H. N >= 0. H H is COMPLEX*16 array, dimension (LDH,N) The upper Hessenberg matrix H. If a NaN is detected in H, the routine will return with INFO=6. LDH LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N). W W is COMPLEX*16 array, dimension (N) On entry, the eigenvalues of H. On exit, the real parts of W may have been altered since close eigenvalues are perturbed slightly in searching for independent eigenvectors. VL VL is COMPLEX*16 array, dimension (LDVL,MM) On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must contain starting vectors for the inverse iteration for the left eigenvectors; the starting vector for each eigenvector must be in the same column in which the eigenvector will be stored. On exit, if SIDE = 'L' or 'B', the left eigenvectors specified by SELECT will be stored consecutively in the columns of VL, in the same order as their eigenvalues. If SIDE = 'R', VL is not referenced. LDVL LDVL is INTEGER The leading dimension of the array VL. LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise. VR VR is COMPLEX*16 array, dimension (LDVR,MM) On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must contain starting vectors for the inverse iteration for the right eigenvectors; the starting vector for each eigenvector must be in the same column in which the eigenvector will be stored. On exit, if SIDE = 'R' or 'B', the right eigenvectors specified by SELECT will be stored consecutively in the columns of VR, in the same order as their eigenvalues. If SIDE = 'L', VR is not referenced. LDVR LDVR is INTEGER The leading dimension of the array VR. LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise. MM MM is INTEGER The number of columns in the arrays VL and/or VR. MM >= M. M M is INTEGER The number of columns in the arrays VL and/or VR required to store the eigenvectors (= the number of .TRUE. elements in SELECT). WORK WORK is COMPLEX*16 array, dimension (N*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) IFAILL IFAILL is INTEGER array, dimension (MM) If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left eigenvector in the ith column of VL (corresponding to the eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the eigenvector converged satisfactorily. If SIDE = 'R', IFAILL is not referenced. IFAILR IFAILR is INTEGER array, dimension (MM) If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right eigenvector in the ith column of VR (corresponding to the eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the eigenvector converged satisfactorily. If SIDE = 'L', IFAILR is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, i is the number of eigenvectors which failed to converge; see IFAILL and IFAILR for further details. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be x+y. Definition at line 242 of file zhsein.f. subroutine zhseqr (character JOB, character COMPZ, integer N, integer ILO, integer IHI, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZHSEQRPurpose: ZHSEQR computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = Q*H*Q**H = (QZ)*T*(QZ)**H. Parameters JOB
JOB is CHARACTER*1 = 'E': compute eigenvalues only; = 'S': compute eigenvalues and the Schur form T. COMPZ COMPZ is CHARACTER*1 = 'N': no Schur vectors are computed; = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = 'V': Z must contain an unitary matrix Q on entry, and the product Q*Z is returned. N N is INTEGER The order of the matrix H. N >= 0. ILO ILO is INTEGER IHI IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGEBAL, and then passed to ZGEHRD when the matrix output by ZGEBAL 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 COMPLEX*16 array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and JOB = 'S', H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) Unlike earlier versions of ZHSEQR, this subroutine may explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO1 or j = IHI+1, IHI+2, ... N. LDH LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N). W W is COMPLEX*16 array, dimension (N) The computed eigenvalues. If JOB = 'S', the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i). Z Z is COMPLEX*16 array, dimension (LDZ,N) If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the unitary matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry Z must contain an NbyN matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z. Normally Q is the unitary matrix generated by ZUNGHR after the call to ZGEHRD 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 COMPLEX*16 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 ZHSEQR does a workspace query. In this case, ZHSEQR 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 ith argument had an illegal value > 0: if INFO = i, ZHSEQR failed to compute all of the eigenvalues. Elements 1:ilo1 and i+1:n of W 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 a unitary matrix. The final value of H is upper Hessenberg and 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 unitary 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 unitary matrix in (*) (regard less of the value of JOB.) If INFO > 0 and COMPZ = 'N', then Z is not accessed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
Further Details: Default values supplied by ILAENV(ISPEC,'ZHSEQR',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 ZLAHQR vs ZLAQR0 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 (IHIILO+1) <= 500 is NS. The default for (IHIILO+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 IHIILO+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 ZLAHQR and this parameter is ignored. See ISPEC=12 above and comments in IPARMQ for details. (**) The asterisks (**) indicate an adhoc 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 MultiShift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929947, 2002. K. Braman, R. Byers and R. Mathias, The MultiShift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948973, 2002. Definition at line 297 of file zhseqr.f. subroutine zla_lin_berr (integer N, integer NZ, integer NRHS, complex*16, dimension( n, nrhs ) RES, double precision, dimension( n, nrhs ) AYB, double precision, dimension( nrhs ) BERR)ZLA_LIN_BERR computes a componentwise relative backward error.Purpose: ZLA_LIN_BERR computes componentwise relative backward error from the formula max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) where abs(Z) is the componentwise absolute value of the matrix or vector Z. Parameters N
N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. NZ NZ is INTEGER We add (NZ+1)*SLAMCH( 'Safe minimum' ) to R(i) in the numerator to guard against spuriously zero residuals. Default value is N. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices AYB, RES, and BERR. NRHS >= 0. RES RES is COMPLEX*16 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 zla_gerfsx_extended.f). BERR BERR is DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error from the formula above. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 100 of file zla_lin_berr.f. subroutine zla_wwaddw (integer N, complex*16, dimension( * ) X, complex*16, dimension( * ) Y, complex*16, dimension( * ) W)ZLA_WWADDW adds a vector into a doubledsingle vector.Purpose: ZLA_WWADDW adds a vector W into a doubledsingle 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 COMPLEX*16 array, dimension (N) The first part of the doubledsingle accumulation vector. Y Y is COMPLEX*16 array, dimension (N) The second part of the doubledsingle accumulation vector. W W is COMPLEX*16 array, dimension (N) The vector to be added. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 80 of file zla_wwaddw.f. subroutine zlaed0 (integer QSIZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldqs, * ) QSTORE, integer LDQS, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)ZLAED0 used by ZSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.Purpose: Using the divide and conquer method, ZLAED0 computes all eigenvalues of a symmetric tridiagonal matrix which is one diagonal block of those from reducing a dense or band Hermitian matrix and corresponding eigenvectors of the dense or band matrix. Parameters QSIZ
QSIZ is INTEGER The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. N N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. D D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, the eigenvalues in ascending order. E E is DOUBLE PRECISION array, dimension (N1) On entry, the offdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, Q must contain an QSIZ x N matrix whose columns unitarily orthonormal. It is a part of the unitary matrix that reduces the full dense Hermitian matrix to a (reducible) symmetric tridiagonal matrix. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N). IWORK IWORK is INTEGER array, the dimension of IWORK must be at least 6 + 6*N + 5*N*lg N ( lg( N ) = smallest integer k such that 2^k >= N ) RWORK RWORK is DOUBLE PRECISION array, dimension (1 + 3*N + 2*N*lg N + 3*N**2) ( lg( N ) = smallest integer k such that 2^k >= N ) QSTORE QSTORE is COMPLEX*16 array, dimension (LDQS, N) Used to store parts of the eigenvector matrix when the updating matrix multiplies take place. LDQS LDQS is INTEGER The leading dimension of the array QSTORE. LDQS >= max(1,N). INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1). Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 143 of file zlaed0.f. subroutine zlaed7 (integer N, integer CUTPNT, integer QSIZ, integer TLVLS, integer CURLVL, integer CURPBM, double precision, dimension( * ) D, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision RHO, integer, dimension( * ) INDXQ, double precision, dimension( * ) QSTORE, integer, dimension( * ) QPTR, integer, dimension( * ) PRMPTR, integer, dimension( * ) PERM, integer, dimension( * ) GIVPTR, integer, dimension( 2, * ) GIVCOL, double precision, dimension( 2, * ) GIVNUM, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rankone symmetric matrix. Used when the original matrix is dense.Purpose: ZLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rankone symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense or banded Hermitian matrix that has been reduced to tridiagonal form. T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out) where Z = Q**Hu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem. Parameters N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. CUTPNT CUTPNT is INTEGER Contains the location of the last eigenvalue in the leading submatrix. min(1,N) <= CUTPNT <= N. QSIZ QSIZ is INTEGER The dimension of the unitary matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N. TLVLS TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree. CURLVL CURLVL is INTEGER The current level in the overall merge routine, 0 <= curlvl <= tlvls. CURPBM CURPBM is INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right). D D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the rank1perturbed matrix. On exit, the eigenvalues of the repaired matrix. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, the eigenvectors of the rank1perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N). RHO RHO is DOUBLE PRECISION Contains the subdiagonal element used to create the rank1 modification. INDXQ INDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, ie. D( INDXQ( I = 1, N ) ) will be in ascending order. IWORK IWORK is INTEGER array, dimension (4*N) RWORK RWORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N) WORK WORK is COMPLEX*16 array, dimension (QSIZ*N) QSTORE QSTORE is DOUBLE PRECISION array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices. QPTR QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top. PRMPTR PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1)  PRMPTR(i) indicates the size of the permutation and also the size of the full, nondeflated problem. PERM PERM is INTEGER array, dimension (N lg N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock. GIVPTR GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1)  GIVPTR(i) indicates the number of Givens rotations. GIVCOL GIVCOL is INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. GIVNUM GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation. INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 245 of file zlaed7.f. subroutine zlaed8 (integer K, integer N, integer QSIZ, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) D, double precision RHO, integer CUTPNT, double precision, dimension( * ) Z, double precision, dimension( * ) DLAMDA, complex*16, dimension( ldq2, * ) Q2, integer LDQ2, double precision, dimension( * ) W, integer, dimension( * ) INDXP, integer, dimension( * ) INDX, integer, dimension( * ) INDXQ, integer, dimension( * ) PERM, integer GIVPTR, integer, dimension( 2, * ) GIVCOL, double precision, dimension( 2, * ) GIVNUM, integer INFO)ZLAED8 used by ZSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.Purpose: ZLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. Parameters K
K is INTEGER Contains the number of nondeflated eigenvalues. This is the order of the related secular equation. N N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. QSIZ QSIZ is INTEGER The dimension of the unitary matrix used to reduce the dense or band matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (NK) updated eigenvectors (those which were deflated) in its last NK columns. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max( 1, N ). D D is DOUBLE PRECISION array, dimension (N) On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (NK) updated eigenvalues (those which were deflated) sorted into increasing order. RHO RHO is DOUBLE PRECISION Contains the off diagonal element associated with the rank1 cut which originally split the two submatrices which are now being recombined. RHO is modified during the computation to the value required by DLAED3. CUTPNT CUTPNT is INTEGER Contains the location of the last eigenvalue in the leading submatrix. MIN(1,N) <= CUTPNT <= N. Z Z is DOUBLE PRECISION array, dimension (N) On input this vector contains the updating vector (the last row of the first subeigenvector matrix and the first row of the second subeigenvector matrix). The contents of Z are destroyed during the updating process. DLAMDA DLAMDA is DOUBLE PRECISION array, dimension (N) Contains a copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation. Q2 Q2 is COMPLEX*16 array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, Contains a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors. LDQ2 LDQ2 is INTEGER The leading dimension of the array Q2. LDQ2 >= max( 1, N ). W W is DOUBLE PRECISION array, dimension (N) This will hold the first k values of the final deflationaltered zvector and will be passed to DLAED3. INDXP INDXP is INTEGER array, dimension (N) This will contain the permutation used to place deflated values of D at the end of the array. On output INDXP(1:K) points to the nondeflated Dvalues and INDXP(K+1:N) points to the deflated eigenvalues. INDX INDX is INTEGER array, dimension (N) This will contain the permutation used to sort the contents of D into ascending order. INDXQ INDXQ is INTEGER array, dimension (N) This contains the permutation which separately sorts the two subproblems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate. PERM PERM is INTEGER array, dimension (N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock. GIVPTR GIVPTR is INTEGER Contains the number of Givens rotations which took place in this subproblem. GIVCOL GIVCOL is INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. GIVNUM GIVNUM is DOUBLE PRECISION array, dimension (2, N) Each number indicates the S value to be used in the corresponding Givens rotation. INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 225 of file zlaed8.f. subroutine zlals0 (integer ICOMPQ, integer NL, integer NR, integer SQRE, integer NRHS, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, 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( * ) RWORK, integer INFO)ZLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.Purpose: ZLALS0 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 divideandconquer 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 Svalues 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 Jth 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 NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE. NRHS NRHS is INTEGER The number of columns of B and BX. NRHS must be at least 1. B B is COMPLEX*16 array, dimension ( LDB, NRHS ) On input, B contains the right hand sides of the least squares problem in rows 1 through M. On output, B contains the solution X in rows 1 through N. LDB LDB is INTEGER The leading dimension of B. LDB must be at least max(1,MAX( M, N ) ). BX BX is COMPLEX*16 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 Ith updated (undeflated) singular value and the Ith (undeflated) old singular value. DIFR DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). On entry, DIFR(I, 1) contains the distances between Ith updated (undeflated) singular value and the I+1th (undeflated) old singular value. And DIFR(I, 2) is the normalizing factor for the Ith right singular vector. Z Z is DOUBLE PRECISION array, dimension ( K ) Contain the components of the deflationadjusted updating row vector. K K is INTEGER Contains the dimension of the nondeflated 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 Cvalue 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 Svalue of a Givens rotation related to the right null space if SQRE = 1. RWORK RWORK is DOUBLE PRECISION array, dimension ( K*(1+NRHS) + 2*NRHS ) INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Ming Gu and RenCang Li, Computer Science Division,
University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
Definition at line 267 of file zlals0.f. subroutine zlalsa (integer ICOMPQ, integer SMLSIZ, integer N, integer NRHS, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, 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( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)ZLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.Purpose: ZLALSA 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, ZLALSA applies the inverse of the left singular vector matrix of an upper bidiagonal matrix to the right hand side; and if ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the right hand side. The singular vector matrices were generated in compact form by ZLALSA. Parameters ICOMPQ
ICOMPQ is INTEGER Specifies whether the left or the right singular vector matrix is involved. = 0: Left singular vector matrix = 1: Right singular vector matrix SMLSIZ SMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree. N N is INTEGER The row and column dimensions of the upper bidiagonal matrix. NRHS NRHS is INTEGER The number of columns of B and BX. NRHS must be at least 1. B B is COMPLEX*16 array, dimension ( LDB, NRHS ) On input, B contains the right hand sides of the least squares problem in rows 1 through M. On output, B contains the solution X in rows 1 through N. LDB LDB is INTEGER The leading dimension of B in the calling subprogram. LDB must be at least max(1,MAX( M, N ) ). BX BX is COMPLEX*16 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**H 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 Ith 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 Ith 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 Ith 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 Ith level. GIVPTR GIVPTR is INTEGER array, dimension ( N ). On entry, GIVPTR( I ) records the number of Givens rotations performed on the Ith 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 Ith 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 Ith 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 Ith level on the computation tree. C C is DOUBLE PRECISION array, dimension ( N ). On entry, if the Ith subproblem is not square, C( I ) contains the Cvalue of a Givens rotation related to the right null space of the Ith subproblem. S S is DOUBLE PRECISION array, dimension ( N ). On entry, if the Ith subproblem is not square, S( I ) contains the Svalue of a Givens rotation related to the right null space of the Ith subproblem. RWORK RWORK is DOUBLE PRECISION array, dimension at least MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ). IWORK IWORK is INTEGER array, dimension (3*N) INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Ming Gu and RenCang Li, Computer Science Division,
University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
Definition at line 263 of file zlalsa.f. subroutine zlalsd (character UPLO, integer SMLSIZ, integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldb, * ) B, integer LDB, double precision RCOND, integer RANK, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)ZLALSD uses the singular value decomposition of A to solve the least squares problem.Purpose: ZLALSD 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*XB, where A is NbyN upper bidiagonal, and X and B are NbyNRHS. 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 (N1) Contains the superdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed. B B is COMPLEX*16 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 COMPLEX*16 array, dimension (N * NRHS) RWORK RWORK is DOUBLE PRECISION array, dimension at least (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ), where NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) IWORK IWORK is INTEGER array, dimension at least (3*N*NLVL + 11*N). INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: The algorithm failed to compute a singular value while working on the submatrix lying in rows and columns INFO/(N+1) through MOD(INFO,N+1). Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Ming Gu and RenCang Li, Computer Science Division,
University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
Definition at line 185 of file zlalsd.f. double precision function zlanhf (character NORM, character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) A, double precision, dimension( 0: * ) WORK)ZLANHF returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.Purpose: ZLANHF returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix A in RFP format. Returns ZLANHF
ZLANHF = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. Parameters NORM
NORM is CHARACTER Specifies the value to be returned in ZLANHF as described above. TRANSR TRANSR is CHARACTER Specifies whether the RFP format of A is normal or conjugatetransposed format. = 'N': RFP format is Normal = 'C': RFP format is Conjugatetransposed UPLO UPLO is CHARACTER On entry, UPLO specifies whether the RFP matrix A came from an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' RFP A came from an upper triangular matrix UPLO = 'L' or 'l' RFP A came from a lower triangular matrix N N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANHF is set to zero. A A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the matrix A in RFP Format. RFP Format is described by TRANSR, UPLO and N as follows: If TRANSR='N' then RFP A is (0:N,0:K1) when N is even; K=N/2. RFP A is (0:N1,0:K) when N is odd; K=N/2. If TRANSR = 'C' then RFP is the Conjugatetranspose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the ( N*(N+1)/2 ) elements of upper packed A either in normal or conjugatetranspose Format. If UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) elements of lower packed A either in normal or conjugatetranspose Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and is N when is odd. See the Note below for more details. Unchanged on exit. WORK WORK is DOUBLE PRECISION array, dimension (LWORK), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 245 of file zlanhf.f. subroutine zlarscl2 (integer M, integer N, double precision, dimension( * ) D, complex*16, dimension( ldx, * ) X, integer LDX)ZLARSCL2 performs reciprocal diagonal scaling on a vector.Purpose: ZLARSCL2 performs a reciprocal diagonal scaling on an vector: x < inv(D) * x where the DOUBLE PRECISION diagonal matrix D is stored as a vector. Eventually to be replaced by BLAS_zge_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 COMPLEX*16 array, dimension (LDX,N) On entry, the vector X to be scaled by D. On exit, the scaled vector. LDX LDX is INTEGER The leading dimension of the vector X. LDX >= M. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 90 of file zlarscl2.f. subroutine zlarz (character SIDE, integer M, integer N, integer L, complex*16, dimension( * ) V, integer INCV, complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK)ZLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.Purpose: ZLARZ applies a complex elementary reflector H to a complex MbyN matrix C, from either the left or the right. H is represented in the form H = I  tau * v * v**H where tau is a complex scalar and v is a complex vector. If tau = 0, then H is taken to be the unit matrix. To apply H**H (the conjugate transpose of H), supply conjg(tau) instead tau. H is a product of k elementary reflectors as returned by ZTZRZF. Parameters SIDE
SIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * H M M is INTEGER The number of rows of the matrix C. N N is INTEGER The number of columns of the matrix C. L L is INTEGER The number of entries of the vector V containing the meaningful part of the Householder vectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. V V is COMPLEX*16 array, dimension (1+(L1)*abs(INCV)) The vector v in the representation of H as returned by ZTZRZF. V is not used if TAU = 0. INCV INCV is INTEGER The increment between elements of v. INCV <> 0. TAU TAU is COMPLEX*16 The value tau in the representation of H. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R' Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: Definition at line 146 of file zlarz.f. subroutine zlarzb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldwork, * ) WORK, integer LDWORK)ZLARZB applies a block reflector or its conjugatetranspose to a general matrix.Purpose: ZLARZB applies a complex block reflector H or its transpose H**H to a complex distributed MbyN C from the left or the right. Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply H or H**H from the Left = 'R': apply H or H**H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'C': apply H**H (Conjugate transpose) DIRECT DIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) = 'B': H = H(k) . . . H(2) H(1) (Backward) STOREV STOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columnwise (not supported yet) = 'R': Rowwise M M is INTEGER The number of rows of the matrix C. N N is INTEGER The number of columns of the matrix C. K K is INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). L L is INTEGER The number of columns of the matrix V containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. V V is COMPLEX*16 array, dimension (LDV,NV). If STOREV = 'C', NV = K; if STOREV = 'R', NV = L. LDV LDV is INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K. T T is COMPLEX*16 array, dimension (LDT,K) The triangular KbyK matrix T in the representation of the block reflector. LDT LDT is INTEGER The leading dimension of the array T. LDT >= K. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (LDWORK,K) LDWORK LDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= max(1,N); if SIDE = 'R', LDWORK >= max(1,M). Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: Definition at line 181 of file zlarzb.f. subroutine zlarzt (character DIRECT, character STOREV, integer N, integer K, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension( ldt, * ) T, integer LDT)ZLARZT forms the triangular factor T of a block reflector H = I  vtvH.Purpose: ZLARZT forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the ith column of the array V, and H = I  V * T * V**H If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the ith row of the array V, and H = I  V**H * T * V Currently, only STOREV = 'R' and DIRECT = 'B' are supported. Parameters DIRECT
DIRECT is CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) = 'B': H = H(k) . . . H(2) H(1) (Backward) STOREV STOREV is CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): = 'C': columnwise (not supported yet) = 'R': rowwise N N is INTEGER The order of the block reflector H. N >= 0. K K is INTEGER The order of the triangular factor T (= the number of elementary reflectors). K >= 1. V V is COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details. LDV LDV is INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i). T T is COMPLEX*16 array, dimension (LDT,K) The k by k triangular factor T of the block reflector. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. The rest of the array is not used. LDT LDT is INTEGER The leading dimension of the array T. LDT >= K. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ______V_____ ( v1 v2 v3 ) / ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) ( v1 v2 v3 ) . . . . . . 1 . . 1 . 1 DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ______V_____ 1 / . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) . . . ( . . 1 . . v3 v3 v3 v3 v3 ) . . . ( v1 v2 v3 ) ( v1 v2 v3 ) V = ( v1 v2 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) Definition at line 184 of file zlarzt.f. subroutine zlascl2 (integer M, integer N, double precision, dimension( * ) D, complex*16, dimension( ldx, * ) X, integer LDX)ZLASCL2 performs diagonal scaling on a vector.Purpose: ZLASCL2 performs a diagonal scaling on a vector: x < D * x where the DOUBLE PRECISION diagonal matrix D is stored as a vector. Eventually to be replaced by BLAS_zge_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 COMPLEX*16 array, dimension (LDX,N) On entry, the vector X to be scaled by D. On exit, the scaled vector. LDX LDX is INTEGER The leading dimension of the vector X. LDX >= M. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 90 of file zlascl2.f. subroutine zlatrz (integer M, integer N, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK)ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations.Purpose: ZLATRZ factors the Mby(M+L) complex upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,NL+1:N) ] as ( R 0 ) * Z by means of unitary transformations, where Z is an (M+L)by(M+L) unitary matrix and, R and A1 are MbyM 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. NM >= L >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the leading MbyN upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading MbyM upper triangular part of A contains the upper triangular matrix R, and elements NL+1 to N of the first M rows of A, with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (M) The scalar factors of the elementary reflectors. WORK WORK is COMPLEX*16 array, dimension (M) Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the ( m  k + 1 )th row of A, is given in the form Z( k ) = ( I 0 ), ( 0 T( k ) ) where T( k ) = I  tau*u( k )*u( k )**H, u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an l element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of A2. The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of A2, such that the elements of z( k ) are in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A1. Z is given by Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). Definition at line 139 of file zlatrz.f. subroutine zpbcon (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZPBCONPurpose: ZPBCON estimates the reciprocal of the condition number (in the 1norm) of a complex Hermitian positive definite band matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPBTRF. 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 COMPLEX*16 array, dimension (LDAB,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, stored in the first KD+1 rows of the array. The jth column of U or L is stored in the jth column of the array AB as follows: if UPLO ='U', AB(kd+1+ij,j) = U(i,j) for max(1,jkd)<=i<=j; if UPLO ='L', AB(1+ij,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 1norm (or infinitynorm) of the Hermitian 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 1norm of inv(A) computed in this routine. WORK WORK is COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 131 of file zpbcon.f. subroutine zpbequ (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)ZPBEQUPurpose: ZPBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the twonorm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangular of A is stored; = 'L': Lower triangular of A is stored. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,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 ith argument had an illegal value. > 0: if INFO = i, the ith diagonal element is nonpositive. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 129 of file zpbequ.f. subroutine zpbrfs (character UPLO, integer N, integer KD, integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldafb, * ) AFB, integer LDAFB, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZPBRFSPurpose: ZPBRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded, and provides error bounds and backward error estimates for the solution. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd). LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. AFB AFB is COMPLEX*16 array, dimension (LDAFB,N) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A as computed by ZPBTRF, in the same storage format as A (see AB). LDAFB LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= KD+1. B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZPBTRS. 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 jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Internal Parameters: ITMAX is the maximum number of steps of iterative refinement. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 187 of file zpbrfs.f. subroutine zpbstf (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, integer INFO)ZPBSTFPurpose: ZPBSTF computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A. This routine is designed to be used in conjunction with ZHBGST. The factorization has the form A = S**H*S where S is a band matrix of the same bandwidth as A and the following structure: S = ( U ) ( M L ) where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order nm. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first kd+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the factor S from the split Cholesky factorization A = S**H*S. See Further Details. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The band storage scheme is illustrated by the following example, when N = 7, KD = 2: S = ( s11 s12 s13 ) ( s22 s23 s24 ) ( s33 s34 ) ( s44 ) ( s53 s54 s55 ) ( s64 s65 s66 ) ( s75 s76 s77 ) If UPLO = 'U', the array AB holds: on entry: on exit: * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 If UPLO = 'L', the array AB holds: on entry: on exit: a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 * a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * * Array elements marked * are not used by the routine; s12**H denotes conjg(s12); the diagonal elements of S are real. Definition at line 152 of file zpbstf.f. subroutine zpbtf2 (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, integer INFO)ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).Purpose: ZPBTF2 computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. The factorization has the form A = U**H * U , if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix, U**H is the conjugate transpose of U, and L is lower triangular. This is the unblocked version of the algorithm, calling Level 2 BLAS. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, in the same storage format as A. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = k, the kth argument had an illegal value > 0: if INFO = k, the leading minor of order k is not positive definite, and the factorization could not be completed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine. Definition at line 141 of file zpbtf2.f. subroutine zpbtrf (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, integer INFO)ZPBTRFPurpose: ZPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. The factorization has the form A = U**H * U, if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine. Contributors: Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March
23, 1989
Definition at line 141 of file zpbtrf.f. subroutine zpbtrs (character UPLO, integer N, integer KD, integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZPBTRSPurpose: ZPBTRS solves a system of linear equations A*X = B with a Hermitian positive definite band matrix A using the Cholesky factorization A = U**H *U or A = L*L**H computed by ZPBTRF. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangular factor stored in AB; = 'L': Lower triangular factor stored in AB. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) The triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, stored in the first KD+1 rows of the array. The jth column of U or L is stored in the jth column of the array AB as follows: if UPLO ='U', AB(kd+1+ij,j) = U(i,j) for max(1,jkd)<=i<=j; if UPLO ='L', AB(1+ij,j) = L(i,j) for j<=i<=min(n,j+kd). LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. B B is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 120 of file zpbtrs.f. subroutine zpftrf (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) A, integer INFO)ZPFTRFPurpose: ZPFTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A. The factorization has the form A = U**H * U, if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. This is the block version of the algorithm, calling Level 3 BLAS. Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugatetranspose TRANSR of RFP A is stored. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of RFP A is stored; = 'L': Lower triangle of RFP A is stored. N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the Hermitian matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' then RFP A is (0:N,0:k1) when N is even; k=N/2. RFP A is (0:N1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is the Conjugatetranspose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the nt elements of upper packed A. If UPLO = 'L' the RFP A contains the elements of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N is odd. See the Note below for more details. On exit, if INFO = 0, the factor U or L from the Cholesky factorization RFP A = U**H*U or RFP A = L*L**H. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed. Further Notes on RFP Format: ============================ We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 210 of file zpftrf.f. subroutine zpftri (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) A, integer INFO)ZPFTRIPurpose: ZPFTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPFTRF. Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugatetranspose TRANSR of RFP A is stored. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the Hermitian matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' then RFP A is (0:N,0:k1) when N is even; k=N/2. RFP A is (0:N1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is the Conjugatetranspose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the nt elements of upper packed A. If UPLO = 'L' the RFP A contains the elements of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N is odd. See the Note below for more details. On exit, the Hermitian inverse of the original matrix, in the same storage format. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 211 of file zpftri.f. subroutine zpftrs (character TRANSR, character UPLO, integer N, integer NRHS, complex*16, dimension( 0: * ) A, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZPFTRSPurpose: ZPFTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPFTRF. Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugatetranspose TRANSR of RFP A is stored. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of RFP A is stored; = 'L': Lower triangle of RFP A is stored. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. A A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); The triangular factor U or L from the Cholesky factorization of RFP A = U**H*U or RFP A = L*L**H, as computed by ZPFTRF. See note below for more details about RFP A. B B is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 219 of file zpftrs.f. subroutine zppcon (character UPLO, integer N, complex*16, dimension( * ) AP, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZPPCONPurpose: ZPPCON estimates the reciprocal of the condition number (in the 1norm) of a complex Hermitian positive definite packed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPPTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, packed columnwise in a linear array. The jth column of U or L is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = U(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = L(i,j) for j<=i<=n. ANORM ANORM is DOUBLE PRECISION The 1norm (or infinitynorm) of the Hermitian 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 1norm of inv(A) computed in this routine. WORK WORK is COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 117 of file zppcon.f. subroutine zppequ (character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)ZPPEQUPurpose: ZPPEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A in packed storage and reduce its condition number (with respect to the twonorm). S contains the scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/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 ith argument had an illegal value > 0: if INFO = i, the ith diagonal element is nonpositive. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 116 of file zppequ.f. subroutine zpprfs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( * ) AFP, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZPPRFSPurpose: ZPPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and packed, and provides error bounds and backward error estimates for the solution. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. AFP AFP is COMPLEX*16 array, dimension (N*(N+1)/2) The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF, packed columnwise in a linear array in the same format as A (see AP). B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZPPTRS. 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 jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Internal Parameters: ITMAX is the maximum number of steps of iterative refinement. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 169 of file zpprfs.f. subroutine zpptrf (character UPLO, integer N, complex*16, dimension( * ) AP, integer INFO)ZPPTRFPurpose: ZPPTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A stored in packed format. The factorization has the form A = U**H * U, if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. See below for further details. On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, in the same storage format as A. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U': Twodimensional storage of the Hermitian matrix A: a11 a12 a13 a14 a22 a23 a24 a33 a34 (aij = conjg(aji)) a44 Packed storage of the upper triangle of A: AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] Definition at line 118 of file zpptrf.f. subroutine zpptri (character UPLO, integer N, complex*16, dimension( * ) AP, integer INFO)ZPPTRIPurpose: ZPPTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPPTRF. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangular factor is stored in AP; = 'L': Lower triangular factor is stored in AP. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, packed columnwise as a linear array. The jth column of U or L is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = U(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = L(i,j) for j<=i<=n. On exit, the upper or lower triangle of the (Hermitian) inverse of A, overwriting the input factor U or L. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 92 of file zpptri.f. subroutine zpptrs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZPPTRSPurpose: ZPPTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A in packed storage using the Cholesky factorization A = U**H * U or A = L * L**H computed by ZPPTRF. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The triangular factor U or L from the Cholesky factorization A = U**H * U or A = L * L**H, packed columnwise in a linear array. The jth column of U or L is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = U(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = L(i,j) for j<=i<=n. B B is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 107 of file zpptrs.f. subroutine zpstf2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, double precision TOL, double precision, dimension( 2*n ) WORK, integer INFO)ZPSTF2 computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.Purpose: ZPSTF2 computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix A. The factorization has the form P**T * A * P = U**H * U , if UPLO = 'U', P**T * A * P = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular, and P is stored as vector PIV. This algorithm does not attempt to check that A is positive semidefinite. This version of the algorithm calls level 2 BLAS. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 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 (K1)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 Kth argument had an illegal value, = 0: algorithm completed successfully, and > 0: the matrix A is either rank deficient with computed rank as returned in RANK, or is not positive semidefinite. See Section 7 of LAPACK Working Note #161 for further information. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 141 of file zpstf2.f. subroutine zpstrf (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, double precision TOL, double precision, dimension( 2*n ) WORK, integer INFO)ZPSTRF computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.Purpose: ZPSTRF computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix A. The factorization has the form P**T * A * P = U**H * U , if UPLO = 'U', P**T * A * P = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular, and P is stored as vector PIV. This algorithm does not attempt to check that A is positive semidefinite. This version of the algorithm calls level 3 BLAS. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 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 (K1)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 Kth argument had an illegal value, = 0: algorithm completed successfully, and > 0: the matrix A is either rank deficient with computed rank as returned in RANK, or is not positive semidefinite. See Section 7 of LAPACK Working Note #161 for further information. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 141 of file zpstrf.f. subroutine zspcon (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)ZSPCONPurpose: ZSPCON estimates the reciprocal of the condition number (in the 1norm) of a complex symmetric packed matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 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 ZSPTRF, 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 ZSPTRF. ANORM ANORM is DOUBLE PRECISION The 1norm 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 1norm of inv(A) computed in this routine. WORK WORK is COMPLEX*16 array, dimension (2*N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 117 of file zspcon.f. subroutine zsprfs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( * ) AFP, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZSPRFSPurpose: ZSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2*nj)/2) = A(i,j) for j<=i<=n. AFP AFP is COMPLEX*16 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 ZSPTRF, 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 ZSPTRF. B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZSPTRS. 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 jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Internal Parameters: ITMAX is the maximum number of steps of iterative refinement. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 178 of file zsprfs.f. subroutine zsptrf (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)ZSPTRFPurpose: ZSPTRF computes the factorization of a complex symmetric matrix A stored in packed format using the BunchKaufman 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 1by1 and 2by2 diagonal blocks. Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 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 jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/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 1by1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 diagonal block. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: 596  Based on modifications by J. Lewis, Boeing Computer Services Company If UPLO = 'U', then A = U*D*U**T, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1by1 and 2by2 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 ) ks U(k) = ( 0 I 0 ) s ( 0 0 I ) nk ks s nk If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper triangle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1: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 1by1 and 2by2 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 ) k1 L(k) = ( 0 I 0 ) s ( 0 v I ) nks+1 k1 s nks+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). Definition at line 157 of file zsptrf.f. subroutine zsptri (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)ZSPTRIPurpose: ZSPTRI computes the inverse of a complex symmetric indefinite matrix A in packed storage using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 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 ZSPTRF, 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 jth column of inv(A) is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = inv(A)(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/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 ZSPTRF. WORK WORK is COMPLEX*16 array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 108 of file zsptri.f. subroutine zsptrs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZSPTRSPurpose: ZSPTRS solves a system of linear equations A*X = B with a complex symmetric matrix A stored in packed format using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. AP AP is COMPLEX*16 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 ZSPTRF, 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 ZSPTRF. B B is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 114 of file zsptrs.f. subroutine zstedc (character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)ZSTEDCPurpose: ZSTEDC computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band complex Hermitian matrix can also be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this matrix to tridiagonal form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C90, or Cray2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See DLAED3 for details. Parameters COMPZ
COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'I': Compute eigenvectors of tridiagonal matrix also. = 'V': Compute eigenvectors of original Hermitian matrix also. On entry, Z contains the unitary matrix used to reduce the original matrix to tridiagonal form. N N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0. D D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order. E E is DOUBLE PRECISION array, dimension (N1) On entry, the subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. Z Z is COMPLEX*16 array, dimension (LDZ,N) On entry, if COMPZ = 'V', then Z contains the unitary matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original Hermitian matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If eigenvectors are desired, then LDZ >= max(1,N). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1. If COMPZ = 'V' and N > 1, LWORK must be at least N*N. Note that for COMPZ = 'V', then if N is less than or equal to the minimum divide size, usually 25, then LWORK need only be 1. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. RWORK RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. LRWORK LRWORK is INTEGER The dimension of the array RWORK. If COMPZ = 'N' or N <= 1, LRWORK must be at least 1. If COMPZ = 'V' and N > 1, LRWORK must be at least 1 + 3*N + 2*N*lg N + 4*N**2 , where lg( N ) = smallest integer k such that 2**k >= N. If COMPZ = 'I' and N > 1, LRWORK must be at least 1 + 4*N + 2*N**2 . Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LRWORK need only be max(1,2*(N1)). If LRWORK = 1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. IWORK IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK LIWORK is INTEGER The dimension of the array IWORK. If COMPZ = 'N' or N <= 1, LIWORK must be at least 1. If COMPZ = 'V' or N > 1, LIWORK must be at least 6 + 6*N + 5*N*lg N. If COMPZ = 'I' or N > 1, LIWORK must be at least 3 + 5*N . Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LIWORK need only be 1. If LIWORK = 1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1). Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Definition at line 210 of file zstedc.f. subroutine zstegr (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, complex*16, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)ZSTEGRPurpose: ZSTEGR 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. ZSTEGR is a compatibility wrapper around the improved ZSTEMR routine. See ZSTEMR for further details. One important change is that the ABSTOL parameter no longer provides any benefit and hence is no longer used. Note : ZSTEGR and ZSTEMR work only on machines which follow IEEE754 floatingpoint 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 IEEE754 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 halfopen interval (VL,VU] will be found. = 'I': the ILth through IUth 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 (N1) subdiagonal elements of the tridiagonal matrix T in elements 1 to N1 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 = IUIL+1. W W is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. Z Z is COMPLEX*16 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 ith 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 ith computed eigenvector is nonzero only in elements ISUPPZ( 2*i1 ) 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 ith argument had an illegal value > 0: if INFO = 1X, internal error in DLARRE, if INFO = 2X, internal error in ZLARRV. Here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code returned by DLARRE or ZLARRV, respectively. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, LBNL/NERSC, USA
Definition at line 262 of file zstegr.f. subroutine zstein (integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, integer M, double precision, dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, complex*16, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)ZSTEINPurpose: ZSTEIN computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. The maximum number of iterations allowed for each eigenvector is specified by an internal parameter MAXITS (currently set to 5). Although the eigenvectors are real, they are stored in a complex array, which may be passed to ZUNMTR or ZUPMTR for back transformation to the eigenvectors of a complex Hermitian matrix which was reduced to tridiagonal form. Parameters N
N is INTEGER The order of the matrix. N >= 0. D D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. E E is DOUBLE PRECISION array, dimension (N1) The (n1) subdiagonal elements of the tridiagonal matrix T, stored in elements 1 to N1. 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 splitoff 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 COMPLEX*16 array, dimension (LDZ, M) The computed eigenvectors. The eigenvector associated with the eigenvalue W(i) is stored in the ith column of Z. Any vector which fails to converge is set to its current iterate after MAXITS iterations. The imaginary parts of the eigenvectors are set to zero. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N). WORK WORK is 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 ith argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in MAXITS iterations. Their indices are stored in array IFAIL. Internal Parameters: MAXITS INTEGER, default = 5 The maximum number of iterations performed. EXTRA INTEGER, default = 2 The number of iterations performed after norm growth criterion is satisfied, should be at least 1. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 180 of file zstein.f. subroutine zstemr (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, complex*16, 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)ZSTEMRPurpose: ZSTEMR 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. 128, 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/CSD97971, UC Berkeley, May 1997. Further Details 1.ZSTEMR works only on machines which follow IEEE754 floatingpoint standard in their handling of infinities and NaNs. This permits the use of efficient inner loops avoiding a check for zero divisors. 2. LAPACK routines can be used to reduce a complex Hermitean matrix to real symmetric tridiagonal form. (Any complex Hermitean tridiagonal matrix has real values on its diagonal and potentially complex numbers on its offdiagonals. By applying a similarity transform with an appropriate diagonal matrix diag(1,e^{i hy_1}, 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 halfopen interval (VL,VU] will be found. = 'I': the ILth through IUth 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 (N1) subdiagonal elements of the tridiagonal matrix T in elements 1 to N1 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 = IUIL+1. W W is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. Z Z is COMPLEX*16 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 ith 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 >= IUIL+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 ith computed eigenvector is nonzero only in elements ISUPPZ( 2*i1 ) 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 relativeaccuracy 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 ith argument had an illegal value > 0: if INFO = 1X, internal error in DLARRE, if INFO = 2X, internal error in ZLARRV. Here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code returned by DLARRE or ZLARRV, respectively. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: Beresford Parlett, University of California, Berkeley,
USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
Definition at line 335 of file zstemr.f. subroutine zsteqr (character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)ZSTEQRPurpose: ZSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band complex Hermitian matrix can also be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this matrix to tridiagonal form. Parameters COMPZ
COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original Hermitian matrix. On entry, Z must contain the unitary matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix. N N is INTEGER The order of the matrix. N >= 0. D D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order. E E is DOUBLE PRECISION array, dimension (N1) On entry, the (n1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. Z Z is COMPLEX*16 array, dimension (LDZ, N) On entry, if COMPZ = 'V', then Z contains the unitary matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original Hermitian matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N). WORK WORK is DOUBLE PRECISION array, dimension (max(1,2*N2)) If COMPZ = 'N', then WORK is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is unitarily similar to the original matrix. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 131 of file zsteqr.f. subroutine ztbcon (character NORM, character UPLO, character DIAG, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision RCOND, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTBCONPurpose: ZTBCON estimates the reciprocal of the condition number of a triangular band matrix A, in either the 1norm or the infinitynorm. 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 1norm condition number or the infinitynorm condition number is required: = '1' or 'O': 1norm; = 'I': Infinitynorm. UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first kd+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 141 of file ztbcon.f. subroutine ztbrfs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTBRFSPurpose: ZTBRFS 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 ZTBTRS or some other means before entering this routine. ZTBRFS does not do iterative refinement because doing so cannot improve the backward error. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. TRANS TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first kd+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) The solution matrix X. LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). FERR FERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 186 of file ztbrfs.f. subroutine ztbtrs (character UPLO, character TRANS, character DIAG, integer N, integer KD, integer NRHS, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZTBTRSPurpose: ZTBTRS solves a triangular system of the form A * X = B, A**T * X = B, or A**H * X = B, where A is a triangular band matrix of order N, and B is an NbyNRHS matrix. A check is made to verify that A is nonsingular. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. TRANS TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of superdiagonals or subdiagonals of the triangular band matrix A. KD >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. AB AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first kd+1 rows of AB. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd). If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. B B is COMPLEX*16 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 ith argument had an illegal value > 0: if INFO = i, the ith diagonal element of A is zero, indicating that the matrix is singular and the solutions X have not been computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 144 of file ztbtrs.f. subroutine ztfsm (character TRANSR, character SIDE, character UPLO, character TRANS, character DIAG, integer M, integer N, complex*16 ALPHA, complex*16, dimension( 0: * ) A, complex*16, dimension( 0: ldb1, 0: * ) B, integer LDB)ZTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).Purpose: Level 3 BLAS like routine for A in RFP Format. ZTFSM 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 nonunit, upper or lower triangular matrix and op( A ) is one of op( A ) = A or op( A ) = A**H. A is in Rectangular Full Packed (RFP) Format. The matrix X is overwritten on B. Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': The Normal Form of RFP A is stored; = 'C': The Conjugatetranspose Form of RFP A is stored. SIDE SIDE is CHARACTER*1 On entry, SIDE specifies whether op( A ) appears on the left or right of X as follows: SIDE = 'L' or 'l' op( A )*X = alpha*B. SIDE = 'R' or 'r' X*op( A ) = alpha*B. Unchanged on exit. UPLO UPLO is CHARACTER*1 On entry, UPLO specifies whether the RFP matrix A came from an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' RFP A came from an upper triangular matrix UPLO = 'L' or 'l' RFP A came from a lower triangular matrix Unchanged on exit. TRANS TRANS is CHARACTER*1 On entry, TRANS specifies the form of op( A ) to be used in the matrix multiplication as follows: TRANS = 'N' or 'n' op( A ) = A. TRANS = 'C' or 'c' op( A ) = conjg( A' ). Unchanged on exit. DIAG DIAG is CHARACTER*1 On entry, DIAG specifies whether or not RFP A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. Unchanged on exit. M M is INTEGER On entry, M specifies the number of rows of B. M must be at least zero. Unchanged on exit. N N is INTEGER On entry, N specifies the number of columns of B. N must be at least zero. Unchanged on exit. ALPHA ALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry. Unchanged on exit. A A is COMPLEX*16 array, dimension (N*(N+1)/2) NT = N*(N+1)/2. On entry, the matrix A in RFP Format. RFP Format is described by TRANSR, UPLO and N as follows: If TRANSR='N' then RFP A is (0:N,0:K1) when N is even; K=N/2. RFP A is (0:N1,0:K) when N is odd; K=N/2. If TRANSR = 'C' then RFP is the Conjugatetranspose 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 conjugatetranspose Format. If UPLO = 'L' the RFP A contains the NT elements of lower packed A either in normal or conjugatetranspose Format. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and is N when is odd. See the Note below for more details. Unchanged on exit. B B is COMPLEX*16 array, dimension (LDB,N) Before entry, the leading m by n part of the array B must contain the righthand side matrix B, and on exit is overwritten by the solution matrix X. LDB LDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. LDB must be at least max( 1, m ). Unchanged on exit. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 296 of file ztfsm.f. subroutine ztftri (character TRANSR, character UPLO, character DIAG, integer N, complex*16, dimension( 0: * ) A, integer INFO)ZTFTRIPurpose: ZTFTRI computes the inverse of a triangular matrix A stored in RFP format. This is a Level 3 BLAS version of the algorithm. Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugatetranspose TRANSR of RFP A is stored. UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the triangular matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' then RFP A is (0:N,0:k1) when N is even; k=N/2. RFP A is (0:N1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is the Conjugatetranspose of RFP A as defined when TRANSR = 'N'. The contents of RFP A are defined by UPLO as follows: If UPLO = 'U' the RFP A contains the nt elements of upper packed A; If UPLO = 'L' the RFP A contains the nt elements of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N is odd. See the Note below for more details. On exit, the (triangular) inverse of the original matrix, in the same storage format. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero. The triangular matrix is singular and its inverse can not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 220 of file ztftri.f. subroutine ztfttp (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) ARF, complex*16, dimension( 0: * ) AP, integer INFO)ZTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).Purpose: ZTFTTP copies a triangular matrix A from rectangular full packed format (TF) to standard packed format (TP). Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': ARF is in Normal format; = 'C': ARF is in Conjugatetranspose format; UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. N N is INTEGER The order of the matrix A. N >= 0. ARF ARF is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On entry, the upper or lower triangular matrix A stored in RFP format. For a further discussion see Notes below. AP AP is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On exit, the upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 207 of file ztfttp.f. subroutine ztfttr (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) ARF, complex*16, dimension( 0: lda1, 0: * ) A, integer LDA, integer INFO)ZTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).Purpose: ZTFTTR copies a triangular matrix A from rectangular full packed format (TF) to standard full format (TR). Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': ARF is in Normal format; = 'C': ARF is in Conjugatetranspose format; UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. N N is INTEGER The order of the matrix A. N >= 0. ARF ARF is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On entry, the upper or lower triangular matrix A stored in RFP format. For a further discussion see Notes below. A A is COMPLEX*16 array, dimension ( LDA, N ) On exit, the triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 215 of file ztfttr.f. subroutine ztgsen (integer IJOB, logical WANTQ, logical WANTZ, logical, dimension( * ) SELECT, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer M, double precision PL, double precision PR, double precision, dimension( * ) DIF, complex*16, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)ZTGSENPurpose: ZTGSEN reorders the generalized Schur decomposition of a complex matrix pair (A, B) (in terms of an unitary equivalence trans formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the pair (A,B). The leading columns of Q and Z form unitary bases of the corresponding left and right eigenspaces (deflating subspaces). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. ZTGSEN also computes the generalized eigenvalues w(j)= ALPHA(j) / BETA(j) of the reordered matrix pair (A, B). Optionally, the routine computes estimates of reciprocal condition numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of "projections" onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)block. Parameters IJOB
IJOB is INTEGER Specifies whether condition numbers are required for the cluster of eigenvalues (PL and PR) or the deflating subspaces (Difu and Difl): =0: Only reorder w.r.t. SELECT. No extras. =1: Reciprocal of norms of "projections" onto left and right eigenspaces w.r.t. the selected cluster (PL and PR). =2: Upper bounds on Difu and Difl. Fnormbased estimate (DIF(1:2)). =3: Estimate of Difu and Difl. 1normbased estimate (DIF(1:2)). About 5 times as expensive as IJOB = 2. =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version to get it all. =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) WANTQ WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. WANTZ WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. SELECT SELECT is LOGICAL array, dimension (N) SELECT specifies the eigenvalues in the selected cluster. To select an eigenvalue w(j), SELECT(j) must be set to .TRUE.. N N is INTEGER The order of the matrices A and B. N >= 0. A A is COMPLEX*16 array, dimension(LDA,N) On entry, the upper triangular matrix A, in generalized Schur canonical form. On exit, A is overwritten by the reordered matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension(LDB,N) On entry, the upper triangular matrix B, in generalized Schur canonical form. On exit, B is overwritten by the reordered matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). ALPHA ALPHA is COMPLEX*16 array, dimension (N) BETA BETA is COMPLEX*16 array, dimension (N) The diagonal elements of A and B, respectively, when the pair (A,B) has been reduced to generalized Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if WANTQ = .TRUE., Q is an NbyN matrix. On exit, Q has been postmultiplied by the left unitary transformation matrix which reorder (A, B); The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTQ = .FALSE., Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N. Z Z is COMPLEX*16 array, dimension (LDZ,N) On entry, if WANTZ = .TRUE., Z is an NbyN matrix. On exit, Z has been postmultiplied by the left unitary transformation matrix which reorder (A, B); The leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTZ = .FALSE., Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N. M M is INTEGER The dimension of the specified pair of left and right eigenspaces, (deflating subspaces) 0 <= M <= N. PL PL is 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 eigenspace with respect to the selected cluster. 0 < PL, PR <= 1. If M = 0 or M = N, PL = PR = 1. If IJOB = 0, 2 or 3 PL, PR are not referenced. DIF DIF is 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 Fnormbased upper bounds on Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1normbased estimates of Difu and Difl, computed using reversed communication with ZLACN2. If M = 0 or N, DIF(1:2) = Fnorm([A, B]). If IJOB = 0 or 1, DIF is not referenced. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1 If IJOB = 1, 2 or 4, LWORK >= 2*M*(NM) If IJOB = 3 or 5, LWORK >= 4*M*(NM) If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. IWORK IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. LIWORK LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2 or 4, LIWORK >= N+2; If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(NM)); 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 ith 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 illconditioned. (A, B) may have been partially reordered. If requested, 0 is returned in DIF(*), PL and PR. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: ZTGSEN first collects the selected eigenvalues by computing unitary U and W that move them to the top left corner of (A, B). In other words, the selected eigenvalues are the eigenvalues of (A11, B11) in U**H*(A, B)*W = (A11 A12) (B11 B12) n1 ( 0 A22),( 0 B22) n2 n1 n2 n1 n2 where N = n1+n2 and U**H means the conjugate transpose of U. The first n1 columns of U and W span the specified pair of left and right eigenspaces (deflating subspaces) of (A, B). If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the reordered generalized Schur form of (C, D) is given by (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H, and the first n1 columns of Q*U and Z*W span the corresponding deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). Note that if the selected eigenvalue is sufficiently illconditioned, 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)] = sigmamin( Zu ) and Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], where sigmamin(Zu) is the smallest singular value of the (2*n1*n2)by(2*n1*n2) matrix Zu = [ kron(In2, A11) kron(A22**H, In1) ] [ kron(In2, B11) kron(B22**H, In1) ]. Here, Inx is the identity matrix of size nx and A22**H is the conjugate 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 = (Fnorm(L)**2+1)**(1/2) and PR = (Fnorm(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 Fnorm(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 = ( Fnorm(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 maxangle(L, L') <= arctan( y * PL / (1  y * (1  PL * PL)**(1/2)) maxangle(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 Frobeniusnorm based estimate DIF is not wanted (see ZLATDF), then the parameter IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF (IJOB = 2 will be used)). See ZTGSYL for more details. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing
Science, Umea University, S901 87 Umea, Sweden.
References: [1] B. Kagstrom; A Direct Method for Reordering
Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A,
B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and RealTime
Applications, Kluwer Academic Publ. 1993, pp 195218.
[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, S901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACKStyle 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, S901 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.
Definition at line 430 of file ztgsen.f. subroutine ztgsja (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, double precision TOLA, double precision TOLB, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, complex*16, dimension( ldu, * ) U, integer LDU, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( * ) WORK, integer NCYCLE, integer INFO)ZTGSJAPurpose: ZTGSJA computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine ZGGSVP from a general MbyN matrix A and PbyN matrix B: NKL K L A = K ( 0 A12 A13 ) if MKL >= 0; L ( 0 0 A23 ) MKL ( 0 0 0 ) NKL K L A = K ( 0 A12 A13 ) if MKL < 0; MK ( 0 0 A23 ) NKL K L B = L ( 0 0 B13 ) PL ( 0 0 0 ) where the KbyK matrix A12 and LbyL matrix B13 are nonsingular upper triangular; A23 is LbyL upper triangular if MKL >= 0, otherwise A23 is (MK)byL upper trapezoidal. On exit, U**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ), where U, V and Q are unitary matrices. R is a nonsingular upper triangular matrix, and D1 and D2 are ``diagonal'' matrices, which are of the following structures: If MKL >= 0, K L D1 = K ( I 0 ) L ( 0 C ) MKL ( 0 0 ) K L D2 = L ( 0 S ) PL ( 0 0 ) NKL 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,NKL+1:N) on exit. If MKL < 0, K MK K+LM D1 = K ( I 0 0 ) MK ( 0 C 0 ) K MK K+LM D2 = MK ( 0 S 0 ) K+LM ( 0 0 I ) PL ( 0 0 0 ) NKL K MK K+LM ( 0 R ) = K ( 0 R11 R12 R13 ) MK ( 0 0 R22 R23 ) K+LM ( 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, NKL+1:N) and R33 is stored ( 0 R22 R23 ) in B(MK+1:L,N+MKL+1:N) on exit. The computation of the unitary transformation matrices U, V or Q is optional. These matrices may either be formed explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1. Parameters JOBU
JOBU is CHARACTER*1 = 'U': U must contain a unitary matrix U1 on entry, and the product U1*U is returned; = 'I': U is initialized to the unit matrix, and the unitary matrix U is returned; = 'N': U is not computed. JOBV JOBV is CHARACTER*1 = 'V': V must contain a unitary matrix V1 on entry, and the product V1*V is returned; = 'I': V is initialized to the unit matrix, and the unitary matrix V is returned; = 'N': V is not computed. JOBQ JOBQ is CHARACTER*1 = 'Q': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'N': Q is not computed. M M is INTEGER The number of rows of the matrix A. M >= 0. P P is INTEGER The number of rows of the matrix B. P >= 0. N N is INTEGER The number of columns of the matrices A and B. N >= 0. K K is INTEGER L L is INTEGER K and L specify the subblocks in the input matrices A and B: A23 = A(K+1:MIN(K+L,M),NL+1:N) and B13 = B(1:L,,NL+1:N) of A and B, whose GSVD is going to be computed by ZTGSJA. See Further Details. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the MbyN matrix A. On exit, A(NK+1:N,1:MIN(K+L,M) ) contains the triangular matrix R or part of R. See Purpose for details. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). B B is COMPLEX*16 array, dimension (LDB,N) On entry, the PbyN matrix B. On exit, if necessary, B(MK+1:L,N+MKL+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 MKL >= 0, ALPHA(K+1:K+L) = diag(C), BETA(K+1:K+L) = diag(S), or if MKL < 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 COMPLEX*16 array, dimension (LDU,M) On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by ZGGSVP). On exit, if JOBU = 'I', U contains the unitary matrix U; if JOBU = 'U', U contains the product U1*U. If JOBU = 'N', U is not referenced. LDU LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. V V is COMPLEX*16 array, dimension (LDV,P) On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by ZGGSVP). On exit, if JOBV = 'I', V contains the unitary matrix V; if JOBV = 'V', V contains the product V1*V. If JOBV = 'N', V is not referenced. LDV LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by ZGGSVP). On exit, if JOBQ = 'I', Q contains the unitary matrix Q; if JOBQ = 'Q', Q contains the product Q1*Q. If JOBQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. WORK WORK is COMPLEX*16 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 ith argument had an illegal value. = 1: the procedure does not converge after MAXIT cycles. Internal Parameters: MAXIT INTEGER MAXIT specifies the total loops that the iterative procedure may take. If after MAXIT cycles, the routine fails to converge, we return INFO = 1. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce min(L,MK)byL triangular (or trapezoidal) matrix A23 and LbyL matrix B13 to the form: U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1, where U1, V1 and Q1 are unitary matrix. C1 and S1 are diagonal matrices satisfying C1**2 + S1**2 = I, and R1 is an LbyL nonsingular upper triangular matrix. Definition at line 376 of file ztgsja.f. subroutine ztgsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, double precision, dimension( * ) S, double precision, dimension( * ) DIF, integer MM, integer M, complex*16, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)ZTGSNAPurpose: ZTGSNA estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. Parameters JOB
JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (DIF): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (DIF); = 'B': for both eigenvalues and eigenvectors (S and DIF). HOWMNY HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT. SELECT SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the corresponding jth eigenvalue and/or eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced. N N is INTEGER The order of the square matrix pair (A, B). N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) The upper triangular matrix A in the pair (A,B). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB,N) The upper triangular matrix B in the pair (A, B). LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). VL VL is COMPLEX*16 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 ZTGEVC. If JOB = 'V', VL is not referenced. LDVL LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; and If JOB = 'E' or 'B', LDVL >= N. VR VR is COMPLEX*16 array, dimension (LDVR,M) IF JOB = 'E' or 'B', VR must contain right eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZTGEVC. 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. 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. If the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. For each eigenvalue/vector specified by SELECT, DIF stores a Frobenius normbased estimate of Difl. If JOB = 'E', DIF is not referenced. MM MM is INTEGER The number of elements in the arrays S and DIF. MM >= M. M M is INTEGER The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected eigenvalue one element is used. If HOWMNY = 'A', M is set to N. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N). If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). IWORK IWORK is INTEGER array, dimension (N+2) If JOB = 'E', IWORK is not referenced. INFO INFO is INTEGER = 0: Successful exit < 0: If INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The reciprocal of the condition number of the ith generalized eigenvalue w = (a, b) is defined as S(I) = (v**HAu**2 + v**HBu**2)**(1/2) / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of (A, B) corresponding to w; z denotes the absolute value of the complex number, and norm(u) denotes the 2norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the matrix pair (A, B). If both a and b equal zero, then (A,B) is singular and S(I) = 1 is returned. An approximate error bound on the chordal distance between the ith computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is chord(w, lambda) <= EPS * norm(A, B) / S(I), where EPS is the machine precision. The reciprocal of the condition number of the right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows. Suppose (A, B) = ( a * ) ( b * ) 1 ( 0 A22 ),( 0 B22 ) n1 1 n1 1 n1 Then the reciprocal condition number DIF(I) is Difl[(a, b), (A22, B22)] = sigmamin( Zl ) where sigmamin(Zl) denotes the smallest singular value of Zl = [ kron(a, In1) kron(1, A22) ] [ kron(b, In1) kron(1, B22) ]. Here In1 is the identity matrix of size n1 and X**H is the conjugate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. We approximate the smallest singular value of Zl with an upper bound. This is done by ZLATDF. An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by EPS * norm(A, B) / DIF(i). See ref. [23] for more details and further references. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing
Science, Umea University, S901 87 Umea, Sweden.
References: [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and RealTime Applications, Kluwer Academic Publ. 1993, pp 195218. [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, S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. [3] B. Kagstrom and P. Poromaa, LAPACKStyle 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, S901 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. Definition at line 308 of file ztgsna.f. subroutine ztpcon (character NORM, character UPLO, character DIAG, integer N, complex*16, dimension( * ) AP, double precision RCOND, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTPCONPurpose: ZTPCON estimates the reciprocal of the condition number of a packed triangular matrix A, in either the 1norm or the infinitynorm. 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 1norm condition number or the infinitynorm condition number is required: = '1' or 'O': 1norm; = 'I': Infinitynorm. UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 128 of file ztpcon.f. subroutine ztpmqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer L, integer NB, 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)ZTPMQRTPurpose: ZTPMQRT applies a complex orthogonal matrix Q obtained from a "triangularpentagonal" complex block reflector H to a general complex matrix C, which consists of two blocks A and B. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. L L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details. NB NB is INTEGER The block size used for the storage of T. K >= NB >= 1. This must be the same value of NB used to generate T in CTPQRT. V V is COMPLEX*16 array, dimension (LDV,K) The ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CTPQRT in B. See Further Details. LDV LDV is INTEGER The leading dimension of the array V. If SIDE = 'L', LDV >= max(1,M); if SIDE = 'R', LDV >= max(1,N). T T is COMPLEX*16 array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by CTPQRT, stored as a NBbyK matrix. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. A A is COMPLEX*16 array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the KbyN or MbyK 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*16 array, dimension (LDB,N) On entry, the MbyN 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*NB if SIDE = 'L', or M*NB if SIDE = 'R'. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2: V = [V1] [V2]. The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L rows of a KbyK 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 KbyN, B is MbyN and V is MbyK. [B] If SIDE = 'R': C = [A B] where A is MbyK, B is MbyN and V is NbyK. The complex orthogonal matrix Q is formed from V and T. If TRANS='N' and SIDE='L', C is on exit replaced with Q * C. If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C. If TRANS='N' and SIDE='R', C is on exit replaced with C * Q. If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H. Definition at line 214 of file ztpmqrt.f. subroutine ztpqrt (integer M, integer N, integer L, integer NB, 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)ZTPQRTPurpose: ZTPQRT computes a blocked QR factorization of a complex "triangularpentagonal" matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Parameters M
M is INTEGER The number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. L L is INTEGER The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. NB NB is INTEGER The block size to be used in the blocked QR. N >= NB >= 1. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the upper triangular NbyN matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB,N) On entry, the pentagonal MbyN matrix B. The first ML rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). T T is COMPLEX*16 array, dimension (LDT,N) The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. WORK WORK is COMPLEX*16 array, dimension (NB*N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The input matrix C is a (N+M)byN matrix C = [ A ] [ B ] where A is an upper triangular NbyN matrix, and B is MbyN pentagonal matrix consisting of a (ML)byN rectangular matrix B1 on top of a LbyN upper trapezoidal matrix B2: B = [ B1 ] < (ML)byN rectangular [ B2 ] < LbyN upper trapezoidal. The upper trapezoidal matrix B2 consists of the first L rows of a NbyN upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular MbyN; if M=L=N, B is upper triangular. The matrix W stores the elementary reflectors H(i) in the ith column below the diagonal (of A) in the (N+M)byN input matrix C C = [ A ] < upper triangular NbyN [ B ] < MbyN pentagonal so that W can be represented as W = [ I ] < identity, NbyN [ V ] < MbyN, 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 ] < (ML)byN rectangular [ V2 ] < LbyN 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  (B1)*NB. For each of the B blocks, a upper triangular block reflector factor is computed: T1, T2, ..., TB. The NBbyNB (and IBbyIB for the last block) T's are stored in the NBbyN matrix T as T = [T1 T2 ... TB]. Definition at line 187 of file ztpqrt.f. subroutine ztpqrt2 (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)ZTPQRT2 computes a QR factorization of a real or complex 'triangularpentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.Purpose: ZTPQRT2 computes a QR factorization of a complex "triangularpentagonal" matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Parameters M
M is INTEGER The total number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. L L is INTEGER The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the upper triangular NbyN matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB,N) On entry, the pentagonal MbyN matrix B. The first ML rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). T T is COMPLEX*16 array, dimension (LDT,N) The NbyN 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The input matrix C is a (N+M)byN matrix C = [ A ] [ B ] where A is an upper triangular NbyN matrix, and B is MbyN pentagonal matrix consisting of a (ML)byN rectangular matrix B1 on top of a LbyN upper trapezoidal matrix B2: B = [ B1 ] < (ML)byN rectangular [ B2 ] < LbyN upper trapezoidal. The upper trapezoidal matrix B2 consists of the first L rows of a NbyN upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular MbyN; if M=L=N, B is upper triangular. The matrix W stores the elementary reflectors H(i) in the ith column below the diagonal (of A) in the (N+M)byN input matrix C C = [ A ] < upper triangular NbyN [ B ] < MbyN pentagonal so that W can be represented as W = [ I ] < identity, NbyN [ V ] < MbyN, 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 ] < (ML)byN rectangular [ V2 ] < LbyN upper trapezoidal. The columns of V represent the vectors which define the H(i)'s. The (M+N)by(M+N) block reflector H is then given by H = I  W * T * W**H where W**H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Definition at line 172 of file ztpqrt2.f. subroutine ztprfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTPRFSPurpose: ZTPRFS 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 ZTPTRS or some other means before entering this routine. ZTPRFS does not do iterative refinement because doing so cannot improve the backward error. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. TRANS TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1. B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) The solution matrix X. LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). FERR FERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 172 of file ztprfs.f. subroutine ztptri (character UPLO, character DIAG, integer N, complex*16, dimension( * ) AP, integer INFO)ZTPTRIPurpose: ZTPTRI computes the inverse of a complex upper or lower triangular matrix A stored in packed format. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangular matrix A, stored columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*((2*nj)/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 ith argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero. The triangular matrix is singular and its inverse can not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: 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+I1) = A(I,J) AP(JC+IJ) = A(I,J) 1 CONTINUE 1 CONTINUE JC = JC + J JC = JC + N  J + 1 2 CONTINUE 2 CONTINUE Definition at line 116 of file ztptri.f. subroutine ztptrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZTPTRSPurpose: ZTPTRS solves a triangular system of the form A * X = B, A**T * X = B, or A**H * X = B, where A is a triangular matrix of order N stored in packed format, and B is an NbyNRHS matrix. A check is made to verify that A is nonsingular. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. TRANS TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2*nj)/2) = A(i,j) for j<=i<=n. B B is COMPLEX*16 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 ith argument had an illegal value > 0: if INFO = i, the ith diagonal element of A is zero, indicating that the matrix is singular and the solutions X have not been computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 129 of file ztptrs.f. subroutine ztpttf (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) AP, complex*16, dimension( 0: * ) ARF, integer INFO)ZTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).Purpose: ZTPTTF copies a triangular matrix A from standard packed format (TP) to rectangular full packed format (TF). Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': ARF in Normal format is wanted; = 'C': ARF in Conjugatetranspose format is wanted. UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On entry, the upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. ARF ARF is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 206 of file ztpttf.f. subroutine ztpttr (character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)ZTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).Purpose: ZTPTTR copies a triangular matrix A from standard packed format (TP) to standard full format (TR). Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular. = 'L': A is lower triangular. N N is INTEGER The order of the matrix A. N >= 0. AP AP is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On entry, the upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. A A is COMPLEX*16 array, dimension ( LDA, N ) On exit, the triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 103 of file ztpttr.f. subroutine ztrcon (character NORM, character UPLO, character DIAG, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision RCOND, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTRCONPurpose: ZTRCON estimates the reciprocal of the condition number of a triangular matrix A, in either the 1norm or the infinitynorm. 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 1norm condition number or the infinitynorm condition number is required: = '1' or 'O': 1norm; = 'I': Infinitynorm. UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 135 of file ztrcon.f. subroutine ztrevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTREVCPurpose: ZTREVC computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T. Matrices of this type are produced by the Schur factorization of a complex general matrix: A = Q*T*Q**H, as computed by ZHSEQR. The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by: T*x = w*x, (y**H)*T = w*(y**H) where y**H denotes the conjugate transpose of the vector y. The eigenvalues are not input to this routine, but are read directly from the diagonal of T. This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix. If Q is the unitary factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A. Parameters SIDE
SIDE is CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. HOWMNY HOWMNY is CHARACTER*1 = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, backtransformed using the matrices supplied in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, as indicated by the logical array SELECT. SELECT SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed. The eigenvector corresponding to the jth eigenvalue is computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = 'A' or 'B'. N N is INTEGER The order of the matrix T. N >= 0. T T is COMPLEX*16 array, dimension (LDT,N) The upper triangular matrix T. T is modified, but restored on exit. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). VL VL is COMPLEX*16 array, dimension (LDVL,MM) On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an NbyN matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. Not referenced if SIDE = 'R'. LDVL LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N. VR VR is COMPLEX*16 array, dimension (LDVR,MM) On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an NbyN matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. Not referenced if SIDE = 'L'. LDVR LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B'; LDVR >= N. MM MM is INTEGER The number of columns in the arrays VL and/or VR. MM >= M. M M is INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected eigenvector occupies one column. WORK WORK is COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow. Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be x + y. Definition at line 216 of file ztrevc.f. subroutine ztrevc3 (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer INFO)ZTREVC3Purpose: ZTREVC3 computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T. Matrices of this type are produced by the Schur factorization of a complex general matrix: A = Q*T*Q**H, as computed by ZHSEQR. The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by: T*x = w*x, (y**H)*T = w*(y**H) where y**H denotes the conjugate transpose of the vector y. The eigenvalues are not input to this routine, but are read directly from the diagonal of T. This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix. If Q is the unitary factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A. This uses a Level 3 BLAS version of the back transformation. Parameters SIDE
SIDE is CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. HOWMNY HOWMNY is CHARACTER*1 = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, backtransformed using the matrices supplied in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, as indicated by the logical array SELECT. SELECT SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed. The eigenvector corresponding to the jth eigenvalue is computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = 'A' or 'B'. N N is INTEGER The order of the matrix T. N >= 0. T T is COMPLEX*16 array, dimension (LDT,N) The upper triangular matrix T. T is modified, but restored on exit. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). VL VL is COMPLEX*16 array, dimension (LDVL,MM) On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an NbyN matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. Not referenced if SIDE = 'R'. LDVL LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N. VR VR is COMPLEX*16 array, dimension (LDVR,MM) On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an NbyN matrix Q (usually the unitary matrix Q of Schur vectors returned by ZHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. Not referenced if SIDE = 'L'. LDVR LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N. MM MM is INTEGER The number of columns in the arrays VL and/or VR. MM >= M. M M is INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected eigenvector occupies one column. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) LWORK LWORK is INTEGER The dimension of array WORK. LWORK >= max(1,2*N). For optimum performance, LWORK >= N + 2*N*NB, where NB is the optimal blocksize. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK RWORK is DOUBLE PRECISION array, dimension (LRWORK) LRWORK LRWORK is INTEGER The dimension of array RWORK. LRWORK >= max(1,N). If LRWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow. Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be x + y. Definition at line 242 of file ztrevc3.f. subroutine ztrexc (character COMPQ, integer N, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldq, * ) Q, integer LDQ, integer IFST, integer ILST, integer INFO)ZTREXCPurpose: ZTREXC reorders the Schur factorization of a complex matrix A = Q*T*Q**H, so that the diagonal element of T with row index IFST is moved to row ILST. The Schur form T is reordered by a unitary similarity transformation Z**H*T*Z, and optionally the matrix Q of Schur vectors is updated by postmultplying it with Z. Parameters COMPQ
COMPQ is CHARACTER*1 = 'V': update the matrix Q of Schur vectors; = 'N': do not update Q. N N is INTEGER The order of the matrix T. N >= 0. If N == 0 arguments ILST and IFST may be any value. T T is COMPLEX*16 array, dimension (LDT,N) On entry, the upper triangular matrix T. On exit, the reordered upper triangular matrix. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if COMPQ = 'V', the matrix Q of Schur vectors. On exit, if COMPQ = 'V', Q has been postmultiplied by the unitary transformation matrix Z which reorders T. If COMPQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1, and if COMPQ = 'V', LDQ >= max(1,N). IFST IFST is INTEGER ILST ILST is INTEGER Specify the reordering of the diagonal elements of T: The element with row index IFST is moved to row ILST by a sequence of transpositions between adjacent elements. 1 <= IFST <= N; 1 <= ILST <= N. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 125 of file ztrexc.f. subroutine ztrrfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)ZTRRFSPurpose: ZTRRFS 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 ZTRTRS or some other means before entering this routine. ZTRRFS does not do iterative refinement because doing so cannot improve the backward error. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. TRANS TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. A A is COMPLEX*16 array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB,NRHS) The right hand side matrix B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). X X is COMPLEX*16 array, dimension (LDX,NRHS) The solution matrix X. LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). FERR FERR is DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the jth 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 COMPLEX*16 array, dimension (2*N) RWORK RWORK is DOUBLE PRECISION array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 180 of file ztrrfs.f. subroutine ztrsen (character JOB, character COMPQ, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( * ) W, integer M, double precision S, double precision SEP, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZTRSENPurpose: ZTRSEN reorders the Schur factorization of a complex matrix A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in the leading positions on the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. Optionally the routine computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace. Parameters JOB
JOB is CHARACTER*1 Specifies whether condition numbers are required for the cluster of eigenvalues (S) or the invariant subspace (SEP): = 'N': none; = 'E': for eigenvalues only (S); = 'V': for invariant subspace only (SEP); = 'B': for both eigenvalues and invariant subspace (S and SEP). COMPQ COMPQ is CHARACTER*1 = 'V': update the matrix Q of Schur vectors; = 'N': do not update Q. SELECT SELECT is LOGICAL array, dimension (N) SELECT specifies the eigenvalues in the selected cluster. To select the jth eigenvalue, SELECT(j) must be set to .TRUE.. N N is INTEGER The order of the matrix T. N >= 0. T T is COMPLEX*16 array, dimension (LDT,N) On entry, the upper triangular matrix T. On exit, T is overwritten by the reordered matrix T, with the selected eigenvalues as the leading diagonal elements. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). Q Q is COMPLEX*16 array, dimension (LDQ,N) On entry, if COMPQ = 'V', the matrix Q of Schur vectors. On exit, if COMPQ = 'V', Q has been postmultiplied by the unitary transformation matrix which reorders T; the leading M columns of Q form an orthonormal basis for the specified invariant subspace. If COMPQ = 'N', Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; and if COMPQ = 'V', LDQ >= N. W W is COMPLEX*16 array, dimension (N) The reordered eigenvalues of T, in the same order as they appear on the diagonal of T. M M is INTEGER The dimension of the specified invariant subspace. 0 <= M <= N. S S is DOUBLE PRECISION If JOB = 'E' or 'B', S is a lower bound on the reciprocal condition number for the selected cluster of eigenvalues. S cannot underestimate the true reciprocal condition number by more than a factor of sqrt(N). If M = 0 or N, S = 1. If JOB = 'N' or 'V', S is not referenced. SEP SEP is DOUBLE PRECISION If JOB = 'V' or 'B', SEP is the estimated reciprocal condition number of the specified invariant subspace. If M = 0 or N, SEP = norm(T). If JOB = 'N' or 'E', SEP is not referenced. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If JOB = 'N', LWORK >= 1; if JOB = 'E', LWORK = max(1,M*(NM)); if JOB = 'V' or 'B', LWORK >= max(1,2*M*(NM)). 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: ZTRSEN first collects the selected eigenvalues by computing a unitary transformation Z to move them to the top left corner of T. In other words, the selected eigenvalues are the eigenvalues of T11 in: Z**H * T * Z = ( T11 T12 ) n1 ( 0 T22 ) n2 n1 n2 where N = n1+n2. The first n1 columns of Z span the specified invariant subspace of T. If T has been obtained from the Schur factorization of a matrix A = Q*T*Q**H, then the reordered Schur factorization of A is given by A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, and the first n1 columns of Q*Z span the corresponding invariant subspace of A. The reciprocal condition number of the average of the eigenvalues of T11 may be returned in S. S lies between 0 (very badly conditioned) and 1 (very well conditioned). It is computed as follows. First we compute R so that P = ( I R ) n1 ( 0 0 ) n2 n1 n2 is the projector on the invariant subspace associated with T11. R is the solution of the Sylvester equation: T11*R  R*T22 = T12. Let Fnorm(M) denote the Frobeniusnorm of M and 2norm(M) denote the twonorm of M. Then S is computed as the lower bound (1 + Fnorm(R)**2)**(1/2) on the reciprocal of 2norm(P), the true reciprocal condition number. S cannot underestimate 1 / 2norm(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 ) = sigmamin( C ) where sigmamin(C) is the smallest singular value of the n1*n2byn1*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 sigmamin(C) by the reciprocal of an estimate of the 1norm of inverse(C). The true reciprocal 1norm of inverse(C) cannot differ from sigmamin(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 Definition at line 262 of file ztrsen.f. subroutine ztrsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, double precision, dimension( * ) S, double precision, dimension( * ) SEP, integer MM, integer M, complex*16, dimension( ldwork, * ) WORK, integer LDWORK, double precision, dimension( * ) RWORK, integer INFO)ZTRSNAPurpose: ZTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary). Parameters JOB
JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (SEP); = 'B': for both eigenvalues and eigenvectors (S and SEP). HOWMNY HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT. SELECT SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the jth eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced. N N is INTEGER The order of the matrix T. N >= 0. T T is COMPLEX*16 array, dimension (LDT,N) The upper triangular matrix T. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). VL VL is COMPLEX*16 array, dimension (LDVL,M) If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by ZHSEIN or ZTREVC. If JOB = 'V', VL is not referenced. LDVL LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. VR VR is COMPLEX*16 array, dimension (LDVR,M) If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by ZHSEIN or ZTREVC. 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. Thus S(j), SEP(j), and the jth columns of VL and VR all correspond to the same eigenpair (but not in general the jth 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. If JOB = 'E', SEP is not referenced. MM MM is INTEGER The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M. M M is INTEGER The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N. WORK WORK is COMPLEX*16 array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced. LDWORK LDWORK is INTEGER The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. RWORK RWORK is DOUBLE PRECISION array, dimension (N) If JOB = 'E', RWORK is not referenced. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The reciprocal of the condition number of an eigenvalue lambda is defined as S(lambda) = v**H*u / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of T corresponding to lambda; v**H denotes the conjugate transpose of v, and norm(u) denotes the Euclidean norm. These reciprocal condition numbers always lie between zero (very badly conditioned) and one (very well conditioned). If n = 1, S(lambda) is defined to be 1. An approximate error bound for a computed eigenvalue W(i) is given by EPS * norm(T) / S(i) where EPS is the machine precision. The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose T = ( lambda c ) ( 0 T22 ) Then the reciprocal condition number is SEP( lambda, T22 ) = sigmamin( T22  lambda*I ) where sigmamin denotes the smallest singular value. We approximate the smallest singular value by the reciprocal of an estimate of the onenorm 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) Definition at line 246 of file ztrsna.f. subroutine ztrti2 (character UPLO, character DIAG, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)ZTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).Purpose: ZTRTI2 computes the inverse of a complex upper or lower triangular matrix. This is the Level 2 BLAS version of the algorithm. Parameters UPLO
UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Nonunit triangular = 'U': Unit triangular N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 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 kth argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 109 of file ztrti2.f. subroutine ztrtri (character UPLO, character DIAG, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)ZTRTRIPurpose: ZTRTRI computes the inverse of a complex upper or lower triangular matrix A. This is the Level 3 BLAS version of the algorithm. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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 ith argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero. The triangular matrix is singular and its inverse can not be computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 108 of file ztrtri.f. subroutine ztrtrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)ZTRTRSPurpose: ZTRTRS solves a triangular system of the form A * X = B, A**T * X = B, or A**H * X = B, where A is a triangular matrix of order N, and B is an NbyNRHS matrix. A check is made to verify that A is nonsingular. Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. TRANS TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) DIAG DIAG is CHARACTER*1 = 'N': A is nonunit triangular; = 'U': A is unit triangular. N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. A A is COMPLEX*16 array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 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 ith argument had an illegal value > 0: if INFO = i, the ith diagonal element of A is zero, indicating that the matrix is singular and the solutions X have not been computed. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 138 of file ztrtrs.f. subroutine ztrttf (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: lda1, 0: * ) A, integer LDA, complex*16, dimension( 0: * ) ARF, integer INFO)ZTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).Purpose: ZTRTTF copies a triangular matrix A from standard full format (TR) to rectangular full packed format (TF) . Parameters TRANSR
TRANSR is CHARACTER*1 = 'N': ARF in Normal mode is wanted; = 'C': ARF in Conjugate Transpose mode is wanted; UPLO UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. N N is INTEGER The order of the matrix A. N >= 0. A A is COMPLEX*16 array, dimension ( LDA, N ) On entry, the triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. LDA LDA is INTEGER The leading dimension of the matrix A. LDA >= max(1,N). ARF ARF is COMPLEX*16 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: We first consider Standard Packed Format when N is even. We give an example where N = 6. AP is Upper AP is Lower 00 01 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last three columns of AP upper. The lower triangle A(4:6,0:2) consists of conjugatetranspose 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 conjugatetranspose of the last three columns of AP lower. To denote conjugate we place  above the element. This covers the case N even and TRANSR = 'N'. RFP A RFP A    03 04 05 33 43 53   13 14 15 00 44 54  23 24 25 10 11 55 33 34 35 20 21 22  00 44 45 30 31 32   01 11 55 40 41 42    02 12 22 50 51 52 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A           03 13 23 33 00 01 02 33 00 10 20 30 40 50           04 14 24 34 44 11 12 43 44 11 21 31 41 51           05 15 25 35 45 55 22 53 54 55 22 32 42 52 We next consider Standard Packed Format when N is odd. We give an example where N = 5. AP is Upper AP is Lower 00 01 02 03 04 00 11 12 13 14 10 11 22 23 24 20 21 22 33 34 30 31 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP holds AP as follows: For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last three columns of AP upper. The lower triangle A(3:4,0:1) consists of conjugatetranspose 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 conjugatetranspose of the last two columns of AP lower. To denote conjugate we place  above the element. This covers the case N odd and TRANSR = 'N'. RFP A RFP A   02 03 04 00 33 43  12 13 14 10 11 44 22 23 24 20 21 22  00 33 34 30 31 32   01 11 44 40 41 42 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate transpose of RFP A above. One therefore gets: RFP A RFP A          02 12 22 00 01 00 10 20 30 40 50          03 13 23 33 11 33 11 21 31 41 51          04 14 24 34 44 43 44 22 32 42 52 Definition at line 215 of file ztrttf.f. subroutine ztrttp (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) AP, integer INFO)ZTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).Purpose: ZTRTTP copies a triangular matrix A from full format (TR) to standard packed format (TP). Parameters UPLO
UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. N N is INTEGER The order of the matrices AP and A. N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the triangular matrix A. If UPLO = 'U', the leading NbyN 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 NbyN lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). AP AP is COMPLEX*16 array, dimension ( N*(N+1)/2 ), On exit, the upper or lower triangular matrix A, packed columnwise in a linear array. The jth column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j1)*(2nj)/2) = A(i,j) for j<=i<=n. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 103 of file ztrttp.f. subroutine ztzrzf (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZTZRZFPurpose: ZTZRZF reduces the MbyN ( M<=N ) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations. The upper trapezoidal matrix A is factored as A = ( R 0 ) * Z, where Z is an NbyN unitary matrix and R is an MbyM upper triangular matrix. Parameters M
M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. N >= M. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the leading MbyN upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading MbyM upper triangular part of A contains the upper triangular matrix R, and elements M+1 to N of the first M rows of A, with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (M) The scalar factors of the elementary reflectors. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: The NbyN matrix Z can be computed by Z = Z(1)*Z(2)* ... *Z(M) where each NbyN Z(k) is given by Z(k) = I  tau(k)*v(k)*v(k)**H with v(k) is the kth row vector of the MbyN matrix V = ( I A(:,M+1:N) ) I is the MbyM identity matrix, A(:,M+1:N) is the output stored in A on exit from ZTZRZF, and tau(k) is the kth element of the array TAU. Definition at line 150 of file ztzrzf.f. subroutine zunbdb (character TRANS, character SIGNS, integer M, integer P, integer Q, complex*16, dimension( ldx11, * ) X11, integer LDX11, complex*16, dimension( ldx12, * ) X12, integer LDX12, complex*16, dimension( ldx21, * ) X21, integer LDX21, complex*16, dimension( ldx22, * ) X22, integer LDX22, double precision, dimension( * ) THETA, double precision, dimension( * ) PHI, complex*16, dimension( * ) TAUP1, complex*16, dimension( * ) TAUP2, complex*16, dimension( * ) TAUQ1, complex*16, dimension( * ) TAUQ2, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNBDBPurpose: ZUNBDB simultaneously bidiagonalizes the blocks of an MbyM partitioned unitary matrix X: [ B11  B12 0 0 ] [ X11  X12 ] [ P1  ] [ 0  0 I 0 ] [ Q1  ]**H X = [] = [] [] [] . [ X21  X22 ] [  P2 ] [ B21  B22 0 0 ] [  Q2 ] [ 0  0 0 I ] X11 is PbyQ. Q must be no larger than P, MP, or MQ. (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 ZUNCSD for details.) The unitary matrices P1, P2, Q1, and Q2 are PbyP, (MP)by (MP), QbyQ, and (MQ)by(MQ), respectively. They are represented implicitly by Householder vectors. B11, B12, B21, and B22 are QbyQ bidiagonal matrices represented implicitly by angles THETA, PHI. Parameters TRANS
TRANS is CHARACTER = 'T': X, U1, U2, V1T, and V2T are stored in rowmajor order; otherwise: X, U1, U2, V1T, and V2T are stored in column major order. SIGNS SIGNS is CHARACTER = 'O': The lowerleft block is made nonpositive (the "other" convention); otherwise: The upperright 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,MP,MQ). X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, the topleft block of the unitary matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the columns of tril(X11) specify reflectors for P1, the rows of triu(X11,1) specify reflectors for Q1; else TRANS = 'T', and the rows of triu(X11) specify reflectors for P1, the columns of tril(X11,1) specify reflectors for Q1. LDX11 LDX11 is INTEGER The leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q. X12 X12 is COMPLEX*16 array, dimension (LDX12,MQ) On entry, the topright block of the unitary matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the rows of triu(X12) specify the first P reflectors for Q2; else TRANS = 'T', and the columns of tril(X12) specify the first P reflectors for Q2. LDX12 LDX12 is INTEGER The leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= MQ. X21 X21 is COMPLEX*16 array, dimension (LDX21,Q) On entry, the bottomleft block of the unitary matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the columns of tril(X21) specify reflectors for P2; else TRANS = 'T', and the rows of triu(X21) specify reflectors for P2. LDX21 LDX21 is INTEGER The leading dimension of X21. If TRANS = 'N', then LDX21 >= MP; else LDX21 >= Q. X22 X22 is COMPLEX*16 array, dimension (LDX22,MQ) On entry, the bottomright block of the unitary matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the rows of triu(X22(Q+1:MP,P+1:MQ)) specify the last MPQ reflectors for Q2, else TRANS = 'T', and the columns of tril(X22(P+1:MQ,Q+1:MP)) specify the last MPQ reflectors for P2. LDX22 LDX22 is INTEGER The leading dimension of X22. If TRANS = 'N', then LDX22 >= MP; else LDX22 >= MQ. 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 (Q1) The entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. See Further Details. TAUP1 TAUP1 is COMPLEX*16 array, dimension (P) The scalar factors of the elementary reflectors that define P1. TAUP2 TAUP2 is COMPLEX*16 array, dimension (MP) The scalar factors of the elementary reflectors that define P2. TAUQ1 TAUQ1 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define Q1. TAUQ2 TAUQ2 is COMPLEX*16 array, dimension (MQ) The scalar factors of the elementary reflectors that define Q2. WORK WORK is COMPLEX*16 array, dimension (LWORK) LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MQ. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The bidiagonal blocks B11, B12, B21, and B22 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q1). 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 ZUNCSD for details. P1, P2, Q1, and Q2 are represented as products of elementary reflectors. See ZUNCSD for details on generating P1, P2, Q1, and Q2 using ZUNGQR and ZUNGLQ. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Definition at line 284 of file zunbdb.f. subroutine zunbdb1 (integer M, integer P, integer Q, complex*16, dimension(ldx11,*) X11, integer LDX11, complex*16, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, double precision, dimension(*) PHI, complex*16, dimension(*) TAUP1, complex*16, dimension(*) TAUP2, complex*16, dimension(*) TAUQ1, complex*16, dimension(*) WORK, integer LWORK, integer INFO)ZUNBDB1Purpose: ZUNBDB1 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 PbyQ, and X21 is (MP)byQ. Q must be no larger than P, MP, or MQ. Routines ZUNBDB2, ZUNBDB3, and ZUNBDB4 handle cases in which Q is not the minimum dimension. The unitary matrices P1, P2, and Q1 are PbyP, (MP)by(MP), and (MQ)by(MQ), respectively. They are represented implicitly by Householder vectors. B11 and B12 are QbyQ 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,MP,MQ). X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1. LDX11 LDX11 is INTEGER The leading dimension of X11. LDX11 >= P. X21 X21 is COMPLEX*16 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 >= MP. 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 (Q1) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details. TAUP1 TAUP1 is COMPLEX*16 array, dimension (P) The scalar factors of the elementary reflectors that define P1. TAUP2 TAUP2 is COMPLEX*16 array, dimension (MP) The scalar factors of the elementary reflectors that define P2. TAUQ1 TAUQ1 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define Q1. WORK WORK is COMPLEX*16 array, dimension (LWORK) LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MQ. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The upperbidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q1). 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 ZUNCSD for details. P1, P2, and Q1 are represented as products of elementary reflectors. See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR and ZUNGLQ. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Definition at line 201 of file zunbdb1.f. subroutine zunbdb2 (integer M, integer P, integer Q, complex*16, dimension(ldx11,*) X11, integer LDX11, complex*16, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, double precision, dimension(*) PHI, complex*16, dimension(*) TAUP1, complex*16, dimension(*) TAUP2, complex*16, dimension(*) TAUQ1, complex*16, dimension(*) WORK, integer LWORK, integer INFO)ZUNBDB2Purpose: ZUNBDB2 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 PbyQ, and X21 is (MP)byQ. P must be no larger than MP, Q, or MQ. Routines ZUNBDB1, ZUNBDB3, and ZUNBDB4 handle cases in which P is not the minimum dimension. The unitary matrices P1, P2, and Q1 are PbyP, (MP)by(MP), and (MQ)by(MQ), respectively. They are represented implicitly by Householder vectors. B11 and B12 are PbyP 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(MP,Q,MQ). Q Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M. X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1. LDX11 LDX11 is INTEGER The leading dimension of X11. LDX11 >= P. X21 X21 is COMPLEX*16 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 >= MP. 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 (Q1) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details. TAUP1 TAUP1 is COMPLEX*16 array, dimension (P) The scalar factors of the elementary reflectors that define P1. TAUP2 TAUP2 is COMPLEX*16 array, dimension (MP) The scalar factors of the elementary reflectors that define P2. TAUQ1 TAUQ1 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define Q1. WORK WORK is COMPLEX*16 array, dimension (LWORK) LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MQ. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The upperbidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q1). 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 ZUNCSD for details. P1, P2, and Q1 are represented as products of elementary reflectors. See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR and ZUNGLQ. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Definition at line 199 of file zunbdb2.f. subroutine zunbdb3 (integer M, integer P, integer Q, complex*16, dimension(ldx11,*) X11, integer LDX11, complex*16, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, double precision, dimension(*) PHI, complex*16, dimension(*) TAUP1, complex*16, dimension(*) TAUP2, complex*16, dimension(*) TAUQ1, complex*16, dimension(*) WORK, integer LWORK, integer INFO)ZUNBDB3Purpose: ZUNBDB3 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 PbyQ, and X21 is (MP)byQ. MP must be no larger than P, Q, or MQ. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB4 handle cases in which MP is not the minimum dimension. The unitary matrices P1, P2, and Q1 are PbyP, (MP)by(MP), and (MQ)by(MQ), respectively. They are represented implicitly by Householder vectors. B11 and B12 are (MP)by(MP) 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. MP <= min(P,Q,MQ). Q Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M. X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1. LDX11 LDX11 is INTEGER The leading dimension of X11. LDX11 >= P. X21 X21 is COMPLEX*16 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 >= MP. 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 (Q1) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details. TAUP1 TAUP1 is COMPLEX*16 array, dimension (P) The scalar factors of the elementary reflectors that define P1. TAUP2 TAUP2 is COMPLEX*16 array, dimension (MP) The scalar factors of the elementary reflectors that define P2. TAUQ1 TAUQ1 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define Q1. WORK WORK is COMPLEX*16 array, dimension (LWORK) LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MQ. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The upperbidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q1). 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 ZUNCSD for details. P1, P2, and Q1 are represented as products of elementary reflectors. See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR and ZUNGLQ. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Definition at line 199 of file zunbdb3.f. subroutine zunbdb4 (integer M, integer P, integer Q, complex*16, dimension(ldx11,*) X11, integer LDX11, complex*16, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, double precision, dimension(*) PHI, complex*16, dimension(*) TAUP1, complex*16, dimension(*) TAUP2, complex*16, dimension(*) TAUQ1, complex*16, dimension(*) PHANTOM, complex*16, dimension(*) WORK, integer LWORK, integer INFO)ZUNBDB4Purpose: ZUNBDB4 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 PbyQ, and X21 is (MP)byQ. MQ must be no larger than P, MP, or Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB3 handle cases in which MQ is not the minimum dimension. The unitary matrices P1, P2, and Q1 are PbyP, (MP)by(MP), and (MQ)by(MQ), respectively. They are represented implicitly by Householder vectors. B11 and B12 are (MQ)by(MQ) 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 MQ <= min(P,MP,Q). X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1. LDX11 LDX11 is INTEGER The leading dimension of X11. LDX11 >= P. X21 X21 is COMPLEX*16 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 >= MP. 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 (Q1) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details. TAUP1 TAUP1 is COMPLEX*16 array, dimension (P) The scalar factors of the elementary reflectors that define P1. TAUP2 TAUP2 is COMPLEX*16 array, dimension (MP) The scalar factors of the elementary reflectors that define P2. TAUQ1 TAUQ1 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define Q1. PHANTOM PHANTOM is COMPLEX*16 array, dimension (M) The routine computes an Mby1 column vector Y that is orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and Y(P+1:M), respectively. WORK WORK is COMPLEX*16 array, dimension (LWORK) LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= MQ. 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: The upperbidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q1). 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 ZUNCSD for details. P1, P2, and Q1 are represented as products of elementary reflectors. See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR and ZUNGLQ. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Definition at line 210 of file zunbdb4.f. subroutine zunbdb5 (integer M1, integer M2, integer N, complex*16, dimension(*) X1, integer INCX1, complex*16, dimension(*) X2, integer INCX2, complex*16, dimension(ldq1,*) Q1, integer LDQ1, complex*16, dimension(ldq2,*) Q2, integer LDQ2, complex*16, dimension(*) WORK, integer LWORK, integer INFO)ZUNBDB5Purpose: ZUNBDB5 orthogonalizes the column vector X = [ X1 ] [ X2 ] with respect to the columns of Q = [ Q1 ] . [ Q2 ] The columns of Q must be orthonormal. If the projection is zero according to Kahan's "twice is enough" criterion, then some other vector from the orthogonal complement is returned. This vector is chosen in an arbitrary but deterministic way. Parameters M1
M1 is INTEGER The dimension of X1 and the number of rows in Q1. 0 <= M1. M2 M2 is INTEGER The dimension of X2 and the number of rows in Q2. 0 <= M2. N N is INTEGER The number of columns in Q1 and Q2. 0 <= N. X1 X1 is COMPLEX*16 array, dimension (M1) On entry, the top part of the vector to be orthogonalized. On exit, the top part of the projected vector. INCX1 INCX1 is INTEGER Increment for entries of X1. X2 X2 is COMPLEX*16 array, dimension (M2) On entry, the bottom part of the vector to be orthogonalized. On exit, the bottom part of the projected vector. INCX2 INCX2 is INTEGER Increment for entries of X2. Q1 Q1 is COMPLEX*16 array, dimension (LDQ1, N) The top part of the orthonormal basis matrix. LDQ1 LDQ1 is INTEGER The leading dimension of Q1. LDQ1 >= M1. Q2 Q2 is COMPLEX*16 array, dimension (LDQ2, N) The bottom part of the orthonormal basis matrix. LDQ2 LDQ2 is INTEGER The leading dimension of Q2. LDQ2 >= M2. WORK WORK is COMPLEX*16 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 154 of file zunbdb5.f. subroutine zunbdb6 (integer M1, integer M2, integer N, complex*16, dimension(*) X1, integer INCX1, complex*16, dimension(*) X2, integer INCX2, complex*16, dimension(ldq1,*) Q1, integer LDQ1, complex*16, dimension(ldq2,*) Q2, integer LDQ2, complex*16, dimension(*) WORK, integer LWORK, integer INFO)ZUNBDB6Purpose: ZUNBDB6 orthogonalizes the column vector X = [ X1 ] [ X2 ] with respect to the columns of Q = [ Q1 ] . [ Q2 ] The columns of Q must be orthonormal. If the projection is zero according to Kahan's "twice is enough" criterion, then the zero vector is returned. Parameters M1
M1 is INTEGER The dimension of X1 and the number of rows in Q1. 0 <= M1. M2 M2 is INTEGER The dimension of X2 and the number of rows in Q2. 0 <= M2. N N is INTEGER The number of columns in Q1 and Q2. 0 <= N. X1 X1 is COMPLEX*16 array, dimension (M1) On entry, the top part of the vector to be orthogonalized. On exit, the top part of the projected vector. INCX1 INCX1 is INTEGER Increment for entries of X1. X2 X2 is COMPLEX*16 array, dimension (M2) On entry, the bottom part of the vector to be orthogonalized. On exit, the bottom part of the projected vector. INCX2 INCX2 is INTEGER Increment for entries of X2. Q1 Q1 is COMPLEX*16 array, dimension (LDQ1, N) The top part of the orthonormal basis matrix. LDQ1 LDQ1 is INTEGER The leading dimension of Q1. LDQ1 >= M1. Q2 Q2 is COMPLEX*16 array, dimension (LDQ2, N) The bottom part of the orthonormal basis matrix. LDQ2 LDQ2 is INTEGER The leading dimension of Q2. LDQ2 >= M2. WORK WORK is COMPLEX*16 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 ith argument had an illegal value. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 152 of file zunbdb6.f. recursive subroutine zuncsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, character SIGNS, integer M, integer P, integer Q, complex*16, dimension( ldx11, * ) X11, integer LDX11, complex*16, dimension( ldx12, * ) X12, integer LDX12, complex*16, dimension( ldx21, * ) X21, integer LDX21, complex*16, dimension( ldx22, * ) X22, integer LDX22, double precision, dimension( * ) THETA, complex*16, dimension( ldu1, * ) U1, integer LDU1, complex*16, dimension( ldu2, * ) U2, integer LDU2, complex*16, dimension( ldv1t, * ) V1T, integer LDV1T, complex*16, dimension( ldv2t, * ) V2T, integer LDV2T, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer INFO)ZUNCSDPurpose: ZUNCSD computes the CS decomposition of an MbyM partitioned unitary matrix X: [ I 0 0  0 0 0 ] [ 0 C 0  0 S 0 ] [ X11  X12 ] [ U1  ] [ 0 0 0  0 0 I ] [ V1  ]**H X = [] = [] [] [] . [ X21  X22 ] [  U2 ] [ 0 0 0  I 0 0 ] [  V2 ] [ 0 S 0  0 C 0 ] [ 0 0 I  0 0 0 ] X11 is PbyQ. The unitary matrices U1, U2, V1, and V2 are PbyP, (MP)by(MP), QbyQ, and (MQ)by(MQ), respectively. C and S are RbyR nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which R = MIN(P,MP,Q,MQ). 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 rowmajor order; otherwise: X, U1, U2, V1T, and V2T are stored in column major order. SIGNS SIGNS is CHARACTER = 'O': The lowerleft block is made nonpositive (the "other" convention); otherwise: The upperright block is made nonpositive (the "default" convention). M M is INTEGER The number of rows and columns in X. P P is INTEGER The number of rows in X11 and X12. 0 <= P <= M. Q Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M. X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, part of the unitary matrix whose CSD is desired. LDX11 LDX11 is INTEGER The leading dimension of X11. LDX11 >= MAX(1,P). X12 X12 is COMPLEX*16 array, dimension (LDX12,MQ) On entry, part of the unitary matrix whose CSD is desired. LDX12 LDX12 is INTEGER The leading dimension of X12. LDX12 >= MAX(1,P). X21 X21 is COMPLEX*16 array, dimension (LDX21,Q) On entry, part of the unitary matrix whose CSD is desired. LDX21 LDX21 is INTEGER The leading dimension of X11. LDX21 >= MAX(1,MP). X22 X22 is COMPLEX*16 array, dimension (LDX22,MQ) On entry, part of the unitary matrix whose CSD is desired. LDX22 LDX22 is INTEGER The leading dimension of X11. LDX22 >= MAX(1,MP). THETA THETA is DOUBLE PRECISION array, dimension (R), in which R = MIN(P,MP,Q,MQ). C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). U1 U1 is COMPLEX*16 array, dimension (LDU1,P) If JOBU1 = 'Y', U1 contains the PbyP unitary matrix U1. LDU1 LDU1 is INTEGER The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P). U2 U2 is COMPLEX*16 array, dimension (LDU2,MP) If JOBU2 = 'Y', U2 contains the (MP)by(MP) unitary matrix U2. LDU2 LDU2 is INTEGER The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,MP). V1T V1T is COMPLEX*16 array, dimension (LDV1T,Q) If JOBV1T = 'Y', V1T contains the QbyQ matrix unitary matrix V1**H. LDV1T LDV1T is INTEGER The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q). V2T V2T is COMPLEX*16 array, dimension (LDV2T,MQ) If JOBV2T = 'Y', V2T contains the (MQ)by(MQ) unitary matrix V2**H. LDV2T LDV2T is INTEGER The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= MAX(1,MQ). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the work array, and no error message related to LWORK is issued by XERBLA. RWORK RWORK is DOUBLE PRECISION array, dimension MAX(1,LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. If INFO > 0 on exit, RWORK(2:R) contains the values PHI(1), ..., PHI(R1) that, together with THETA(1), ..., THETA(R), define the matrix in intermediate bidiagonalblock form remaining after nonconvergence. INFO specifies the number of nonzero PHI's. LRWORK LRWORK is INTEGER The dimension of the array RWORK. If LRWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the RWORK array, returns this value as the first entry of the work array, and no error message related to LRWORK is issued by XERBLA. IWORK IWORK is INTEGER array, dimension (MMIN(P,MP,Q,MQ)) INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: ZBBCSD did not converge. See the description of RWORK above for details. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 314 of file zuncsd.f. subroutine zuncsd2by1 (character JOBU1, character JOBU2, character JOBV1T, integer M, integer P, integer Q, complex*16, dimension(ldx11,*) X11, integer LDX11, complex*16, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, complex*16, dimension(ldu1,*) U1, integer LDU1, complex*16, dimension(ldu2,*) U2, integer LDU2, complex*16, dimension(ldv1t,*) V1T, integer LDV1T, complex*16, dimension(*) WORK, integer LWORK, double precision, dimension(*) RWORK, integer LRWORK, integer, dimension(*) IWORK, integer INFO)ZUNCSD2BY1Purpose: ZUNCSD2BY1 computes the CS decomposition of an MbyQ matrix X with orthonormal columns that has been partitioned into a 2by1 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 PbyQ. The unitary matrices U1, U2, and V1 are PbyP, (MP)by(MP), and QbyQ, respectively. C and S are RbyR nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which R = MIN(P,MP,Q,MQ). I1 is a K1byK1 identity matrix and I2 is a K2byK2 identity matrix, where K1 = MAX(Q+PM,0), K2 = MAX(QP,0). Parameters JOBU1
JOBU1 is CHARACTER = 'Y': U1 is computed; otherwise: U1 is not computed. JOBU2 JOBU2 is CHARACTER = 'Y': U2 is computed; otherwise: U2 is not computed. JOBV1T JOBV1T is CHARACTER = 'Y': V1T is computed; otherwise: V1T is not computed. M M is INTEGER The number of rows in X. P P is INTEGER The number of rows in X11. 0 <= P <= M. Q Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M. X11 X11 is COMPLEX*16 array, dimension (LDX11,Q) On entry, part of the unitary matrix whose CSD is desired. LDX11 LDX11 is INTEGER The leading dimension of X11. LDX11 >= MAX(1,P). X21 X21 is COMPLEX*16 array, dimension (LDX21,Q) On entry, part of the unitary matrix whose CSD is desired. LDX21 LDX21 is INTEGER The leading dimension of X21. LDX21 >= MAX(1,MP). THETA THETA is DOUBLE PRECISION array, dimension (R), in which R = MIN(P,MP,Q,MQ). C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). U1 U1 is COMPLEX*16 array, dimension (P) If JOBU1 = 'Y', U1 contains the PbyP unitary matrix U1. LDU1 LDU1 is INTEGER The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= MAX(1,P). U2 U2 is COMPLEX*16 array, dimension (MP) If JOBU2 = 'Y', U2 contains the (MP)by(MP) unitary matrix U2. LDU2 LDU2 is INTEGER The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= MAX(1,MP). V1T V1T is COMPLEX*16 array, dimension (Q) If JOBV1T = 'Y', V1T contains the QbyQ matrix unitary matrix V1**T. LDV1T LDV1T is INTEGER The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= MAX(1,Q). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK and RWORK arrays, returns this value as the first entry of the WORK and RWORK array, respectively, and no error message related to LWORK or LRWORK is issued by XERBLA. RWORK RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. If INFO > 0 on exit, RWORK(2:R) contains the values PHI(1), ..., PHI(R1) that, together with THETA(1), ..., THETA(R), define the matrix in intermediate bidiagonalblock form remaining after nonconvergence. INFO specifies the number of nonzero PHI's. LRWORK LRWORK is INTEGER The dimension of the array RWORK. If LRWORK=1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK and RWORK arrays, returns this value as the first entry of the WORK and RWORK array, respectively, and no error message related to LWORK or LRWORK is issued by XERBLA. IWORK IWORK is INTEGER array, dimension (MMIN(P,MP,Q,MQ)) INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: ZBBCSD did not converge. See the description of WORK above for details. References: [1] Brian D. Sutton. Computing the complete CS
decomposition. Numer. Algorithms, 50(1):3365, 2009.
Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 252 of file zuncsd2by1.f. subroutine zung2l (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)ZUNG2L generates all or part of the unitary matrix Q from a QL factorization determined by cgeqlf (unblocked algorithm).Purpose: ZUNG2L generates an m by n complex matrix Q with orthonormal columns, which is defined as the last n columns of a product of k elementary reflectors of order m Q = H(k) . . . H(2) H(1) as returned by ZGEQLF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. M >= N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the (nk+i)th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQLF in the last k columns of its array argument A. On exit, the mbyn matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQLF. WORK WORK is COMPLEX*16 array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 113 of file zung2l.f. subroutine zung2r (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)ZUNG2RPurpose: ZUNG2R generates an m by n complex matrix Q with orthonormal columns, which is defined as the first n columns of a product of k elementary reflectors of order m Q = H(1) H(2) . . . H(k) as returned by ZGEQRF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. M >= N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQRF in the first k columns of its array argument A. On exit, the m by n matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQRF. WORK WORK is COMPLEX*16 array, dimension (N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 113 of file zung2r.f. subroutine zunghr (integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGHRPurpose: ZUNGHR generates a complex unitary matrix Q which is defined as the product of IHIILO elementary reflectors of order N, as returned by ZGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi1). 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 ZGEHRD. Q is equal to the unit matrix except in the submatrix Q(ilo+1:ihi,ilo+1:ihi). 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the vectors which define the elementary reflectors, as returned by ZGEHRD. On exit, the NbyN unitary matrix Q. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). TAU TAU is COMPLEX*16 array, dimension (N1) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEHRD. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= IHIILO. For optimum performance LWORK >= (IHIILO)*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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 125 of file zunghr.f. subroutine zungl2 (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)ZUNGL2 generates all or part of the unitary matrix Q from an LQ factorization determined by cgelqf (unblocked algorithm).Purpose: ZUNGL2 generates an mbyn complex matrix Q with orthonormal rows, which is defined as the first m rows of a product of k elementary reflectors of order n Q = H(k)**H . . . H(2)**H H(1)**H as returned by ZGELQF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. N >= M. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGELQF in the first k rows of its array argument A. On exit, the m by n matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGELQF. WORK WORK is COMPLEX*16 array, dimension (M) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 112 of file zungl2.f. subroutine zunglq (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGLQPurpose: ZUNGLQ generates an MbyN complex matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N Q = H(k)**H . . . H(2)**H H(1)**H as returned by ZGELQF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. N >= M. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGELQF in the first k rows of its array argument A. On exit, the MbyN matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGELQF. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 126 of file zunglq.f. subroutine zungql (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGQLPurpose: ZUNGQL generates an MbyN complex matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M Q = H(k) . . . H(2) H(1) as returned by ZGEQLF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. M >= N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the (nk+i)th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQLF in the last k columns of its array argument A. On exit, the MbyN matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQLF. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 127 of file zungql.f. subroutine zungqr (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGQRPurpose: ZUNGQR generates an MbyN complex matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M Q = H(1) H(2) . . . H(k) as returned by ZGEQRF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. M >= N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQRF in the first k columns of its array argument A. On exit, the MbyN matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQRF. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 127 of file zungqr.f. subroutine zungr2 (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer INFO)ZUNGR2 generates all or part of the unitary matrix Q from an RQ factorization determined by cgerqf (unblocked algorithm).Purpose: ZUNGR2 generates an m by n complex matrix Q with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors of order n Q = H(1)**H H(2)**H . . . H(k)**H as returned by ZGERQF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. N >= M. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the (mk+i)th row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGERQF in the last k rows of its array argument A. On exit, the mbyn matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGERQF. WORK WORK is COMPLEX*16 array, dimension (M) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 113 of file zungr2.f. subroutine zungrq (integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGRQPurpose: ZUNGRQ generates an MbyN complex matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N Q = H(1)**H H(2)**H . . . H(k)**H as returned by ZGERQF. Parameters M
M is INTEGER The number of rows of the matrix Q. M >= 0. N N is INTEGER The number of columns of the matrix Q. N >= M. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the (mk+i)th row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGERQF in the last k rows of its array argument A. On exit, the MbyN matrix Q. LDA LDA is INTEGER The first dimension of the array A. LDA >= max(1,M). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGERQF. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. 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 ith argument has an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 127 of file zungrq.f. subroutine zungtr (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGTRPurpose: ZUNGTR generates a complex unitary matrix Q which is defined as the product of n1 elementary reflectors of order N, as returned by ZHETRD: if UPLO = 'U', Q = H(n1) . . . H(2) H(1), if UPLO = 'L', Q = H(1) H(2) . . . H(n1). Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangle of A contains elementary reflectors from ZHETRD; = 'L': Lower triangle of A contains elementary reflectors from ZHETRD. N N is INTEGER The order of the matrix Q. N >= 0. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the vectors which define the elementary reflectors, as returned by ZHETRD. On exit, the NbyN unitary matrix Q. LDA LDA is INTEGER The leading dimension of the array A. LDA >= N. TAU TAU is COMPLEX*16 array, dimension (N1) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZHETRD. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= N1. For optimum performance LWORK >= (N1)*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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 122 of file zungtr.f. subroutine zungtsqr (integer M, integer N, integer MB, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGTSQRPurpose: ZUNGTSQR generates an MbyN complex matrix Q_out with orthonormal columns, which are the first N columns of a product of comlpex unitary matrices of order M which are returned by ZLATSQR Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ). See the documentation for ZLATSQR. 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. M >= N >= 0. MB MB is INTEGER The row block size used by ZLATSQR to return arrays A and T. MB > N. (Note that if MB > M, then M is used instead of MB as the row block size). NB NB is INTEGER The column block size used by ZLATSQR to return arrays A and T. NB >= 1. (Note that if NB > N, then N is used instead of NB as the column block size). A A is COMPLEX*16 array, dimension (LDA,N) On entry: The elements on and above the diagonal are not accessed. The elements below the diagonal represent the unit lowertrapezoidal blocked matrix V computed by ZLATSQR that defines the input matrices Q_in(k) (ones on the diagonal are not stored) (same format as the output A below the diagonal in ZLATSQR). On exit: The array A contains an MbyN orthonormal matrix Q_out, i.e the columns of A are orthogonal unit vectors. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is COMPLEX*16 array, dimension (LDT, N * NIRB) where NIRB = Number_of_input_row_blocks = MAX( 1, CEIL((MN)/(MBN)) ) Let NICB = Number_of_input_col_blocks = CEIL(N/NB) The uppertriangular block reflectors used to define the input matrices Q_in(k), k=(1:NIRB*NICB). The block reflectors are stored in compact form in NIRB block reflector sequences. Each of NIRB block reflector sequences is stored in a larger NBbyN column block of T and consists of NICB smaller NBbyNB uppertriangular column blocks. (same format as the output T in ZLATSQR). LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,min(NB1,N)). WORK (workspace) COMPLEX*16 array, dimension (MAX(2,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK The dimension of the array WORK. LWORK >= (M+NB)*N. If LWORK = 1, then a workspace query is assumed. The routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley Definition at line 173 of file zungtsqr.f. subroutine zungtsqr_row (integer M, integer N, integer MB, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNGTSQR_ROWPurpose: ZUNGTSQR_ROW generates an MbyN complex matrix Q_out with orthonormal columns from the output of ZLATSQR. These N orthonormal columns are the first N columns of a product of complex unitary matrices Q(k)_in of order M, which are returned by ZLATSQR in a special format. Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ). The input matrices Q(k)_in are stored in row and column blocks in A. See the documentation of ZLATSQR for more details on the format of Q(k)_in, where each Q(k)_in is represented by block Householder transformations. This routine calls an auxiliary routine ZLARFB_GETT, where the computation is performed on each individual block. The algorithm first sweeps NBsized column blocks from the right to left starting in the bottom row block and continues to the top row block (hence _ROW in the routine name). This sweep is in reverse order of the order in which ZLATSQR generates the output blocks. 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. M >= N >= 0. MB MB is INTEGER The row block size used by ZLATSQR to return arrays A and T. MB > N. (Note that if MB > M, then M is used instead of MB as the row block size). NB NB is INTEGER The column block size used by ZLATSQR to return arrays A and T. NB >= 1. (Note that if NB > N, then N is used instead of NB as the column block size). A A is COMPLEX*16 array, dimension (LDA,N) On entry: The elements on and above the diagonal are not used as input. The elements below the diagonal represent the unit lowertrapezoidal blocked matrix V computed by ZLATSQR that defines the input matrices Q_in(k) (ones on the diagonal are not stored). See ZLATSQR for more details. On exit: The array A contains an MbyN orthonormal matrix Q_out, i.e the columns of A are orthogonal unit vectors. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is COMPLEX*16 array, dimension (LDT, N * NIRB) where NIRB = Number_of_input_row_blocks = MAX( 1, CEIL((MN)/(MBN)) ) Let NICB = Number_of_input_col_blocks = CEIL(N/NB) The uppertriangular block reflectors used to define the input matrices Q_in(k), k=(1:NIRB*NICB). The block reflectors are stored in compact form in NIRB block reflector sequences. Each of the NIRB block reflector sequences is stored in a larger NBbyN column block of T and consists of NICB smaller NBbyNB uppertriangular column blocks. See ZLATSQR for more details on the format of T. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,min(NB,N)). WORK (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK The dimension of the array WORK. LWORK >= NBLOCAL * MAX(NBLOCAL,(NNBLOCAL)), where NBLOCAL=MIN(NB,N). If LWORK = 1, then a workspace query is assumed. The routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: November 2020, Igor Kozachenko, Computer Science Division, University of California, Berkeley Definition at line 186 of file zungtsqr_row.f. subroutine zunhr_col (integer M, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) D, integer INFO)ZUNHR_COLPurpose: ZUNHR_COL takes an MbyN complex matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR), i.e. reconstructs Householder vectors V(i) implicitly representing another MbyN matrix Q_out, with the property that Q_in = Q_out*S, where S is an NbyN diagonal matrix with diagonal entries equal to +1 or 1. The Householder vectors (columns V(i) of V) are stored in A on output, and the diagonal entries of S are stored in D. Block reflectors are also returned in T (same output format as ZGEQRT). 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. M >= N >= 0. NB NB is INTEGER The column block size to be used in the reconstruction of Householder column vector blocks in the array A and corresponding block reflectors in the array T. NB >= 1. (Note that if NB > N, then N is used instead of NB as the column block size.) A A is COMPLEX*16 array, dimension (LDA,N) On entry: The array A contains an MbyN orthonormal matrix Q_in, i.e the columns of A are orthogonal unit vectors. On exit: The elements below the diagonal of A represent the unit lowertrapezoidal matrix V of Householder column vectors V(i). The unit diagonal entries of V are not stored (same format as the output below the diagonal in A from ZGEQRT). The matrix T and the matrix V stored on output in A implicitly define Q_out. The elements above the diagonal contain the factor U of the "modified" LUdecomposition: Q_in  ( S ) = V * U ( 0 ) where 0 is a (MN)by(MN) zero matrix. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is COMPLEX*16 array, dimension (LDT, N) Let NOCB = Number_of_output_col_blocks = CEIL(N/NB) On exit, T(1:NB, 1:N) contains NOCB uppertriangular block reflectors used to define Q_out stored in compact form as a sequence of uppertriangular NBbyNB column blocks (same format as the output T in ZGEQRT). The matrix T and the matrix V stored on output in A implicitly define Q_out. NOTE: The lower triangles below the uppertriangular blocks will be filled with zeros. See Further Details. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,min(NB,N)). D D is COMPLEX*16 array, dimension min(M,N). The elements can be only plus or minus one. D(i) is constructed as D(i) = SIGN(Q_in_i(i,i)), where 1 <= i <= min(M,N), and Q_in_i is Q_in after performing i1 steps of “modified” Gaussian elimination. See Further Details. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Further Details: The computed MbyM unitary factor Q_out is defined implicitly as a product of unitary matrices Q_out(i). Each Q_out(i) is stored in the compact WYrepresentation format in the corresponding blocks of matrices V (stored in A) and T. The MbyN unit lowertrapezoidal matrix V stored in the MbyN matrix A contains the column vectors V(i) in NBsize column blocks VB(j). For example, VB(1) contains the columns V(1), V(2), ... V(NB). NOTE: The unit entries on the diagonal of Y are not stored in A. The number of column blocks is NOCB = Number_of_output_col_blocks = CEIL(N/NB) where each block is of order NB except for the last block, which is of order LAST_NB = N  (NOCB1)*NB. For example, if M=6, N=5 and NB=2, the matrix V is V = ( VB(1), VB(2), VB(3) ) = = ( 1 ) ( v21 1 ) ( v31 v32 1 ) ( v41 v42 v43 1 ) ( v51 v52 v53 v54 1 ) ( v61 v62 v63 v54 v65 ) For each of the column blocks VB(i), an uppertriangular block reflector TB(i) is computed. These blocks are stored as a sequence of uppertriangular column blocks in the NBbyN matrix T. The size of each TB(i) block is NBbyNB, except for the last block, whose size is LAST_NBbyLAST_NB. For example, if M=6, N=5 and NB=2, the matrix T is T = ( TB(1), TB(2), TB(3) ) = = ( t11 t12 t13 t14 t15 ) ( t22 t24 ) The MbyM factor Q_out is given as a product of NOCB unitary MbyM matrices Q_out(i). Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB), where each matrix Q_out(i) is given by the WYrepresentation using corresponding blocks from the matrices V and T: Q_out(i) = I  VB(i) * TB(i) * (VB(i))**T, where I is the identity matrix. Here is the formula with matrix dimensions: Q(i){MbyM} = I{MbyM}  VB(i){MbyINB} * TB(i){INBbyINB} * (VB(i))**T {INBbyM}, where INB = NB, except for the last block NOCB for which INB=LAST_NB. ===== NOTE: ===== If Q_in is the result of doing a QR factorization B = Q_in * R_in, then: B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out. So if one wants to interpret Q_out as the result of the QR factorization of B, then the corresponding R_out should be equal to R_out = S * R_in, i.e. some rows of R_in should be multiplied by 1. For the details of the algorithm, see [1]. [1] "Reconstructing Householder vectors from tallskinny QR", G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 331, 2015. Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley Definition at line 258 of file zunhr_col.f. subroutine zunm2l (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)ZUNM2L multiplies a general matrix by the unitary matrix from a QL factorization determined by cgeqlf (unblocked algorithm).Purpose: ZUNM2L overwrites the general complex mbyn matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or Q**H* C if SIDE = 'L' and TRANS = 'C', or C * Q if SIDE = 'R' and TRANS = 'N', or C * Q**H if SIDE = 'R' and TRANS = 'C', where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(k) . . . H(2) H(1) as returned by ZGEQLF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left = 'R': apply Q or Q**H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply Q (No transpose) = 'C': apply Q**H (Conjugate transpose) M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,K) The ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQLF in the last k columns of its array argument A. A is modified by the routine but restored on exit. LDA LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQLF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the mbyn matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L', (M) if SIDE = 'R' INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 157 of file zunm2l.f. subroutine zunm2r (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)ZUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf (unblocked algorithm).Purpose: ZUNM2R overwrites the general complex mbyn matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or Q**H* C if SIDE = 'L' and TRANS = 'C', or C * Q if SIDE = 'R' and TRANS = 'N', or C * Q**H if SIDE = 'R' and TRANS = 'C', where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by ZGEQRF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left = 'R': apply Q or Q**H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply Q (No transpose) = 'C': apply Q**H (Conjugate transpose) M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,K) The ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQRF in the first k columns of its array argument A. A is modified by the routine but restored on exit. LDA LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQRF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the mbyn matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L', (M) if SIDE = 'R' INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 157 of file zunm2r.f. subroutine zunmbr (character VECT, character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMBRPurpose: If VECT = 'Q', ZUNMBR overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H If VECT = 'P', ZUNMBR overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': P * C C * P TRANS = 'C': P**H * C C * P**H Here Q and P**H are the unitary matrices determined by ZGEBRD when reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. Q and P**H are defined as products of elementary reflectors H(i) and G(i) respectively. Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the order of the unitary matrix Q or P**H that is applied. If VECT = 'Q', A is assumed to have been an NQbyK matrix: if nq >= k, Q = H(1) H(2) . . . H(k); if nq < k, Q = H(1) H(2) . . . H(nq1). If VECT = 'P', A is assumed to have been a KbyNQ matrix: if k < nq, P = G(1) G(2) . . . G(k); if k >= nq, P = G(1) G(2) . . . G(nq1). Parameters VECT
VECT is CHARACTER*1 = 'Q': apply Q or Q**H; = 'P': apply P or P**H. SIDE SIDE is CHARACTER*1 = 'L': apply Q, Q**H, P or P**H from the Left; = 'R': apply Q, Q**H, P or P**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q or P; = 'C': Conjugate transpose, apply Q**H or P**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER If VECT = 'Q', the number of columns in the original matrix reduced by ZGEBRD. If VECT = 'P', the number of rows in the original matrix reduced by ZGEBRD. K >= 0. A A is COMPLEX*16 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 ZGEBRD. LDA LDA is INTEGER The leading dimension of the array A. If VECT = 'Q', LDA >= max(1,nq); if VECT = 'P', LDA >= max(1,min(nq,K)). TAU TAU is COMPLEX*16 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 ZGEBRD in the array argument TAUQ or TAUP. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q or P*C or P**H*C or C*P or C*P**H. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M); if N = 0 or M = 0, LWORK >= 1. For optimum performance LWORK >= max(1,N*NB) if SIDE = 'L', and LWORK >= max(1,M*NB) if SIDE = 'R', where NB is the optimal blocksize. (NB = 0 if M = 0 or N = 0.) If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 194 of file zunmbr.f. subroutine zunmhr (character SIDE, character TRANS, integer M, integer N, integer ILO, integer IHI, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMHRPurpose: ZUNMHR overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHIILO elementary reflectors, as returned by ZGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi1). Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': apply Q (No transpose) = 'C': apply Q**H (Conjugate transpose) M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. ILO ILO is INTEGER IHI IHI is INTEGER ILO and IHI must have the same values as in the previous call of ZGEHRD. Q is equal to the unit matrix except in the submatrix Q(ilo+1:ihi,ilo+1:ihi). If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and ILO = 1 and IHI = 0, if M = 0; if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and ILO = 1 and IHI = 0, if N = 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L' (LDA,N) if SIDE = 'R' The vectors which define the elementary reflectors, as returned by ZGEHRD. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'. TAU TAU is COMPLEX*16 array, dimension (M1) if SIDE = 'L' (N1) if SIDE = 'R' TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEHRD. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For optimum performance LWORK >= 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 176 of file zunmhr.f. subroutine zunml2 (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)ZUNML2 multiplies a general matrix by the unitary matrix from a LQ factorization determined by cgelqf (unblocked algorithm).Purpose: ZUNML2 overwrites the general complex mbyn matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or Q**H* C if SIDE = 'L' and TRANS = 'C', or C * Q if SIDE = 'R' and TRANS = 'N', or C * Q**H if SIDE = 'R' and TRANS = 'C', where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(k)**H . . . H(2)**H H(1)**H as returned by ZGELQF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left = 'R': apply Q or Q**H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply Q (No transpose) = 'C': apply Q**H (Conjugate transpose) M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L', (LDA,N) if SIDE = 'R' The ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGELQF in the first k rows of its array argument A. A is modified by the routine but restored on exit. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,K). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGELQF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the mbyn matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L', (M) if SIDE = 'R' INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 157 of file zunml2.f. subroutine zunmlq (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMLQPurpose: ZUNMLQ overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(k)**H . . . H(2)**H H(1)**H as returned by ZGELQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L', (LDA,N) if SIDE = 'R' The ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGELQF in the first k rows of its array argument A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,K). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGELQF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 165 of file zunmlq.f. subroutine zunmql (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMQLPurpose: ZUNMQL overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(k) . . . H(2) H(1) as returned by ZGEQLF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,K) The ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQLF in the last k columns of its array argument A. LDA LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQLF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 165 of file zunmql.f. subroutine zunmqr (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMQRPurpose: ZUNMQR overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by ZGEQRF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,K) The ith column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQRF in the first k columns of its array argument A. LDA LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,M); if SIDE = 'R', LDA >= max(1,N). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQRF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 165 of file zunmqr.f. subroutine zunmr2 (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)ZUNMR2 multiplies a general matrix by the unitary matrix from a RQ factorization determined by cgerqf (unblocked algorithm).Purpose: ZUNMR2 overwrites the general complex mbyn matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or Q**H* C if SIDE = 'L' and TRANS = 'C', or C * Q if SIDE = 'R' and TRANS = 'N', or C * Q**H if SIDE = 'R' and TRANS = 'C', where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1)**H H(2)**H . . . H(k)**H as returned by ZGERQF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left = 'R': apply Q or Q**H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply Q (No transpose) = 'C': apply Q**H (Conjugate transpose) M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L', (LDA,N) if SIDE = 'R' The ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGERQF in the last k rows of its array argument A. A is modified by the routine but restored on exit. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,K). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGERQF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the mbyn matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L', (M) if SIDE = 'R' INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 157 of file zunmr2.f. subroutine zunmr3 (character SIDE, character TRANS, integer M, integer N, integer K, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)ZUNMR3 multiplies a general matrix by the unitary matrix from a RZ factorization determined by ctzrzf (unblocked algorithm).Purpose: ZUNMR3 overwrites the general complex m by n matrix C with Q * C if SIDE = 'L' and TRANS = 'N', or Q**H* C if SIDE = 'L' and TRANS = 'C', or C * Q if SIDE = 'R' and TRANS = 'N', or C * Q**H if SIDE = 'R' and TRANS = 'C', where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by ZTZRZF. Q is of order m if SIDE = 'L' and of order n if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left = 'R': apply Q or Q**H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply Q (No transpose) = 'C': apply Q**H (Conjugate transpose) M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. L L is INTEGER The number of columns of the matrix A containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L', (LDA,N) if SIDE = 'R' The ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZTZRZF in the last k rows of its array argument A. A is modified by the routine but restored on exit. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,K). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZTZRZF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the mbyn matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L', (M) if SIDE = 'R' INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: Definition at line 176 of file zunmr3.f. subroutine zunmrq (character SIDE, character TRANS, integer M, integer N, integer K, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMRQPurpose: ZUNMRQ overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1)**H H(2)**H . . . H(k)**H as returned by ZGERQF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L', (LDA,N) if SIDE = 'R' The ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGERQF in the last k rows of its array argument A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,K). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGERQF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 165 of file zunmrq.f. subroutine zunmrz (character SIDE, character TRANS, integer M, integer N, integer K, integer L, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMRZPurpose: ZUNMRZ overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix defined as the product of k elementary reflectors Q = H(1) H(2) . . . H(k) as returned by ZTZRZF. Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. K K is INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0. L L is INTEGER The number of columns of the matrix A containing the meaningful part of the Householder reflectors. If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L', (LDA,N) if SIDE = 'R' The ith row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZTZRZF in the last k rows of its array argument A. A is modified by the routine but restored on exit. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,K). TAU TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZTZRZF. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Contributors: A. Petitet, Computer Science Dept., Univ. of Tenn.,
Knoxville, USA
Further Details: Definition at line 185 of file zunmrz.f. subroutine zunmtr (character SIDE, character UPLO, character TRANS, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)ZUNMTRPurpose: ZUNMTR overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq1 elementary reflectors, as returned by ZHETRD: if UPLO = 'U', Q = H(nq1) . . . H(2) H(1); if UPLO = 'L', Q = H(1) H(2) . . . H(nq1). Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A contains elementary reflectors from ZHETRD; = 'L': Lower triangle of A contains elementary reflectors from ZHETRD. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. A A is COMPLEX*16 array, dimension (LDA,M) if SIDE = 'L' (LDA,N) if SIDE = 'R' The vectors which define the elementary reflectors, as returned by ZHETRD. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'. TAU TAU is COMPLEX*16 array, dimension (M1) if SIDE = 'L' (N1) if SIDE = 'R' TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZHETRD. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. If SIDE = 'L', LWORK >= max(1,N); if SIDE = 'R', LWORK >= max(1,M). For optimum performance LWORK >= 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 ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 169 of file zunmtr.f. subroutine zupgtr (character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) TAU, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16, dimension( * ) WORK, integer INFO)ZUPGTRPurpose: ZUPGTR generates a complex unitary matrix Q which is defined as the product of n1 elementary reflectors H(i) of order n, as returned by ZHPTRD using packed storage: if UPLO = 'U', Q = H(n1) . . . H(2) H(1), if UPLO = 'L', Q = H(1) H(2) . . . H(n1). Parameters UPLO
UPLO is CHARACTER*1 = 'U': Upper triangular packed storage used in previous call to ZHPTRD; = 'L': Lower triangular packed storage used in previous call to ZHPTRD. N N is INTEGER The order of the matrix Q. N >= 0. AP AP is COMPLEX*16 array, dimension (N*(N+1)/2) The vectors which define the elementary reflectors, as returned by ZHPTRD. TAU TAU is COMPLEX*16 array, dimension (N1) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZHPTRD. Q Q is COMPLEX*16 array, dimension (LDQ,N) The NbyN unitary matrix Q. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N). WORK WORK is COMPLEX*16 array, dimension (N1) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 113 of file zupgtr.f. subroutine zupmtr (character SIDE, character UPLO, character TRANS, integer M, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK, integer INFO)ZUPMTRPurpose: ZUPMTR overwrites the general complex MbyN matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H where Q is a complex unitary matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of nq1 elementary reflectors, as returned by ZHPTRD using packed storage: if UPLO = 'U', Q = H(nq1) . . . H(2) H(1); if UPLO = 'L', Q = H(1) H(2) . . . H(nq1). Parameters SIDE
SIDE is CHARACTER*1 = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right. UPLO UPLO is CHARACTER*1 = 'U': Upper triangular packed storage used in previous call to ZHPTRD; = 'L': Lower triangular packed storage used in previous call to ZHPTRD. TRANS TRANS is CHARACTER*1 = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H. M M is INTEGER The number of rows of the matrix C. M >= 0. N N is INTEGER The number of columns of the matrix C. N >= 0. AP AP is COMPLEX*16 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 ZHPTRD. AP is modified by the routine but restored on exit. TAU TAU is COMPLEX*16 array, dimension (M1) if SIDE = 'L' or (N1) if SIDE = 'R' TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZHPTRD. C C is COMPLEX*16 array, dimension (LDC,N) On entry, the MbyN matrix C. On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L' (M) if SIDE = 'R' INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value Author Univ. of Tennessee
Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 148 of file zupmtr.f. AuthorGenerated automatically by Doxygen for LAPACK from the source code.
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