## table of contents

complex16OTHERcomputational(3) | LAPACK | complex16OTHERcomputational(3) |

# NAME¶

complex16OTHERcomputational - complex16

# SYNOPSIS¶

## Functions¶

subroutine **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 pre-processor 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** pre-processor 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 rank-k 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 component-wise relative backward error.
subroutine **zla_wwaddw** (N, X, Y, W)

**ZLA_WWADDW** adds a vector into a doubled-single 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 rank-one 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 1-norm, 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 conjugate-transpose 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
'triangular-pentagonal' 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 Description¶

This is the group of complex16 other Computational routines

# Function Documentation¶

## subroutine 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)¶

**ZBBCSD**

**Purpose:**

ZBBCSD computes the CS decomposition of a unitary matrix in

bidiagonal-block form,

[ B11 | B12 0 0 ]

[ 0 | 0 -I 0 ]

X = [----------------]

[ B21 | B22 0 0 ]

[ 0 | 0 0 I ]

[ C | -S 0 0 ]

[ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**H

= [---------] [---------------] [---------] .

[ | U2 ] [ S | C 0 0 ] [ | V2 ]

[ 0 | 0 0 I ]

X is M-by-M, its top-left block is P-by-Q, and Q must be no larger

than P, M-P, or M-Q. (If Q is not the smallest index, then X must be

transposed and/or permuted. This can be done in constant time using

the TRANS and SIGNS options. See ZUNCSD for details.)

The bidiagonal matrices B11, B12, B21, and B22 are represented

implicitly by angles THETA(1:Q) and PHI(1:Q-1).

The unitary matrices U1, U2, V1T, and V2T are input/output.

The input matrices are pre- or post-multiplied by the appropriate

singular vector matrices.

**Parameters**

*JOBU1*

JOBU1 is CHARACTER

= 'Y': U1 is updated;

otherwise: U1 is not updated.

*JOBU2*

JOBU2 is CHARACTER

= 'Y': U2 is updated;

otherwise: U2 is not updated.

*JOBV1T*

JOBV1T is CHARACTER

= 'Y': V1T is updated;

otherwise: V1T is not updated.

*JOBV2T*

JOBV2T is CHARACTER

= 'Y': V2T is updated;

otherwise: V2T is not updated.

*TRANS*

TRANS is CHARACTER

= 'T': X, U1, U2, V1T, and V2T are stored in row-major

order;

otherwise: X, U1, U2, V1T, and V2T are stored in column-

major order.

*M*

M is INTEGER

The number of rows and columns in X, the unitary matrix in

bidiagonal-block form.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

The number of columns in the top-left block of X.

0 <= Q <= MIN(P,M-P,M-Q).

*THETA*

THETA is DOUBLE PRECISION array, dimension (Q)

On entry, the angles THETA(1),...,THETA(Q) that, along with

PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block

form. On exit, the angles whose cosines and sines define the

diagonal blocks in the CS decomposition.

*PHI*

PHI is DOUBLE PRECISION array, dimension (Q-1)

The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,

THETA(Q), define the matrix in bidiagonal-block form.

*U1*

U1 is COMPLEX*16 array, dimension (LDU1,P)

On entry, a P-by-P matrix. On exit, U1 is postmultiplied

by the left singular vector matrix common to [ B11 ; 0 ] and

[ B12 0 0 ; 0 -I 0 0 ].

*LDU1*

LDU1 is INTEGER

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

*U2*

U2 is COMPLEX*16 array, dimension (LDU2,M-P)

On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is

postmultiplied by the left singular vector matrix common to

[ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].

*LDU2*

LDU2 is INTEGER

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

*V1T*

V1T is COMPLEX*16 array, dimension (LDV1T,Q)

On entry, a Q-by-Q 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,M-Q)

On entry, an (M-Q)-by-(M-Q) 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,M-Q).

*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 top-left

block.

*B11E*

B11E is DOUBLE PRECISION array, dimension (Q-1)

When ZBBCSD converges, B11E contains zeros. If ZBBCSD fails

to converge, then B11E contains the superdiagonal of the

partially reduced top-left 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 top-right

block.

*B12E*

B12E is DOUBLE PRECISION array, dimension (Q-1)

When ZBBCSD converges, B12E contains zeros. If ZBBCSD fails

to converge, then B12E contains the subdiagonal of the

partially reduced top-right 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 bottom-left

block.

*B21E*

B21E is DOUBLE PRECISION array, dimension (Q-1)

When ZBBCSD converges, B21E contains zeros. If ZBBCSD fails

to converge, then B21E contains the subdiagonal of the

partially reduced bottom-left 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 bottom-right

block.

*B22E*

B22E is DOUBLE PRECISION array, dimension (Q-1)

When ZBBCSD converges, B22E contains zeros. If ZBBCSD fails

to converge, then B22E contains the subdiagonal of the

partially reduced bottom-right 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 i-th 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:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZBDSQR**

**Purpose:**

ZBDSQR computes the singular values and, optionally, the right and/or

left singular vectors from the singular value decomposition (SVD) of

a real N-by-N (upper or lower) bidiagonal matrix B using the implicit

zero-shift 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. 873-912, Sept 1990) and

"Accurate singular values and differential qd algorithms," by

B. Parlett and V. Fernando, Technical Report CPAM-554, 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 (N-1)

On entry, the N-1 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 N-by-NCVT 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 NRU-by-N 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 N-by-NCC 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 i-th 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZGGHD3**

**Purpose:**

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 matrix-matrix

multiplications for parts of the computation to enhance performance.

**Parameters**

*COMPQ*

COMPQ is CHARACTER*1

= 'N': do not compute Q;

= 'I': Q is initialized to the unit matrix, and the

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:ILO-1 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 N-by-N general matrix to be reduced.

On exit, the upper triangle and the first subdiagonal of A

are overwritten with the upper Hessenberg matrix H, and the

rest is set to zero.

*LDA*

LDA is INTEGER

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

*B*

B is COMPLEX*16 array, dimension (LDB, N)

On entry, the N-by-N 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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine reduces A to Hessenberg form and maintains B in triangular form

using a blocked variant of Moler and Stewart's original algorithm,

as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti

(BIT 2008).

## subroutine 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)¶

**ZGGHRD**

**Purpose:**

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:ILO-1 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 N-by-N general matrix to be reduced.

On exit, the upper triangle and the first subdiagonal of A

are overwritten with the upper Hessenberg matrix H, and the

rest is set to zero.

*LDA*

LDA is INTEGER

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

*B*

B is COMPLEX*16 array, dimension (LDB, N)

On entry, the N-by-N 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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine reduces A to Hessenberg and B to triangular form by

an unblocked reduction, as described in _Matrix_Computations_,

by Golub and van Loan (Johns Hopkins Press).

## subroutine 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)¶

**ZGGQRF**

**Purpose:**

ZGGQRF computes a generalized QR factorization of an N-by-M matrix A

and an N-by-P matrix B:

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

where Q is an N-by-N unitary matrix, Z is a P-by-P 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 ) N-M N M-N

M

where R11 is upper triangular, and

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

P-N N ( T21 ) P

P

where T12 or T21 is upper triangular.

In particular, if B is square and nonsingular, the GQR factorization

of A and B implicitly gives the QR factorization of inv(B)*A:

inv(B)*A = Z**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 N-by-M matrix A.

On exit, the elements on and above the diagonal of the array

contain the min(N,M)-by-M upper trapezoidal matrix R (R is

upper triangular if N >= M); the elements below the diagonal,

with the array TAUA, represent the 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 N-by-P matrix B.

On exit, if N <= P, the upper triangle of the subarray

B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;

if N > P, the elements on and above the (N-P)-th subdiagonal

contain the N-by-P upper trapezoidal matrix T; the remaining

elements, with the array TAUB, represent the 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 N-by-M matrix, NB2 is the optimal blocksize for the

RQ factorization of an N-by-P matrix, and NB3 is the optimal

blocksize for a call of 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 i-th argument had an illegal value.

**Author**

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:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),

and taua in TAUA(i).

To form Q explicitly, use LAPACK subroutine 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(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in

B(n-k+i,1:p-k+i-1), and taub in TAUB(i).

To form Z explicitly, use LAPACK subroutine ZUNGRQ.

To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.

## 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)¶

**ZGGRQF**

**Purpose:**

ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A

and a P-by-N matrix B:

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

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary

matrix, and R and T assume one of the forms:

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

N-M M ( R21 ) N

N

where R12 or R21 is upper triangular, and

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

( 0 ) P-N P N-P

N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization

of A and B implicitly gives the RQ factorization of A*inv(B):

A*inv(B) = (R*inv(T))*Z**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 M-by-N matrix A.

On exit, if M <= N, the upper triangle of the subarray

A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;

if M > N, the elements on and above the (M-N)-th subdiagonal

contain the M-by-N upper trapezoidal matrix R; the remaining

elements, with the array TAUA, represent the 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 P-by-N matrix B.

On exit, the elements on and above the diagonal of the array

contain the min(P,N)-by-N upper trapezoidal matrix T (T is

upper triangular if P >= N); the elements below the diagonal,

with the array TAUB, represent the 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 M-by-N matrix, NB2 is the optimal blocksize for the

QR factorization of a P-by-N matrix, and NB3 is the optimal

blocksize for a call of 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 i-th argument had an illegal value.

**Author**

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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in

A(m-k+i,1:n-k+i-1), and taua in TAUA(i).

To form Q explicitly, use LAPACK subroutine 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:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),

and taub in TAUB(i).

To form Z explicitly, use LAPACK subroutine ZUNGQR.

To use Z to update another matrix, use LAPACK subroutine ZUNMQR.

## 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)¶

**ZGGSVP3**

**Purpose:**

ZGGSVP3 computes unitary matrices U, V and Q such that

N-K-L K L

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

L ( 0 0 A23 )

M-K-L ( 0 0 0 )

N-K-L K L

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

M-K ( 0 0 A23 )

N-K-L K L

V**H*B*Q = L ( 0 0 B13 )

P-L ( 0 0 0 )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular

upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,

otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective

numerical rank of the (M+P)-by-N matrix (A**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 M-by-N matrix A.

On exit, A contains the triangular (or trapezoidal) matrix

described in the Purpose section.

*LDA*

LDA is INTEGER

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

*B*

B is COMPLEX*16 array, dimension (LDB,N)

On entry, the P-by-N matrix B.

On exit, B contains the triangular matrix described in

the Purpose section.

*LDB*

LDB is INTEGER

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

*TOLA*

TOLA is DOUBLE PRECISION

*TOLB*

TOLB is DOUBLE PRECISION

TOLA and TOLB are the thresholds to determine the effective

numerical rank of matrix B and a subblock of A. Generally,

they are set to

TOLA = MAX(M,N)*norm(A)*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 i-th argument had an illegal value.

**Author**

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.

## 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 pre-processor for the routine zgesvj. **

**Purpose:**

ZGSVJ0 is called from ZGESVJ as a pre-processor 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 N-by-N array V.

(See the description of V.)

= 'A': the product of the Jacobi rotations is accumulated

by postmulyiplying the MV-by-N array V.

(See the descriptions of MV and V.)

= 'N': the Jacobi rotations are not accumulated.

*M*

M is INTEGER

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

*N*

N is INTEGER

The number of columns of the input matrix A.

M >= N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, M-by-N 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)

post-multiplied by a sequence of Jacobi rotations, where the

rotation threshold and the total number of sweeps are given in

TOL and NSWEEP, respectively.

(See the descriptions of D, TOL and NSWEEP.)

*LDA*

LDA is INTEGER

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

*D*

D is 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

post-multiplied by a sequence of Jacobi rotations, where the

rotation threshold and the total number of sweeps are given in

TOL and NSWEEP, respectively.

(See the descriptions of A, TOL and NSWEEP.)

*SVA*

SVA is DOUBLE PRECISION array, dimension (N)

On entry, SVA contains the Euclidean norms of the columns of

the matrix A*diag(D).

On exit, SVA contains the Euclidean norms of the columns of

the matrix A_onexit*diag(D_onexit).

*MV*

MV is INTEGER

If JOBV = 'A', then MV rows of V are post-multipled by a

sequence of Jacobi rotations.

If JOBV = 'N', then MV is not referenced.

*V*

V is COMPLEX*16 array, dimension (LDV,N)

If JOBV = 'V' then N rows of V are post-multipled by a

sequence of Jacobi rotations.

If JOBV = 'A' then MV rows of V are post-multipled by a

sequence of Jacobi rotations.

If JOBV = 'N', then V is not referenced.

*LDV*

LDV is INTEGER

The leading dimension of the array V, LDV >= 1.

If JOBV = 'V', LDV >= N.

If JOBV = 'A', LDV >= MV.

*EPS*

EPS is DOUBLE PRECISION

EPS = DLAMCH('Epsilon')

*SFMIN*

SFMIN is DOUBLE PRECISION

SFMIN = DLAMCH('Safe Minimum')

*TOL*

TOL is DOUBLE PRECISION

TOL is the threshold for Jacobi rotations. For a pair

A(:,p), A(:,q) of pivot columns, the Jacobi rotation is

applied only if 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Contributor: Zlatko Drmac (Zagreb, Croatia)

**Bugs, Examples and Comments:**

## 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** pre-processor for the routine zgesvj, applies Jacobi
rotations targeting only particular pivots.

**Purpose:**

ZGSVJ1 is called from ZGESVJ as a pre-processor 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 M-by-N matrix A. The pivot pairs are taken from the (1,2)

off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The

block-entries (tiles) of the (1,2) off-diagonal block are marked by the

[x]'s in the following scheme:

| * * * [x] [x] [x]|

| * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.

| * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.

|[x] [x] [x] * * * |

|[x] [x] [x] * * * |

|[x] [x] [x] * * * |

In terms of the columns of A, the first N1 columns are rotated 'against'

the remaining N-N1 columns, trying to increase the angle between the

corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is

tiled using quadratic tiles of side KBL. Here, KBL is a 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 N-by-N array V.

(See the description of V.)

= 'A': the product of the Jacobi rotations is accumulated

by postmulyiplying the MV-by-N array V.

(See the descriptions of MV and V.)

= 'N': the Jacobi rotations are not accumulated.

*M*

M is INTEGER

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

*N*

N is INTEGER

The number of columns of the input matrix A.

M >= N >= 0.

*N1*

N1 is INTEGER

N1 specifies the 2 x 2 block partition, the first N1 columns are

rotated 'against' the remaining N-N1 columns of A.

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, M-by-N matrix A, such that A*diag(D) represents

the input matrix.

On exit,

A_onexit * D_onexit represents the input matrix A*diag(D)

post-multiplied by a sequence of Jacobi rotations, where the

rotation threshold and the total number of sweeps are given in

TOL and NSWEEP, respectively.

(See the descriptions of N1, D, TOL and NSWEEP.)

*LDA*

LDA is INTEGER

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

*D*

D is 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

post-multiplied by a sequence of Jacobi rotations, where the

rotation threshold and the total number of sweeps are given in

TOL and NSWEEP, respectively.

(See the descriptions of N1, A, TOL and NSWEEP.)

*SVA*

SVA is DOUBLE PRECISION array, dimension (N)

On entry, SVA contains the Euclidean norms of the columns of

the matrix A*diag(D).

On exit, SVA contains the Euclidean norms of the columns of

the matrix onexit*diag(D_onexit).

*MV*

MV is INTEGER

If JOBV = 'A', then MV rows of V are post-multipled by a

sequence of Jacobi rotations.

If JOBV = 'N', then MV is not referenced.

*V*

V is COMPLEX*16 array, dimension (LDV,N)

If JOBV = 'V' then N rows of V are post-multipled by a

sequence of Jacobi rotations.

If JOBV = 'A' then MV rows of V are post-multipled by a

sequence of Jacobi rotations.

If JOBV = 'N', then V is not referenced.

*LDV*

LDV is INTEGER

The leading dimension of the array V, LDV >= 1.

If JOBV = 'V', LDV >= N.

If JOBV = 'A', LDV >= MV.

*EPS*

EPS is DOUBLE PRECISION

EPS = DLAMCH('Epsilon')

*SFMIN*

SFMIN is DOUBLE PRECISION

SFMIN = DLAMCH('Safe Minimum')

*TOL*

TOL is DOUBLE PRECISION

TOL is the threshold for Jacobi rotations. For a pair

A(:,p), A(:,q) of pivot columns, the Jacobi rotation is

applied only if 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributor:**

## 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)¶

**ZHBGST**

**Purpose:**

ZHBGST reduces a complex Hermitian-definite banded generalized

eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y,

such that C has the same bandwidth as A.

B must have been previously factorized as S**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

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).

On exit, the transformed matrix X**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 n-by-n matrix X.

If VECT = 'N', the array X is not referenced.

*LDX*

LDX is INTEGER

The leading dimension of the array X.

LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

*WORK*

WORK is 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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZHBTRD**

**Purpose:**

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

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, the diagonal elements of AB are overwritten by the

diagonal elements of the tridiagonal matrix T; if KD > 0, the

elements on the first superdiagonal (if UPLO = 'U') or the

first subdiagonal (if UPLO = 'L') are overwritten by the

off-diagonal elements of T; the rest of AB is overwritten by

values generated during the reduction.

*LDAB*

LDAB is INTEGER

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

*D*

D is DOUBLE PRECISION array, dimension (N)

The diagonal elements of the tridiagonal matrix T.

*E*

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

The off-diagonal elements of the tridiagonal matrix T:

E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

*Q*

Q is COMPLEX*16 array, dimension (LDQ,N)

On entry, if VECT = 'U', then Q must contain an N-by-N

matrix X; if VECT = 'N' or 'V', then Q need not be set.

On exit:

if VECT = 'V', Q contains the N-by-N 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Modified by Linda Kaufman, Bell Labs.

## 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 T

**Purpose:**

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

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, the diagonal elements of AB are overwritten by the

diagonal elements of the tridiagonal matrix T; if KD > 0, the

elements on the first superdiagonal (if UPLO = 'U') or the

first subdiagonal (if UPLO = 'L') are overwritten by the

off-diagonal elements of T; the rest of AB is overwritten by

values generated during the reduction.

*LDAB*

LDAB is INTEGER

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

*D*

D is DOUBLE PRECISION array, dimension (N)

The diagonal elements of the tridiagonal matrix T.

*E*

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

The off-diagonal elements of the tridiagonal matrix T:

E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

*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 i-th argument had an illegal value

**Author**

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 fine-grained and memory-aware 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 CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

Volume 28 Issue 2, Pages 196-209, May 2014.

http://hpc.sagepub.com/content/28/2/196

## 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 rank-k operation for matrix in
RFP format.

**Purpose:**

Level 3 BLAS like routine for C in RFP Format.

ZHFRK performs one of the Hermitian rank--k 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 n--by--n Hermitian

matrix and A is an n--by--k matrix in the first case and a k--by--n

matrix in the second case.

**Parameters**

*TRANSR*

TRANSR is CHARACTER*1

= 'N': The Normal Form of RFP A is stored;

= 'C': The Conjugate-transpose Form of RFP A is stored.

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the upper or lower

triangular part of the array C is to be referenced as

follows:

UPLO = 'U' or 'u' Only the upper triangular part of C

is to be referenced.

UPLO = 'L' or 'l' Only the lower triangular part of C

is to be referenced.

Unchanged on exit.

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the operation to be performed as

follows:

TRANS = 'N' or 'n' C := alpha*A*A**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 N--by--K part of

the array A must contain the matrix A, otherwise the leading

K--by--N part of the array A must contain the matrix A.

Unchanged on exit.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. When TRANS = 'N' or 'n'

then LDA must be at least max( 1, n ), otherwise LDA must

be at least max( 1, k ).

Unchanged on exit.

*BETA*

BETA is DOUBLE PRECISION

On entry, BETA specifies the scalar beta.

Unchanged on exit.

*C*

C is 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZHPCON**

**Purpose:**

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 1-norm of the original matrix A.

*RCOND*

RCOND is DOUBLE PRECISION

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an

estimate of the 1-norm of inv(A) computed in this routine.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zhpgst (integer ITYPE, character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, integer INFO)¶

**ZHPGST**

**Purpose:**

ZHPGST reduces a complex Hermitian-definite generalized

eigenproblem to standard form, using packed storage.

If ITYPE = 1, the problem is A*x = lambda*B*x,

and A is overwritten by inv(U**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 j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

On exit, if INFO = 0, the transformed matrix, stored in the

same format as A.

*BP*

BP is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZHPRFS**

**Purpose:**

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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

*AFP*

AFP is 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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zhptrd (character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( * ) TAU, integer INFO)¶

**ZHPTRD**

**Purpose:**

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 j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, if UPLO = 'U', the diagonal and first superdiagonal

of A are overwritten by the corresponding elements of the

tridiagonal matrix T, and the elements above the first

superdiagonal, with the array TAU, represent the 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 (N-1)

The off-diagonal elements of the tridiagonal matrix T:

E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

*TAU*

TAU is COMPLEX*16 array, dimension (N-1)

The scalar factors of the elementary reflectors (see Further

Details).

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

If UPLO = 'U', the matrix Q is represented as a product of elementary

reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**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:i-1) is stored on exit in AP,

overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary

reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**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).

## subroutine zhptrf (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)¶

**ZHPTRF**

**Purpose:**

ZHPTRF computes the factorization of a complex Hermitian packed

matrix A using the Bunch-Kaufman 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

1-by-1 and 2-by-2 diagonal blocks.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*AP*

AP is 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 j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

On exit, the block diagonal matrix D and the multipliers used

to obtain the factor U or L, stored as a packed triangular

matrix overwriting A (see below for further details).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and

columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)

is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =

IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were

interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, D(i,i) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular, and division by zero will occur if it

is used to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),

and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**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 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),

and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

**Contributors:**

## subroutine zhptri (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)¶

**ZHPTRI**

**Purpose:**

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 j-th column

of inv(A) is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;

if UPLO = 'L',

AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZHPTRF.

*WORK*

WORK is COMPLEX*16 array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zhptrs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZHPTRS**

**Purpose:**

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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZHSEIN**

**Purpose:**

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

block-triangular, then the j-th eigenvalue can be

assumed to be an eigenvalue of the block containing

the j-th row/column. This property allows 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': user-supplied initial vectors are stored in the arrays

VL and/or VR.

*SELECT*

SELECT is LOGICAL array, dimension (N)

Specifies the eigenvectors to be computed. To select the

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 i-th 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 i-th 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 i-th argument had an illegal value

> 0: if INFO = i, i is the number of eigenvectors which

failed to converge; see IFAILL and IFAILR for further

details.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Each eigenvector is normalized so that the element of largest

magnitude has magnitude 1; here the magnitude of a complex number

(x,y) is taken to be |x|+|y|.

## subroutine 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)¶

**ZHSEQR**

**Purpose:**

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:ILO-1 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, ... ILO-1

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

N-by-N matrix Q, which is assumed to be equal to the unit

matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,

if INFO = 0, Z contains Q*Z.

Normally Q is the 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 i-th argument had an illegal

value

> 0: if INFO = i, ZHSEQR failed to compute all of

the eigenvalues. Elements 1:ilo-1 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**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 (IHI-ILO+1) <= 500 is NS.

The default for (IHI-ILO+1) > 500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for

details.) Default: 14% of deflation window

size.

ISPEC=15: Number of simultaneous shifts in a multishift

QR iteration.

If IHI-ILO+1 is ...

greater than ...but less ... the

or equal to ... than default is

1 30 NS = 2(+)

30 60 NS = 4(+)

60 150 NS = 10(+)

150 590 NS = **

590 3000 NS = 64

3000 6000 NS = 128

6000 infinity NS = 256

(+) By default some or all matrices of this order

are passed to the implicit double shift routine

ZLAHQR and this parameter is ignored. See

ISPEC=12 above and comments in IPARMQ for

details.

(**) The asterisks (**) indicate an ad-hoc

function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.

If the number of simultaneous shifts (specified

by ISPEC=15) is less than 14, then the default

for ISPEC=16 is 0. Otherwise the default for

ISPEC=16 is 2.

**References:**

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR

Algorithm Part I: Maintaining Well Focused Shifts, and Level 3

Performance, SIAM Journal of Matrix Analysis, volume 23, pages

929--947, 2002.

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

## subroutine 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 component-wise 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zla_wwaddw (integer N, complex*16, dimension( * ) X, complex*16, dimension( * ) Y, complex*16, dimension( * ) W)¶

**ZLA_WWADDW** adds a vector into a doubled-single vector.

**Purpose:**

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

This works for all extant IBM's hex and binary floating point

arithmetic, but not for decimal.

**Parameters**

*N*

N is INTEGER

The length of vectors X, Y, and W.

*X*

X is COMPLEX*16 array, dimension (N)

The first part of the doubled-single accumulation vector.

*Y*

Y is COMPLEX*16 array, dimension (N)

The second part of the doubled-single accumulation vector.

*W*

W is COMPLEX*16 array, dimension (N)

The vector to be added.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 (N-1)

On entry, the off-diagonal 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 i-th 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 rank-one symmetric matrix. Used
when the original matrix is dense.

**Purpose:**

ZLAED7 computes the updated eigensystem of a diagonal

matrix after modification by a rank-one 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

sub-matrix. 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 rank-1-perturbed matrix.

On exit, the eigenvalues of the repaired matrix.

*Q*

Q is COMPLEX*16 array, dimension (LDQ,N)

On entry, the eigenvectors of the rank-1-perturbed 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 rank-1

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, non-deflated 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 i-th argument had an illegal value.

> 0: if INFO = 1, an eigenvalue did not converge

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 non-deflated 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 (N-K) updated eigenvectors

(those which were deflated) in its last N-K 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 (N-K) updated

eigenvalues (those which were deflated) sorted into increasing

order.

*RHO*

RHO is DOUBLE PRECISION

Contains the off diagonal element associated with the rank-1

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

sub-matrix. 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 sub-eigenvector matrix and the first row of

the second sub-eigenvector 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

deflation-altered z-vector 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 D-values 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

sub-problems 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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal

matrices are involved:

(1L) Givens rotations: the number of such rotations is GIVPTR; the

pairs of columns/rows they were applied to are stored in GIVCOL;

and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first

row, and for J=2:N, PERM(J)-th row of B is to be moved to the

J-th row.

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

For the right singular vector matrix, four types of orthogonal

matrices are involved:

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

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

null space.

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

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

**Parameters**

*ICOMPQ*

ICOMPQ is INTEGER

Specifies whether singular vectors are to be computed in

factored form:

= 0: Left singular vector matrix.

= 1: Right singular vector matrix.

*NL*

NL is INTEGER

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

*NR*

NR is INTEGER

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

*SQRE*

SQRE is INTEGER

= 0: the lower block is an NR-by-NR square matrix.

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

The bidiagonal matrix has row dimension N = NL + NR + 1,

and column dimension M = N + SQRE.

*NRHS*

NRHS is INTEGER

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

*B*

B is 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 I-th updated

(undeflated) singular value and the I-th (undeflated) old

singular value.

*DIFR*

DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).

On entry, DIFR(I, 1) contains the distances between I-th

updated (undeflated) singular value and the I+1-th

(undeflated) old singular value. And DIFR(I, 2) is the

normalizing factor for the I-th right singular vector.

*Z*

Z is DOUBLE PRECISION array, dimension ( K )

Contain the components of the deflation-adjusted updating row

vector.

*K*

K is INTEGER

Contains the dimension of the non-deflated matrix,

This is the order of the related secular equation. 1 <= K <=N.

*C*

C is DOUBLE PRECISION

C contains garbage if SQRE =0 and the C-value of a Givens

rotation related to the right null space if SQRE = 1.

*S*

S is DOUBLE PRECISION

S contains garbage if SQRE =0 and the S-value of a Givens

rotation related to the right null space if SQRE = 1.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension

( K*(1+NRHS) + 2*NRHS )

*INFO*

INFO is INTEGER

= 0: successful exit.

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Osni Marques, LBNL/NERSC, USA

## 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 I-th level and

singular values on the (I -1)-th level, and DIFR(*, 2 * I)

record the normalizing factors of the right singular vectors

matrices of subproblems on I-th level.

*Z*

Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).

On entry, Z(1, I) contains the components of the deflation-

adjusted updating row vector for subproblems on the I-th

level.

*POLES*

POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).

On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old

singular values involved in the secular equations on the I-th

level.

*GIVPTR*

GIVPTR is INTEGER array, dimension ( N ).

On entry, GIVPTR( I ) records the number of Givens

rotations performed on the I-th problem on the computation

tree.

*GIVCOL*

GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).

On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the

locations of Givens rotations performed on the I-th level on

the computation tree.

*LDGCOL*

LDGCOL is INTEGER, LDGCOL = > N.

The leading dimension of arrays GIVCOL and PERM.

*PERM*

PERM is INTEGER array, dimension ( LDGCOL, NLVL ).

On entry, PERM(*, I) records permutations done on the I-th

level of the computation tree.

*GIVNUM*

GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).

On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-

values of Givens rotations performed on the I-th level on the

computation tree.

*C*

C is DOUBLE PRECISION array, dimension ( N ).

On entry, if the I-th subproblem is not square,

C( I ) contains the C-value of a Givens rotation related to

the right null space of the I-th subproblem.

*S*

S is DOUBLE PRECISION array, dimension ( N ).

On entry, if the I-th subproblem is not square,

S( I ) contains the S-value of a Givens rotation related to

the right null space of the I-th subproblem.

*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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Osni Marques, LBNL/NERSC, USA

## 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*X-B, where A is N-by-N upper bidiagonal, and X and B

are N-by-NRHS. The solution X overwrites B.

The singular values of A smaller than RCOND times the largest

singular value are treated as zero in solving the least squares

problem; in this case a minimum norm solution is returned.

The actual singular values are returned in D in ascending order.

This code makes very mild assumptions about floating point

arithmetic. It will work on machines with a guard digit in

add/subtract, or on those binary machines without guard digits

which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.

It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': D and E define an upper bidiagonal matrix.

= 'L': D and E define a lower bidiagonal matrix.

*SMLSIZ*

SMLSIZ is INTEGER

The maximum size of the subproblems at the bottom of the

computation tree.

*N*

N is INTEGER

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

*NRHS*

NRHS is INTEGER

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

*D*

D is DOUBLE PRECISION array, dimension (N)

On entry D contains the main diagonal of the bidiagonal

matrix. On exit, if INFO = 0, D contains its singular values.

*E*

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

Contains the super-diagonal entries of the bidiagonal matrix.

On exit, E has been destroyed.

*B*

B is 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 i-th argument had an illegal value.

> 0: The algorithm failed to compute a singular value while

working on the submatrix lying in rows and columns

INFO/(N+1) through MOD(INFO,N+1).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Osni Marques, LBNL/NERSC, USA

## 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 1-norm, 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 = ( 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

conjugate-transposed format.

= 'N': RFP format is Normal

= 'C': RFP format is Conjugate-transposed

*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:K-1) when N is even;

K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If

TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A

as defined when TRANSR = 'N'. The contents of RFP A are

defined by UPLO as follows: If UPLO = 'U' the RFP A

contains the ( N*(N+1)/2 ) elements of upper packed A

either in normal or conjugate-transpose Format. If

UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) elements

of lower packed A either in normal or conjugate-transpose

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 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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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

M-by-N 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+(L-1)*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 M-by-N matrix C.

On exit, C is overwritten by the matrix H * C if SIDE = 'L',

or C * H if SIDE = 'R'.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L'

or (M) if SIDE = 'R'

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

## 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 conjugate-transpose
to a general matrix.

**Purpose:**

ZLARZB applies a complex block reflector H or its transpose H**H

to a complex distributed M-by-N C from the left or the right.

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

**Parameters**

*SIDE*

SIDE is CHARACTER*1

= 'L': apply H or H**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 K-by-K matrix T in the representation of the

block reflector.

*LDT*

LDT is INTEGER

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

*C*

C is COMPLEX*16 array, dimension (LDC,N)

On entry, the M-by-N 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

## 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 i-th 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 i-th 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

The shape of the matrix V and the storage of the vectors which define

the H(i) is best illustrated by the following example with n = 5 and

k = 3. The elements equal to 1 are not stored; the corresponding

array elements are modified but restored on exit. The rest of the

array is not used.

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

______V_____

( v1 v2 v3 ) / ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )

V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )

( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )

( v1 v2 v3 )

. . .

. . .

1 . .

1 .

1

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

______V_____

1 / . 1 ( 1 . . . . v1 v1 v1 v1 v1 )

. . 1 ( . 1 . . . v2 v2 v2 v2 v2 )

. . . ( . . 1 . . v3 v3 v3 v3 v3 )

. . .

( v1 v2 v3 )

( v1 v2 v3 )

V = ( v1 v2 v3 )

( v1 v2 v3 )

( v1 v2 v3 )

## subroutine 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 M-by-(M+L) complex upper trapezoidal matrix

[ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+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 M-by-M upper triangular matrices.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*L*

L is INTEGER

The number of columns of the matrix A containing the

meaningful part of the Householder vectors. N-M >= L >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the leading M-by-N upper trapezoidal part of the

array A must contain the matrix to be factorized.

On exit, the leading M-by-M upper triangular part of A

contains the upper triangular matrix R, and elements N-L+1 to

N of the first M rows of A, with the array TAU, represent the

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

The factorization is obtained by Householder's method. The kth

transformation matrix, Z( k ), which is used to introduce zeros into

the ( m - k + 1 )th row of A, is given in the form

Z( k ) = ( I 0 ),

( 0 T( k ) )

where

T( k ) = I - tau*u( k )*u( k )**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 ).

## 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)¶

**ZPBCON**

**Purpose:**

ZPBCON estimates the reciprocal of the condition number (in the

1-norm) 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 sub-diagonals 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 j-th column of U or L is

stored in the j-th column of the array AB as follows:

if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;

if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).

*LDAB*

LDAB is INTEGER

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

*ANORM*

ANORM is DOUBLE PRECISION

The 1-norm (or infinity-norm) of the 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 1-norm 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZPBEQU**

**Purpose:**

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 two-norm). S contains the scale factors,

S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with

elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This

choice of S puts the condition number of B within a factor N of the

smallest possible condition number over all possible diagonal

scalings.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangular of A is stored;

= 'L': Lower triangular of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is 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 j-th column

of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

*LDAB*

LDAB is INTEGER

The leading dimension of the array A. LDAB >= KD+1.

*S*

S is DOUBLE PRECISION array, dimension (N)

If INFO = 0, S contains the scale factors for A.

*SCOND*

SCOND is DOUBLE PRECISION

If INFO = 0, S contains the ratio of the smallest S(i) to

the largest S(i). If SCOND >= 0.1 and AMAX is neither too

large nor too small, it is not worth scaling by S.

*AMAX*

AMAX is DOUBLE PRECISION

Absolute value of largest matrix element. If AMAX is very

close to overflow or very close to underflow, the matrix

should be scaled.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, the i-th diagonal element is nonpositive.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZPBRFS**

**Purpose:**

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 j-th column

of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

*LDAB*

LDAB is INTEGER

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

*AFB*

AFB is 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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zpbstf (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, integer INFO)¶

**ZPBSTF**

**Purpose:**

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 n-m.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

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

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, if INFO = 0, the factor S from the split Cholesky

factorization A = S**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 i-th argument had an illegal value

> 0: if INFO = i, the factorization could not be completed,

because the updated element a(i,i) was negative; the

matrix A is not positive definite.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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.

## 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 super-diagonals of the matrix A if UPLO = 'U',

or the number of sub-diagonals 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

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, if INFO = 0, the triangular factor U or L from the

Cholesky factorization A = U**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 k-th argument had an illegal value

> 0: if INFO = k, the leading minor of order k is not

positive definite, and the factorization could not be

completed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The band storage scheme is illustrated by the following example, when

N = 6, KD = 2, and UPLO = 'U':

On entry: On exit:

* * a13 a24 a35 a46 * * u13 u24 u35 u46

* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56

a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry: On exit:

a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66

a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *

a31 a42 a53 a64 * * l31 l42 l53 l64 * *

Array elements marked * are not used by the routine.

## subroutine zpbtrf (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, integer INFO)¶

**ZPBTRF**

**Purpose:**

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

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, if INFO = 0, the triangular factor U or L from the

Cholesky factorization A = U**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 i-th argument had an illegal value

> 0: if INFO = i, the leading minor of order i is not

positive definite, and the factorization could not be

completed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The band storage scheme is illustrated by the following example, when

N = 6, KD = 2, and UPLO = 'U':

On entry: On exit:

* * a13 a24 a35 a46 * * u13 u24 u35 u46

* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56

a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry: On exit:

a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66

a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *

a31 a42 a53 a64 * * l31 l42 l53 l64 * *

Array elements marked * are not used by the routine.

**Contributors:**

## subroutine 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)¶

**ZPBTRS**

**Purpose:**

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 j-th column of U or L is

stored in the j-th column of the array AB as follows:

if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;

if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).

*LDAB*

LDAB is INTEGER

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

*B*

B is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zpftrf (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) A, integer INFO)¶

**ZPFTRF**

**Purpose:**

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 Conjugate-transpose TRANSR of RFP A is stored.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of RFP A is stored;

= 'L': Lower triangle of RFP A is stored.

*N*

N is INTEGER

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

*A*

A is 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:k-1) when N is even; k=N/2. RFP A is

(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is

the Conjugate-transpose of RFP A as defined when

TRANSR = 'N'. The contents of RFP A are defined by UPLO as

follows: If UPLO = 'U' the RFP A contains the nt elements of

upper packed A. If UPLO = 'L' the RFP A contains the elements

of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =

'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 i-th argument had an illegal value

> 0: if INFO = i, the leading minor of order i is not

positive definite, and the factorization could not be

completed.

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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zpftri (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) A, integer INFO)¶

**ZPFTRI**

**Purpose:**

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 Conjugate-transpose TRANSR of RFP A is stored.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*A*

A is 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:k-1) when N is even; k=N/2. RFP A is

(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is

the Conjugate-transpose of RFP A as defined when

TRANSR = 'N'. The contents of RFP A are defined by UPLO as

follows: If UPLO = 'U' the RFP A contains the nt elements of

upper packed A. If UPLO = 'L' the RFP A contains the elements

of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =

'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 i-th argument had an illegal value

> 0: if INFO = i, the (i,i) element of the factor U or L is

zero, and the inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## subroutine zpftrs (character TRANSR, character UPLO, integer N, integer NRHS, complex*16, dimension( 0: * ) A, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZPFTRS**

**Purpose:**

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 Conjugate-transpose TRANSR of RFP A is stored.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of RFP A is stored;

= 'L': Lower triangle of RFP A is stored.

*N*

N is INTEGER

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

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is 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 i-th argument had an illegal value

**Author**

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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## 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)¶

**ZPPCON**

**Purpose:**

ZPPCON estimates the reciprocal of the condition number (in the

1-norm) 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 j-th column of U or L is stored in the array AP

as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

*ANORM*

ANORM is DOUBLE PRECISION

The 1-norm (or infinity-norm) of the 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 1-norm 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zppequ (character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)¶

**ZPPEQU**

**Purpose:**

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 two-norm). S contains the

scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix

B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.

This choice of S puts the condition number of B within a factor N of

the smallest possible condition number over all possible diagonal

scalings.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*AP*

AP is 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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*S*

S is DOUBLE PRECISION array, dimension (N)

If INFO = 0, S contains the scale factors for A.

*SCOND*

SCOND is DOUBLE PRECISION

If INFO = 0, S contains the ratio of the smallest S(i) to

the largest S(i). If SCOND >= 0.1 and AMAX is neither too

large nor too small, it is not worth scaling by S.

*AMAX*

AMAX is DOUBLE PRECISION

Absolute value of largest matrix element. If AMAX is very

close to overflow or very close to underflow, the matrix

should be scaled.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, the i-th diagonal element is nonpositive.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZPPRFS**

**Purpose:**

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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*AFP*

AFP is 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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zpptrf (character UPLO, integer N, complex*16, dimension( * ) AP, integer INFO)¶

**ZPPTRF**

**Purpose:**

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 j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

See below for further details.

On exit, if INFO = 0, the triangular factor U or L from the

Cholesky factorization A = U**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 i-th argument had an illegal value

> 0: if INFO = i, the leading minor of order i is not

positive definite, and the factorization could not be

completed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The packed storage scheme is illustrated by the following example

when N = 4, UPLO = 'U':

Two-dimensional 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 ]

## subroutine zpptri (character UPLO, integer N, complex*16, dimension( * ) AP, integer INFO)¶

**ZPPTRI**

**Purpose:**

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 j-th column of U or L is stored in the

array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

On exit, the upper or lower triangle of the (Hermitian)

inverse of A, overwriting the input factor U or L.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, the (i,i) element of the factor U or L is

zero, and the inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zpptrs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZPPTRS**

**Purpose:**

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 j-th column of U or L is stored in the array AP

as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

*B*

B is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 (K-1)st step

if the pivot <= TOL.

*LDA*

LDA is INTEGER

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

*WORK*

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

Work space.

*INFO*

INFO is INTEGER

< 0: If INFO = -K, the K-th argument had an illegal value,

= 0: algorithm completed successfully, and

> 0: the matrix A is either rank deficient with computed rank

as returned in RANK, or is not positive semidefinite. See

Section 7 of LAPACK Working Note #161 for further

information.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 (K-1)st step

if the pivot <= TOL.

*WORK*

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

Work space.

*INFO*

INFO is INTEGER

< 0: If INFO = -K, the K-th argument had an illegal value,

= 0: algorithm completed successfully, and

> 0: the matrix A is either rank deficient with computed rank

as returned in RANK, or is not positive semidefinite. See

Section 7 of LAPACK Working Note #161 for further

information.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZSPCON**

**Purpose:**

ZSPCON estimates the reciprocal of the condition number (in the

1-norm) 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 1-norm of the original matrix A.

*RCOND*

RCOND is DOUBLE PRECISION

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an

estimate of the 1-norm of inv(A) computed in this routine.

*WORK*

WORK is COMPLEX*16 array, dimension (2*N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZSPRFS**

**Purpose:**

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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

*AFP*

AFP is 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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsptrf (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, integer INFO)¶

**ZSPTRF**

**Purpose:**

ZSPTRF computes the factorization of a complex symmetric matrix A

stored in packed format using the Bunch-Kaufman diagonal pivoting

method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, and D is symmetric and block diagonal with

1-by-1 and 2-by-2 diagonal blocks.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*AP*

AP is 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 j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

On exit, the block diagonal matrix D and the multipliers used

to obtain the factor U or L, stored as a packed triangular

matrix overwriting A (see below for further details).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and

columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)

is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =

IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were

interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, D(i,i) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular, and division by zero will occur if it

is used to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

5-96 - 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 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),

and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),

and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

## subroutine zsptri (character UPLO, integer N, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)¶

**ZSPTRI**

**Purpose:**

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 j-th column

of inv(A) is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;

if UPLO = 'L',

AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by ZSPTRF.

*WORK*

WORK is COMPLEX*16 array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zsptrs (character UPLO, integer N, integer NRHS, complex*16, dimension( * ) AP, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZSPTRS**

**Purpose:**

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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZSTEDC**

**Purpose:**

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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.

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 (N-1)

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*(N-1)).

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 i-th 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## 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)¶

**ZSTEGR**

**Purpose:**

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

IEEE-754 floating-point standard in their handling of infinities and

NaNs. Normal execution may create these exceptiona values and hence

may abort due to a floating point exception in environments which

do not conform to the IEEE-754 standard.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found.

= 'I': the IL-th through IU-th eigenvalues will be found.

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is DOUBLE PRECISION array, dimension (N)

On entry, the N diagonal elements of the tridiagonal matrix

T. On exit, D is overwritten.

*E*

E is DOUBLE PRECISION array, dimension (N)

On entry, the (N-1) subdiagonal elements of the tridiagonal

matrix T in elements 1 to N-1 of E. E(N) need not be set on

input, but is used internally as workspace.

On exit, E is overwritten.

*VL*

VL is DOUBLE PRECISION

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is DOUBLE PRECISION

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is DOUBLE PRECISION

Unused. Was the absolute error tolerance for the

eigenvalues/eigenvectors in previous versions.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is DOUBLE PRECISION array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is 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 i-th

column of Z holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

Supplying N columns is always safe.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', then LDZ >= max(1,N).

*ISUPPZ*

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )

The support of the eigenvectors in Z, i.e., the indices

indicating the nonzero elements in Z. The i-th computed eigenvector

is nonzero only in elements ISUPPZ( 2*i-1 ) through

ISUPPZ( 2*i ). This is relevant in the case when the matrix

is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

*WORK*

WORK is DOUBLE PRECISION array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

(and minimal) LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,18*N)

if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (LIWORK)

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= max(1,10*N)

if the eigenvectors are desired, and LIWORK >= max(1,8*N)

if only the eigenvalues are to be computed.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal size of the IWORK array,

returns this value as the first entry of the IWORK array, and

no error message related to LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

On exit, INFO

= 0: successful exit

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

> 0: if INFO = 1X, internal error in DLARRE,

if INFO = 2X, internal error in ZLARRV.

Here, the digit X = ABS( IINFO ) < 10, where IINFO is

the nonzero error code returned by DLARRE or

ZLARRV, respectively.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Osni Marques, LBNL/NERSC, USA

Christof Voemel, LBNL/NERSC, USA

## 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)¶

**ZSTEIN**

**Purpose:**

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 (N-1)

The (n-1) subdiagonal elements of the tridiagonal matrix

T, stored in elements 1 to N-1.

*M*

M is INTEGER

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

*W*

W is DOUBLE PRECISION array, dimension (N)

The first M elements of W contain the eigenvalues for

which eigenvectors are to be computed. The eigenvalues

should be grouped by split-off block and ordered from

smallest to largest within the block. ( The output array

W from DSTEBZ with ORDER = 'B' is expected here. )

*IBLOCK*

IBLOCK is INTEGER array, dimension (N)

The submatrix indices associated with the corresponding

eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to

the first submatrix from the top, =2 if W(i) belongs to

the second submatrix, etc. ( The output array IBLOCK

from DSTEBZ is expected here. )

*ISPLIT*

ISPLIT is INTEGER array, dimension (N)

The splitting points, at which T breaks up into submatrices.

The first submatrix consists of rows/columns 1 to

ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1

through ISPLIT( 2 ), etc.

( The output array ISPLIT from DSTEBZ is expected here. )

*Z*

Z is COMPLEX*16 array, dimension (LDZ, M)

The computed eigenvectors. The eigenvector associated

with the eigenvalue W(i) is stored in the i-th column of

Z. Any vector which fails to converge is set to its current

iterate after MAXITS iterations.

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 i-th argument had an illegal value

> 0: if INFO = i, then i eigenvectors failed to converge

in MAXITS iterations. Their indices are stored in

array IFAIL.

**Internal Parameters:**

MAXITS INTEGER, default = 5

The maximum number of iterations performed.

EXTRA INTEGER, default = 2

The number of iterations performed after norm growth

criterion is satisfied, should be at least 1.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZSTEMR**

**Purpose:**

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. 1-28, August 2004.

- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and

Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,

2004. Also LAPACK Working Note 154.

- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric

tridiagonal eigenvalue/eigenvector problem",

Computer Science Division Technical Report No. UCB/CSD-97-971,

UC Berkeley, May 1997.

Further Details

1.ZSTEMR works only on machines which follow IEEE-754

floating-point standard in their handling of infinities and NaNs.

This permits the use of efficient inner loops avoiding a check for

zero divisors.

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 off-diagonals. By applying a

similarity transform with an appropriate diagonal matrix

diag(1,e^{i hy_1},

... , e^{i hy_{n-1}}),

the complex Hermitean

matrix can be transformed into a real symmetric matrix and complex

arithmetic can be entirely avoided.)

While the eigenvectors of the real symmetric tridiagonal matrix are real,

the eigenvectors of original complex Hermitean matrix have complex entries

in general.

Since LAPACK drivers overwrite the matrix data with the eigenvectors,

ZSTEMR accepts complex workspace to facilitate interoperability

with ZUNMTR or ZUPMTR.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found.

= 'I': the IL-th through IU-th eigenvalues will be found.

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is DOUBLE PRECISION array, dimension (N)

On entry, the N diagonal elements of the tridiagonal matrix

T. On exit, D is overwritten.

*E*

E is DOUBLE PRECISION array, dimension (N)

On entry, the (N-1) subdiagonal elements of the tridiagonal

matrix T in elements 1 to N-1 of E. E(N) need not be set on

input, but is used internally as workspace.

On exit, E is overwritten.

*VL*

VL is DOUBLE PRECISION

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is DOUBLE PRECISION

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0.

Not referenced if RANGE = 'A' or 'V'.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is DOUBLE PRECISION array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is 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 i-th

column of Z holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and can be computed with a workspace

query by setting NZC = -1, see below.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', then LDZ >= max(1,N).

*NZC*

NZC is INTEGER

The number of eigenvectors to be held in the array Z.

If RANGE = 'A', then NZC >= max(1,N).

If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].

If RANGE = 'I', then NZC >= IU-IL+1.

If NZC = -1, then a workspace query is assumed; the

routine calculates the number of columns of the array Z that

are needed to hold the eigenvectors.

This value is returned as the first entry of the Z array, and

no error message related to NZC is issued by XERBLA.

*ISUPPZ*

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )

The support of the eigenvectors in Z, i.e., the indices

indicating the nonzero elements in Z. The i-th computed eigenvector

is nonzero only in elements ISUPPZ( 2*i-1 ) through

ISUPPZ( 2*i ). This is relevant in the case when the matrix

is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

*TRYRAC*

TRYRAC is LOGICAL

If TRYRAC = .TRUE., indicates that the code should check whether

the tridiagonal matrix defines its eigenvalues to high relative

accuracy. If so, the code uses relative-accuracy preserving

algorithms that might be (a bit) slower depending on the matrix.

If the matrix does not define its eigenvalues to high relative

accuracy, the code can uses possibly faster algorithms.

If TRYRAC = .FALSE., the code is not required to guarantee

relatively accurate eigenvalues and can use the fastest possible

techniques.

On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix

does not define its eigenvalues to high relative accuracy.

*WORK*

WORK is DOUBLE PRECISION array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

(and minimal) LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,18*N)

if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (LIWORK)

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= max(1,10*N)

if the eigenvectors are desired, and LIWORK >= max(1,8*N)

if only the eigenvalues are to be computed.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal size of the IWORK array,

returns this value as the first entry of the IWORK array, and

no error message related to LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

On exit, INFO

= 0: successful exit

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

> 0: if INFO = 1X, internal error in DLARRE,

if INFO = 2X, internal error in ZLARRV.

Here, the digit X = ABS( IINFO ) < 10, where IINFO is

the nonzero error code returned by DLARRE or

ZLARRV, respectively.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Jim Demmel, University of California, Berkeley, USA

Inderjit Dhillon, University of Texas, Austin, USA

Osni Marques, LBNL/NERSC, USA

Christof Voemel, University of California, Berkeley, USA

## subroutine 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)¶

**ZSTEQR**

**Purpose:**

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 (N-1)

On entry, the (n-1) 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*N-2))

If COMPZ = 'N', then WORK is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTBCON**

**Purpose:**

ZTBCON estimates the reciprocal of the condition number of a

triangular band matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for

norm(inv(A)), then the reciprocal of the condition number is

computed as

RCOND = 1 / ( norm(A) * norm(inv(A)) ).

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies whether the 1-norm condition number or the

infinity-norm condition number is required:

= '1' or 'O': 1-norm;

= 'I': Infinity-norm.

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals or subdiagonals of the

triangular band matrix A. KD >= 0.

*AB*

AB is 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 j-th column of A is stored

in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

If DIAG = 'U', the diagonal elements of A are not referenced

and are assumed to be 1.

*LDAB*

LDAB is INTEGER

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

*RCOND*

RCOND is DOUBLE PRECISION

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(norm(A) * norm(inv(A))).

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTBRFS**

**Purpose:**

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 non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals or subdiagonals of the

triangular band matrix A. KD >= 0.

*NRHS*

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 j-th column of A is stored

in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

If DIAG = 'U', the diagonal elements of A are not referenced

and are assumed to be 1.

*LDAB*

LDAB is INTEGER

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

*B*

B is 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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTBTRS**

**Purpose:**

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

N-by-NRHS matrix. A check is made to verify that A is nonsingular.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate transpose)

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals or subdiagonals of the

triangular band matrix A. KD >= 0.

*NRHS*

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 j-th column of A is stored

in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

If DIAG = 'U', the diagonal elements of A are not referenced

and are assumed to be 1.

*LDAB*

LDAB is INTEGER

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

*B*

B is 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 i-th argument had an illegal value

> 0: if INFO = i, the i-th diagonal element of A is zero,

indicating that the matrix is singular and the

solutions X have not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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: ldb-1, 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

non-unit, 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 Conjugate-transpose Form of RFP A is stored.

*SIDE*

SIDE is CHARACTER*1

On entry, SIDE specifies whether op( A ) appears on the left

or right of X as follows:

SIDE = 'L' or 'l' op( A )*X = alpha*B.

SIDE = 'R' or 'r' X*op( A ) = alpha*B.

Unchanged on exit.

*UPLO*

UPLO is CHARACTER*1

On entry, UPLO specifies whether the RFP matrix A came from

an upper or lower triangular matrix as follows:

UPLO = 'U' or 'u' RFP A came from an upper triangular matrix

UPLO = 'L' or 'l' RFP A came from a lower triangular matrix

Unchanged on exit.

*TRANS*

TRANS is CHARACTER*1

On entry, TRANS specifies the form of op( A ) to be used

in the matrix multiplication as follows:

TRANS = 'N' or 'n' op( A ) = A.

TRANS = '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:K-1) when N is even;

K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If

TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A as

defined when TRANSR = 'N'. The contents of RFP A are defined

by UPLO as follows: If UPLO = 'U' the RFP A contains the NT

elements of upper packed A either in normal or

conjugate-transpose Format. If UPLO = 'L' the RFP A contains

the NT elements of lower packed A either in normal or

conjugate-transpose 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 right-hand side matrix B, and on exit is

overwritten by the solution matrix X.

*LDB*

LDB is INTEGER

On entry, LDB specifies the first dimension of B as declared

in the calling (sub) program. LDB must be at least

max( 1, m ).

Unchanged on exit.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## subroutine ztftri (character TRANSR, character UPLO, character DIAG, integer N, complex*16, dimension( 0: * ) A, integer INFO)¶

**ZTFTRI**

**Purpose:**

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 Conjugate-transpose TRANSR of RFP A is stored.

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

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:k-1) when N is even; k=N/2. RFP A is

(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is

the Conjugate-transpose of RFP A as defined when

TRANSR = 'N'. The contents of RFP A are defined by UPLO as

follows: If UPLO = 'U' the RFP A contains the nt elements of

upper packed A; If UPLO = 'L' the RFP A contains the nt

elements of lower packed A. The LDA of RFP A is (N+1)/2 when

TRANSR = '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 i-th argument had an illegal value

> 0: if INFO = i, A(i,i) is exactly zero. The triangular

matrix is singular and its inverse can not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## 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 Conjugate-transpose format;

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*N*

N is INTEGER

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

*ARF*

ARF is 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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## subroutine ztfttr (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: * ) ARF, complex*16, dimension( 0: lda-1, 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 Conjugate-transpose format;

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*N*

N is INTEGER

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

*ARF*

ARF is 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 N-by-N upper triangular part of the array A contains

the upper triangular matrix, and the strictly lower

triangular part of A is not referenced. If UPLO = 'L', the

leading N-by-N lower triangular part of the array A contains

the lower triangular matrix, and the strictly upper

triangular part of A is not referenced.

*LDA*

LDA is INTEGER

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

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## 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)¶

**ZTGSEN**

**Purpose:**

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. F-norm-based estimate

(DIF(1:2)).

=3: Estimate of Difu and Difl. 1-norm-based estimate

(DIF(1:2)).

About 5 times as expensive as IJOB = 2.

=4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic

version to get it all.

=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

*WANTQ*

WANTQ is LOGICAL

.TRUE. : update the left transformation matrix Q;

.FALSE.: do not update Q.

*WANTZ*

WANTZ is LOGICAL

.TRUE. : update the right transformation matrix Z;

.FALSE.: do not update Z.

*SELECT*

SELECT is LOGICAL array, dimension (N)

SELECT specifies the eigenvalues in the selected cluster. To

select 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 N-by-N 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 N-by-N 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 F-norm-based upper bounds on

Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based

estimates of Difu and Difl, computed using reversed

communication with ZLACN2.

If M = 0 or N, DIF(1:2) = F-norm([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*(N-M)

If IJOB = 3 or 5, LWORK >= 4*M*(N-M)

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= 1.

If IJOB = 1, 2 or 4, LIWORK >= N+2;

If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal size of the IWORK array,

returns this value as the first entry of the IWORK array, and

no error message related to LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

=0: Successful exit.

<0: If INFO = -i, the i-th argument had an illegal value.

=1: Reordering of (A, B) failed because the transformed

matrix pair (A, B) would be too far from generalized

Schur form; the problem is very ill-conditioned.

(A, B) may have been partially reordered.

If requested, 0 is returned in DIF(*), PL and PR.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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 ill-conditioned,

then its value may differ significantly from its value before

reordering.

The reciprocal condition numbers of the left and right eigenspaces

spanned by the first n1 columns of U and W (or Q*U and Z*W) may

be returned in DIF(1:2), corresponding to Difu and Difl, resp.

The Difu and Difl are defined as:

Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )

and

Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the

(2*n1*n2)-by-(2*n1*n2) matrix

Zu = [ kron(In2, A11) -kron(A22**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 = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).

An approximate (asymptotic) bound on the average absolute error of

the selected eigenvalues is

EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up

to a certain restriction: A lower bound (x) on the smallest

F-norm(E,F) for which an eigenvalue of (A11, B11) may move and

coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),

(i.e. (A + E, B + F), is

x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR.

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed

(L', R') and unperturbed (L, R) left and right deflating subspaces

associated with the selected cluster in the (1,1)-blocks can be

bounded as

max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))

max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User's Guide section 4.11 or the following references

for more information.

Note that if the default method for computing the Frobenius-norm-

based estimate DIF is not wanted (see 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:**

**References:**

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

## subroutine 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)¶

**ZTGSJA**

**Purpose:**

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 M-by-N matrix A and P-by-N matrix B:

N-K-L K L

A = K ( 0 A12 A13 ) if M-K-L >= 0;

L ( 0 0 A23 )

M-K-L ( 0 0 0 )

N-K-L K L

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

M-K ( 0 0 A23 )

N-K-L K L

B = L ( 0 0 B13 )

P-L ( 0 0 0 )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular

upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,

otherwise A23 is (M-K)-by-L upper trapezoidal.

On exit,

U**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 M-K-L >= 0,

K L

D1 = K ( I 0 )

L ( 0 C )

M-K-L ( 0 0 )

K L

D2 = L ( 0 S )

P-L ( 0 0 )

N-K-L K L

( 0 R ) = K ( 0 R11 R12 ) K

L ( 0 0 R22 ) L

where

C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),

S = diag( BETA(K+1), ... , BETA(K+L) ),

C**2 + S**2 = I.

R is stored in A(1:K+L,N-K-L+1:N) on exit.

If M-K-L < 0,

K M-K K+L-M

D1 = K ( I 0 0 )

M-K ( 0 C 0 )

K M-K K+L-M

D2 = M-K ( 0 S 0 )

K+L-M ( 0 0 I )

P-L ( 0 0 0 )

N-K-L K M-K K+L-M

( 0 R ) = K ( 0 R11 R12 R13 )

M-K ( 0 0 R22 R23 )

K+L-M ( 0 0 0 R33 )

where

C = diag( ALPHA(K+1), ... , ALPHA(M) ),

S = diag( BETA(K+1), ... , BETA(M) ),

C**2 + S**2 = I.

R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored

( 0 R22 R23 )

in B(M-K+1:L,N+M-K-L+1:N) on exit.

The computation of the 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),N-L+1:N) and B13 = B(1:L,,N-L+1:N)

of A and B, whose GSVD is going to be computed by ZTGSJA.

See Further Details.

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the M-by-N matrix A.

On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular

matrix R or part of R. See Purpose for details.

*LDA*

LDA is INTEGER

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

*B*

B is COMPLEX*16 array, dimension (LDB,N)

On entry, the P-by-N matrix B.

On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains

a part of R. See Purpose for details.

*LDB*

LDB is INTEGER

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

*TOLA*

TOLA is DOUBLE PRECISION

*TOLB*

TOLB is DOUBLE PRECISION

TOLA and TOLB are the convergence criteria for the Jacobi-

Kogbetliantz iteration procedure. Generally, they are the

same as used in the preprocessing step, say

TOLA = MAX(M,N)*norm(A)*MAZHEPS,

TOLB = MAX(P,N)*norm(B)*MAZHEPS.

*ALPHA*

ALPHA is DOUBLE PRECISION array, dimension (N)

*BETA*

BETA is DOUBLE PRECISION array, dimension (N)

On exit, ALPHA and BETA contain the generalized singular

value pairs of A and B;

ALPHA(1:K) = 1,

BETA(1:K) = 0,

and if M-K-L >= 0,

ALPHA(K+1:K+L) = diag(C),

BETA(K+1:K+L) = diag(S),

or if M-K-L < 0,

ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0

BETA(K+1:M) = S, BETA(M+1:K+L) = 1.

Furthermore, if K+L < N,

ALPHA(K+L+1:N) = 0 and

BETA(K+L+1:N) = 0.

*U*

U is 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 i-th argument had an illegal value.

= 1: the procedure does not converge after MAXIT cycles.

**Internal Parameters:**

MAXIT INTEGER

MAXIT specifies the total loops that the iterative procedure

may take. If after MAXIT cycles, the routine fails to

converge, we return INFO = 1.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce

min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L

matrix B13 to the form:

U1**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 L-by-L nonsingular upper triangular matrix.

## 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)¶

**ZTGSNA**

**Purpose:**

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 j-th 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 norm-based 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The reciprocal of the condition number of the i-th 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 2-norm 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 i-th

computed generalized eigenvalue w and the corresponding exact

eigenvalue lambda is

chord(w, lambda) <= EPS * norm(A, B) / S(I),

where EPS is the machine precision.

The reciprocal of the condition number 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 ) n-1

1 n-1 1 n-1

Then the reciprocal condition number DIF(I) is

Difl[(a, b), (A22, B22)] = sigma-min( Zl )

where sigma-min(Zl) denotes the smallest singular value of

Zl = [ kron(a, In-1) -kron(1, A22) ]

[ kron(b, In-1) -kron(1, B22) ].

Here In-1 is the identity matrix of size n-1 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. [2-3] for more details and further references.

**Contributors:**

**References:**

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the

Generalized Real Schur Form of a Regular Matrix Pair (A, B), in

M.S. Moonen et al (eds), Linear Algebra for Large Scale and

Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified

Eigenvalues of a Regular Matrix Pair (A, B) and Condition

Estimation: Theory, Algorithms and Software, Report

UMINF - 94.04, Department of Computing Science, Umea University,

S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.

To appear in Numerical Algorithms, 1996.

[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

for Solving the Generalized Sylvester Equation and Estimating the

Separation between Regular Matrix Pairs, Report UMINF - 93.23,

Department of Computing Science, Umea University, S-901 87 Umea,

Sweden, December 1993, Revised April 1994, Also as LAPACK Working

Note 75.

To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

## subroutine 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)¶

**ZTPCON**

**Purpose:**

ZTPCON estimates the reciprocal of the condition number of a packed

triangular matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for

norm(inv(A)), then the reciprocal of the condition number is

computed as

RCOND = 1 / ( norm(A) * norm(inv(A)) ).

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies whether the 1-norm condition number or the

infinity-norm condition number is required:

= '1' or 'O': 1-norm;

= 'I': Infinity-norm.

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2)

The upper or lower triangular matrix A, packed columnwise in

a linear array. The j-th column of A is stored in the array

AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

If DIAG = 'U', the diagonal elements of A are not referenced

and are assumed to be 1.

*RCOND*

RCOND is DOUBLE PRECISION

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(norm(A) * norm(inv(A))).

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTPMQRT**

**Purpose:**

ZTPMQRT applies a complex orthogonal matrix Q obtained from a

"triangular-pentagonal" complex block reflector H to a general

complex matrix C, which consists of two blocks A and B.

**Parameters**

*SIDE*

SIDE is CHARACTER*1

= 'L': apply Q or Q**H from the Left;

= 'R': apply Q or Q**H from the Right.

*TRANS*

TRANS is CHARACTER*1

= 'N': No transpose, apply Q;

= 'C': 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 i-th column must contain the vector which defines the

elementary reflector H(i), for i = 1,2,...,k, as returned by

CTPQRT in B. See Further Details.

*LDV*

LDV is INTEGER

The leading dimension of the array V.

If SIDE = 'L', LDV >= max(1,M);

if SIDE = 'R', LDV >= max(1,N).

*T*

T is COMPLEX*16 array, dimension (LDT,K)

The upper triangular factors of the block reflectors

as returned by CTPQRT, stored as a NB-by-K matrix.

*LDT*

LDT is INTEGER

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

*A*

A is COMPLEX*16 array, dimension

(LDA,N) if SIDE = 'L' or

(LDA,K) if SIDE = 'R'

On entry, the K-by-N or M-by-K matrix A.

On exit, A is overwritten by the corresponding block of

Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

*LDA*

LDA is INTEGER

The leading dimension of the array A.

If SIDE = 'L', LDC >= max(1,K);

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

*B*

B is COMPLEX*16 array, dimension (LDB,N)

On entry, the M-by-N matrix B.

On exit, B is overwritten by the corresponding block of

Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

*LDB*

LDB is INTEGER

The leading dimension of the array B.

LDB >= max(1,M).

*WORK*

WORK is COMPLEX*16 array. The dimension of WORK is

N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The columns of the pentagonal matrix V contain the elementary reflectors

H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a

trapezoidal block V2:

V = [V1]

[V2].

The size of the trapezoidal block V2 is determined by the parameter L,

where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L

rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular;

if L=0, there is no trapezoidal block, hence V = V1 is rectangular.

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

[B]

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

The 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.

## 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)¶

**ZTPQRT**

**Purpose:**

ZTPQRT computes a blocked QR factorization of a complex

"triangular-pentagonal" matrix C, which is composed of a

triangular block A and pentagonal block B, using the compact

WY representation for Q.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix B.

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 N-by-N matrix A.

On exit, the elements on and above the diagonal of the array

contain the upper triangular matrix R.

*LDA*

LDA is INTEGER

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

*B*

B is COMPLEX*16 array, dimension (LDB,N)

On entry, the pentagonal M-by-N matrix B. The first M-L rows

are rectangular, and the last L rows are upper trapezoidal.

On exit, B contains the pentagonal matrix V. See Further Details.

*LDB*

LDB is INTEGER

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

*T*

T is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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

C = [ A ]

[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal

matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N

upper trapezoidal matrix B2:

B = [ B1 ] <- (M-L)-by-N rectangular

[ B2 ] <- L-by-N upper trapezoidal.

The upper trapezoidal matrix B2 consists of the first L rows of a

N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,

B is rectangular M-by-N; if M=L=N, B is upper triangular.

The matrix W stores the elementary reflectors H(i) in the i-th column

below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ] <- upper triangular N-by-N

[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ] <- identity, N-by-N

[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which

we call V above. Note that V has the same form as B; that is,

V = [ V1 ] <- (M-L)-by-N rectangular

[ V2 ] <- L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s.

The number of blocks is B = ceiling(N/NB), where each

block is of order NB except for the last block, which is of order

IB = N - (B-1)*NB. For each of the B blocks, a upper triangular block

reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB

for the last block) T's are stored in the NB-by-N matrix T as

T = [T1 T2 ... TB].

## subroutine 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
'triangular-pentagonal' 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 "triangular-pentagonal"

matrix C, which is composed of a triangular block A and pentagonal block B,

using the compact WY representation for Q.

**Parameters**

*M*

M is INTEGER

The total number of rows of the matrix B.

M >= 0.

*N*

N is INTEGER

The number of columns of the matrix B, and the order of

the triangular matrix A.

N >= 0.

*L*

L is INTEGER

The number of rows of the 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 N-by-N matrix A.

On exit, the elements on and above the diagonal of the array

contain the upper triangular matrix R.

*LDA*

LDA is INTEGER

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

*B*

B is COMPLEX*16 array, dimension (LDB,N)

On entry, the pentagonal M-by-N matrix B. The first M-L rows

are rectangular, and the last L rows are upper trapezoidal.

On exit, B contains the pentagonal matrix V. See Further Details.

*LDB*

LDB is INTEGER

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

*T*

T is COMPLEX*16 array, dimension (LDT,N)

The N-by-N upper triangular factor T of the block reflector.

See Further Details.

*LDT*

LDT is INTEGER

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

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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

C = [ A ]

[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal

matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N

upper trapezoidal matrix B2:

B = [ B1 ] <- (M-L)-by-N rectangular

[ B2 ] <- L-by-N upper trapezoidal.

The upper trapezoidal matrix B2 consists of the first L rows of a

N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,

B is rectangular M-by-N; if M=L=N, B is upper triangular.

The matrix W stores the elementary reflectors H(i) in the i-th column

below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ] <- upper triangular N-by-N

[ B ] <- M-by-N pentagonal

so that W can be represented as

W = [ I ] <- identity, N-by-N

[ V ] <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which

we call V above. Note that V has the same form as B; that is,

V = [ V1 ] <- (M-L)-by-N rectangular

[ V2 ] <- L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s.

The (M+N)-by-(M+N) block reflector H is then given by

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

where W**H is the conjugate transpose of W and T is the upper triangular

factor of the block reflector.

## 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)¶

**ZTPRFS**

**Purpose:**

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 non-unit 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 j-th column of A is stored in the array

AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

If DIAG = 'U', the diagonal elements of A are not referenced

and are assumed to be 1.

*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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ztptri (character UPLO, character DIAG, integer N, complex*16, dimension( * ) AP, integer INFO)¶

**ZTPTRI**

**Purpose:**

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 non-unit 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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.

See below for further details.

On exit, the (triangular) inverse of the original matrix, in

the same packed storage format.

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, A(i,i) is exactly zero. The triangular

matrix is singular and its inverse can not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

A triangular matrix A can be transferred to packed storage using one

of the following program segments:

UPLO = 'U': UPLO = 'L':

JC = 1 JC = 1

DO 2 J = 1, N DO 2 J = 1, N

DO 1 I = 1, J DO 1 I = J, N

AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)

1 CONTINUE 1 CONTINUE

JC = JC + J JC = JC + N - J + 1

2 CONTINUE 2 CONTINUE

## subroutine ztptrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS, complex*16, dimension( * ) AP, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**ZTPTRS**

**Purpose:**

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 N-by-NRHS matrix. A check is made to verify that A is

nonsingular.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate transpose)

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*NRHS*

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 j-th column of A is stored in the array

AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

*B*

B is 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 i-th argument had an illegal value

> 0: if INFO = i, the i-th diagonal element of A is zero,

indicating that the matrix is singular and the

solutions X have not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 Conjugate-transpose format is wanted.

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*N*

N is INTEGER

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

*AP*

AP is COMPLEX*16 array, dimension ( N*(N+1)/2 ),

On entry, the upper or lower triangular matrix A, packed

columnwise in a linear array. The j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*ARF*

ARF is 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 i-th argument had an illegal value

**Author**

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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## 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 j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*A*

A is COMPLEX*16 array, dimension ( LDA, N )

On exit, the triangular matrix A. If UPLO = 'U', the leading

N-by-N upper triangular part of A contains the upper

triangular part of the matrix A, and the strictly lower

triangular part of A is not referenced. If UPLO = 'L', the

leading N-by-N lower triangular part of A contains the lower

triangular part of the matrix A, and the strictly upper

triangular part of A is not referenced.

*LDA*

LDA is INTEGER

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

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTRCON**

**Purpose:**

ZTRCON estimates the reciprocal of the condition number of a

triangular matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for

norm(inv(A)), then the reciprocal of the condition number is

computed as

RCOND = 1 / ( norm(A) * norm(inv(A)) ).

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies whether the 1-norm condition number or the

infinity-norm condition number is required:

= '1' or 'O': 1-norm;

= 'I': Infinity-norm.

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*A*

A is COMPLEX*16 array, dimension (LDA,N)

The triangular matrix A. If UPLO = 'U', the leading N-by-N

upper triangular part of the array A contains the upper

triangular matrix, and the strictly lower triangular part of

A is not referenced. If UPLO = 'L', the leading N-by-N lower

triangular part of the array A contains the lower triangular

matrix, and the strictly upper triangular part of A is not

referenced. If DIAG = 'U', the diagonal elements of A are

also not referenced and are assumed to be 1.

*LDA*

LDA is INTEGER

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

*RCOND*

RCOND is DOUBLE PRECISION

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(norm(A) * norm(inv(A))).

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTREVC**

**Purpose:**

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 j-th 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 N-by-N 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 N-by-N 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The algorithm used in this program is basically backward (forward)

substitution, with scaling to make the the code robust against

possible overflow.

Each eigenvector is normalized so that the element of largest

magnitude has magnitude 1; here the magnitude of a complex number

(x,y) is taken to be |x| + |y|.

## subroutine 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)¶

**ZTREVC3**

**Purpose:**

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 j-th 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 N-by-N 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 N-by-N 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The algorithm used in this program is basically backward (forward)

substitution, with scaling to make the the code robust against

possible overflow.

Each eigenvector is normalized so that the element of largest

magnitude has magnitude 1; here the magnitude of a complex number

(x,y) is taken to be |x| + |y|.

## subroutine 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)¶

**ZTREXC**

**Purpose:**

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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTRRFS**

**Purpose:**

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 non-unit 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 N-by-N

upper triangular part of the array A contains the upper

triangular matrix, and the strictly lower triangular part of

A is not referenced. If UPLO = 'L', the leading N-by-N lower

triangular part of the array A contains the lower triangular

matrix, and the strictly upper triangular part of A is not

referenced. If DIAG = 'U', the diagonal elements of A are

also not referenced and are assumed to be 1.

*LDA*

LDA is INTEGER

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

*B*

B is 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 j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is DOUBLE PRECISION array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTRSEN**

**Purpose:**

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 j-th 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*(N-M));

if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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 F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote

the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.

S cannot underestimate 1 / 2-norm(P) by more than a factor of

sqrt(N).

An approximate error bound for the computed average of the

eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace

spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.

SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the

n1*n2-by-n1*n2 matrix

C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker

product. We estimate sigma-min(C) by the reciprocal of an estimate of

the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)

cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in

the invariant subspace. An approximate bound on the maximum angular

error in the computed right invariant subspace is

EPS * norm(T) / SEP

## subroutine 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)¶

**ZTRSNA**

**Purpose:**

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 j-th 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 j-th columns of VL and VR

all correspond to the same eigenpair (but not in general the

j-th eigenpair, unless all eigenpairs are selected).

If JOB = 'V', S is not referenced.

*SEP*

SEP is DOUBLE PRECISION array, dimension (MM)

If JOB = 'V' or 'B', the estimated reciprocal condition

numbers of the selected eigenvectors, stored in consecutive

elements of the array.

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 i-th argument had an illegal value

**Author**

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 ) = sigma-min( T22 - lambda*I )

where sigma-min denotes the smallest singular value. We approximate

the smallest singular value by the reciprocal of an estimate of the

one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is

defined to be abs(T(1,1)).

An approximate error bound for a computed right eigenvector VR(i)

is given by

EPS * norm(T) / SEP(i)

## subroutine 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': Non-unit 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 k-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ztrtri (character UPLO, character DIAG, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)¶

**ZTRTRI**

**Purpose:**

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 non-unit 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 N-by-N upper triangular part of the array A contains

the upper triangular matrix, and the strictly lower

triangular part of A is not referenced. If UPLO = 'L', the

leading N-by-N lower triangular part of the array A contains

the lower triangular matrix, and the strictly upper

triangular part of A is not referenced. If DIAG = 'U', the

diagonal elements of A are also not referenced and are

assumed to be 1.

On exit, the (triangular) inverse of the original matrix, in

the same storage format.

*LDA*

LDA is INTEGER

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

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, A(i,i) is exactly zero. The triangular

matrix is singular and its inverse can not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTRTRS**

**Purpose:**

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 N-by-NRHS

matrix. A check is made to verify that A is nonsingular.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': A is upper triangular;

= 'L': A is lower triangular.

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate transpose)

*DIAG*

DIAG is CHARACTER*1

= 'N': A is non-unit triangular;

= 'U': A is unit triangular.

*N*

N is INTEGER

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

*NRHS*

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 N-by-N

upper triangular part of the array A contains the upper

triangular matrix, and the strictly lower triangular part of

A is not referenced. If UPLO = 'L', the leading N-by-N lower

triangular part of the array A contains the lower triangular

matrix, and the strictly upper triangular part of A is not

referenced. If DIAG = 'U', the diagonal elements of A are

also not referenced and are assumed to be 1.

*LDA*

LDA is INTEGER

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

*B*

B is 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 i-th argument had an illegal value

> 0: if INFO = i, the i-th diagonal element of A is zero,

indicating that the matrix is singular and the solutions

X have not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ztrttf (character TRANSR, character UPLO, integer N, complex*16, dimension( 0: lda-1, 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 N-by-N upper triangular part of the array A contains

the upper triangular matrix, and the strictly lower

triangular part of A is not referenced. If UPLO = 'L', the

leading N-by-N lower triangular part of the array A contains

the lower triangular matrix, and the strictly upper

triangular part of A is not referenced.

*LDA*

LDA is INTEGER

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

*ARF*

ARF is 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 i-th argument had an illegal value

**Author**

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

conjugate-transpose of the first three columns of AP upper.

For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:2,0:2) consists of

conjugate-transpose 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

conjugate-transpose of the first two columns of AP upper.

For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first

three columns of AP lower. The upper triangle A(0:1,1:2) consists of

conjugate-transpose 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

## 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

N-by-N upper triangular part of A contains the upper

triangular part of the matrix A, and the strictly lower

triangular part of A is not referenced. If UPLO = 'L', the

leading N-by-N lower triangular part of A contains the lower

triangular part of the matrix A, and the strictly upper

triangular part of A is not referenced.

*LDA*

LDA is INTEGER

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

*AP*

AP is COMPLEX*16 array, dimension ( N*(N+1)/2 ),

On exit, the upper or lower triangular matrix A, packed

columnwise in a linear array. The j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZTZRZF**

**Purpose:**

ZTZRZF reduces the M-by-N ( 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 N-by-N unitary matrix and R is an M-by-M upper

triangular matrix.

**Parameters**

*M*

M is INTEGER

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

*N*

N is INTEGER

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

*A*

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the leading M-by-N upper trapezoidal part of the

array A must contain the matrix to be factorized.

On exit, the leading M-by-M upper triangular part of A

contains the upper triangular matrix R, and elements M+1 to

N of the first M rows of A, with the array TAU, represent the

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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

The N-by-N matrix Z can be computed by

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

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**H

with v(k) is the kth row vector of the M-by-N matrix

V = ( I A(:,M+1:N) )

I is the M-by-M identity matrix, A(:,M+1:N)

is the output stored in A on exit from ZTZRZF,

and tau(k) is the kth element of the array TAU.

## 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)¶

**ZUNBDB**

**Purpose:**

ZUNBDB simultaneously bidiagonalizes the blocks of an M-by-M

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 P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is

not the case, then X must be transposed and/or permuted. This can be

done in constant time using the TRANS and SIGNS options. See ZUNCSD

for details.)

The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-

(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are

represented implicitly by Householder vectors.

B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented

implicitly by angles THETA, PHI.

**Parameters**

*TRANS*

TRANS is CHARACTER

= 'T': X, U1, U2, V1T, and V2T are stored in row-major

order;

otherwise: X, U1, U2, V1T, and V2T are stored in column-

major order.

*SIGNS*

SIGNS is CHARACTER

= 'O': The lower-left block is made nonpositive (the

"other" convention);

otherwise: The upper-right block is made nonpositive (the

"default" convention).

*M*

M is INTEGER

The number of rows and columns in X.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

MIN(P,M-P,M-Q).

*X11*

X11 is COMPLEX*16 array, dimension (LDX11,Q)

On entry, the top-left 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,M-Q)

On entry, the top-right 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 >= M-Q.

*X21*

X21 is COMPLEX*16 array, dimension (LDX21,Q)

On entry, the bottom-left 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 >=

M-P; else LDX21 >= Q.

*X22*

X22 is COMPLEX*16 array, dimension (LDX22,M-Q)

On entry, the bottom-right 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:M-P,P+1:M-Q)) specify the last

M-P-Q reflectors for Q2,

else TRANS = 'T', and

the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last

M-P-Q reflectors for P2.

*LDX22*

LDX22 is INTEGER

The leading dimension of X22. If TRANS = 'N', then LDX22 >=

M-P; else LDX22 >= M-Q.

*THETA*

THETA is DOUBLE PRECISION array, dimension (Q)

The entries of the bidiagonal blocks B11, B12, B21, B22 can

be computed from the angles THETA and PHI. See Further

Details.

*PHI*

PHI is DOUBLE PRECISION array, dimension (Q-1)

The entries of the bidiagonal blocks B11, B12, B21, B22 can

be computed from the angles THETA and PHI. See Further

Details.

*TAUP1*

TAUP1 is COMPLEX*16 array, dimension (P)

The scalar factors of the elementary reflectors that define

P1.

*TAUP2*

TAUP2 is COMPLEX*16 array, dimension (M-P)

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 (M-Q)

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 >= M-Q.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The bidiagonal blocks B11, B12, B21, and B22 are represented

implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,

PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are

lower bidiagonal. Every entry in each bidiagonal band is a product

of a sine or cosine of a THETA with a sine or cosine of a PHI. See

[1] or 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:**

## 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)¶

**ZUNBDB1**

**Purpose:**

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 P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,

M-P, or M-Q. Routines ZUNBDB2, ZUNBDB3, and ZUNBDB4 handle cases in

which Q is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),

and (M-Q)-by-(M-Q), respectively. They are represented implicitly by

Householder vectors.

B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by

angles THETA, PHI.

**Parameters**

*M*

M is INTEGER

The number of rows X11 plus the number of rows in X21.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

MIN(P,M-P,M-Q).

*X11*

X11 is 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 >= M-P.

*THETA*

THETA is DOUBLE PRECISION array, dimension (Q)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*PHI*

PHI is DOUBLE PRECISION array, dimension (Q-1)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*TAUP1*

TAUP1 is COMPLEX*16 array, dimension (P)

The scalar factors of the elementary reflectors that define

P1.

*TAUP2*

TAUP2 is COMPLEX*16 array, dimension (M-P)

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 >= M-Q.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The upper-bidiagonal blocks B11, B21 are represented implicitly by

angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry

in each bidiagonal band is a product of a sine or cosine of a THETA

with a sine or cosine of a PHI. See [1] or 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:**

## 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)¶

**ZUNBDB2**

**Purpose:**

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 P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,

Q, or M-Q. Routines ZUNBDB1, ZUNBDB3, and ZUNBDB4 handle cases in

which P is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),

and (M-Q)-by-(M-Q), respectively. They are represented implicitly by

Householder vectors.

B11 and B12 are P-by-P bidiagonal matrices represented implicitly by

angles THETA, PHI.

**Parameters**

*M*

M is INTEGER

The number of rows X11 plus the number of rows in X21.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

*X11*

X11 is 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 >= M-P.

*THETA*

THETA is DOUBLE PRECISION array, dimension (Q)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*PHI*

PHI is DOUBLE PRECISION array, dimension (Q-1)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*TAUP1*

TAUP1 is COMPLEX*16 array, dimension (P)

The scalar factors of the elementary reflectors that define

P1.

*TAUP2*

TAUP2 is COMPLEX*16 array, dimension (M-P)

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 >= M-Q.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The upper-bidiagonal blocks B11, B21 are represented implicitly by

angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry

in each bidiagonal band is a product of a sine or cosine of a THETA

with a sine or cosine of a PHI. See [1] or 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:**

## 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)¶

**ZUNBDB3**

**Purpose:**

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 P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,

Q, or M-Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB4 handle cases in

which M-P is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),

and (M-Q)-by-(M-Q), respectively. They are represented implicitly by

Householder vectors.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented

implicitly by angles THETA, PHI.

**Parameters**

*M*

M is INTEGER

The number of rows X11 plus the number of rows in X21.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

*X11*

X11 is 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 >= M-P.

*THETA*

THETA is DOUBLE PRECISION array, dimension (Q)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*PHI*

PHI is DOUBLE PRECISION array, dimension (Q-1)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*TAUP1*

TAUP1 is COMPLEX*16 array, dimension (P)

The scalar factors of the elementary reflectors that define

P1.

*TAUP2*

TAUP2 is COMPLEX*16 array, dimension (M-P)

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 >= M-Q.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The upper-bidiagonal blocks B11, B21 are represented implicitly by

angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry

in each bidiagonal band is a product of a sine or cosine of a THETA

with a sine or cosine of a PHI. See [1] or 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:**

## 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)¶

**ZUNBDB4**

**Purpose:**

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 P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,

M-P, or Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB3 handle cases in

which M-Q is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),

and (M-Q)-by-(M-Q), respectively. They are represented implicitly by

Householder vectors.

B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented

implicitly by angles THETA, PHI.

**Parameters**

*M*

M is INTEGER

The number of rows X11 plus the number of rows in X21.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

M-Q <= min(P,M-P,Q).

*X11*

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 >= M-P.

*THETA*

THETA is DOUBLE PRECISION array, dimension (Q)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*PHI*

PHI is DOUBLE PRECISION array, dimension (Q-1)

The entries of the bidiagonal blocks B11, B21 are defined by

THETA and PHI. See Further Details.

*TAUP1*

TAUP1 is COMPLEX*16 array, dimension (P)

The scalar factors of the elementary reflectors that define

P1.

*TAUP2*

TAUP2 is COMPLEX*16 array, dimension (M-P)

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 M-by-1 column vector Y that is

orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and

PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and

Y(P+1:M), respectively.

*WORK*

WORK is COMPLEX*16 array, dimension (LWORK)

*LWORK*

LWORK is INTEGER

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

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The upper-bidiagonal blocks B11, B21 are represented implicitly by

angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry

in each bidiagonal band is a product of a sine or cosine of a THETA

with a sine or cosine of a PHI. See [1] or 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:**

## 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)¶

**ZUNBDB5**

**Purpose:**

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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNBDB6**

**Purpose:**

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 i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNCSD**

**Purpose:**

ZUNCSD computes the CS decomposition of an M-by-M 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 P-by-Q. The unitary matrices U1, U2, V1, and V2 are P-by-P,

(M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are

R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in

which R = MIN(P,M-P,Q,M-Q).

**Parameters**

*JOBU1*

JOBU1 is CHARACTER

= 'Y': U1 is computed;

otherwise: U1 is not computed.

*JOBU2*

JOBU2 is CHARACTER

= 'Y': U2 is computed;

otherwise: U2 is not computed.

*JOBV1T*

JOBV1T is CHARACTER

= 'Y': V1T is computed;

otherwise: V1T is not computed.

*JOBV2T*

JOBV2T is CHARACTER

= 'Y': V2T is computed;

otherwise: V2T is not computed.

*TRANS*

TRANS is CHARACTER

= 'T': X, U1, U2, V1T, and V2T are stored in row-major

order;

otherwise: X, U1, U2, V1T, and V2T are stored in column-

major order.

*SIGNS*

SIGNS is CHARACTER

= 'O': The lower-left block is made nonpositive (the

"other" convention);

otherwise: The upper-right block is made nonpositive (the

"default" convention).

*M*

M is INTEGER

The number of rows and columns in X.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

*X11*

X11 is 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,M-Q)

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,M-P).

*X22*

X22 is COMPLEX*16 array, dimension (LDX22,M-Q)

On entry, part of the unitary matrix whose CSD is desired.

*LDX22*

LDX22 is INTEGER

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

*THETA*

THETA is DOUBLE PRECISION array, dimension (R), in which R =

MIN(P,M-P,Q,M-Q).

C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and

S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

*U1*

U1 is COMPLEX*16 array, dimension (LDU1,P)

If JOBU1 = 'Y', U1 contains the P-by-P 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,M-P)

If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary

matrix U2.

*LDU2*

LDU2 is INTEGER

The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=

MAX(1,M-P).

*V1T*

V1T is COMPLEX*16 array, dimension (LDV1T,Q)

If JOBV1T = 'Y', V1T contains the Q-by-Q 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,M-Q)

If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary

matrix V2**H.

*LDV2T*

LDV2T is INTEGER

The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=

MAX(1,M-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 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(R-1) that, together with THETA(1), ..., THETA(R),

define the matrix in intermediate bidiagonal-block form

remaining after nonconvergence. INFO specifies the number

of nonzero PHI's.

*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 (M-MIN(P,M-P,Q,M-Q))

*INFO*

INFO is INTEGER

= 0: successful exit.

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

> 0: ZBBCSD did not converge. See the description of RWORK

above for details.

**References:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNCSD2BY1**

**Purpose:**

ZUNCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with

orthonormal columns that has been partitioned into a 2-by-1 block

structure:

[ I1 0 0 ]

[ 0 C 0 ]

[ X11 ] [ U1 | ] [ 0 0 0 ]

X = [-----] = [---------] [----------] V1**T .

[ X21 ] [ | U2 ] [ 0 0 0 ]

[ 0 S 0 ]

[ 0 0 I2]

X11 is P-by-Q. The unitary matrices U1, U2, and V1 are P-by-P,

(M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R

nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which

R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a

K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

**Parameters**

*JOBU1*

JOBU1 is CHARACTER

= 'Y': U1 is computed;

otherwise: U1 is not computed.

*JOBU2*

JOBU2 is CHARACTER

= 'Y': U2 is computed;

otherwise: U2 is not computed.

*JOBV1T*

JOBV1T is CHARACTER

= 'Y': V1T is computed;

otherwise: V1T is not computed.

*M*

M is INTEGER

The number of rows in X.

*P*

P is INTEGER

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

*Q*

Q is INTEGER

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

*X11*

X11 is 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,M-P).

*THETA*

THETA is DOUBLE PRECISION array, dimension (R), in which R =

MIN(P,M-P,Q,M-Q).

C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and

S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

*U1*

U1 is COMPLEX*16 array, dimension (P)

If JOBU1 = 'Y', U1 contains the P-by-P 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 (M-P)

If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary

matrix U2.

*LDU2*

LDU2 is INTEGER

The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=

MAX(1,M-P).

*V1T*

V1T is COMPLEX*16 array, dimension (Q)

If JOBV1T = 'Y', V1T contains the Q-by-Q 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(R-1) that, together with THETA(1), ..., THETA(R),

define the matrix in intermediate bidiagonal-block form

remaining after nonconvergence. INFO specifies the number

of nonzero PHI's.

*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 (M-MIN(P,M-P,Q,M-Q))

*INFO*

INFO is INTEGER

= 0: successful exit.

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

> 0: ZBBCSD did not converge. See the description of WORK

above for details.

**References:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 (n-k+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 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 ZGEQLF.

*WORK*

WORK is COMPLEX*16 array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNG2R**

**Purpose:**

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 i-th 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 i-th argument has an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGHR**

**Purpose:**

ZUNGHR generates a complex unitary matrix Q which is defined as the

product of IHI-ILO elementary reflectors of order N, as returned by

ZGEHRD:

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

**Parameters**

*N*

N is INTEGER

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

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER

ILO and IHI must have the same values as in the previous call

of 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 N-by-N 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 (N-1)

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 >= IHI-ILO.

For optimum performance LWORK >= (IHI-ILO)*NB, where NB is

the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 m-by-n 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 i-th 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 i-th argument has an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGLQ**

**Purpose:**

ZUNGLQ generates an M-by-N 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 i-th 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 (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 i-th argument has an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGQL**

**Purpose:**

ZUNGQL 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 (n-k+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 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 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 i-th argument has an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGQR**

**Purpose:**

ZUNGQR 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 i-th 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 (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 i-th argument has an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 (m-k+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 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 ZGERQF.

*WORK*

WORK is COMPLEX*16 array, dimension (M)

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGRQ**

**Purpose:**

ZUNGRQ 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 (m-k+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 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 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 i-th argument has an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGTR**

**Purpose:**

ZUNGTR generates a complex unitary matrix Q which is defined as the

product of n-1 elementary reflectors of order N, as returned by

ZHETRD:

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

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

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A contains elementary reflectors

from 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 N-by-N unitary matrix Q.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= N.

*TAU*

TAU is COMPLEX*16 array, dimension (N-1)

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 >= N-1.

For optimum performance LWORK >= (N-1)*NB, where NB is

the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNGTSQR**

**Purpose:**

ZUNGTSQR generates an M-by-N 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

lower-trapezoidal 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 M-by-N orthonormal matrix Q_out,

i.e the columns of A are orthogonal unit vectors.

*LDA*

LDA is INTEGER

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

*T*

T is COMPLEX*16 array,

dimension (LDT, N * NIRB)

where NIRB = Number_of_input_row_blocks

= MAX( 1, CEIL((M-N)/(MB-N)) )

Let NICB = Number_of_input_col_blocks

= CEIL(N/NB)

The upper-triangular block reflectors used to define the

input matrices Q_in(k), k=(1:NIRB*NICB). The block

reflectors are stored in compact form in NIRB block

reflector sequences. Each of NIRB block reflector sequences

is stored in a larger NB-by-N column block of T and consists

of NICB smaller NB-by-NB upper-triangular column blocks.

(same format as the output T in 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2019, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

## 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_ROW**

**Purpose:**

ZUNGTSQR_ROW generates an M-by-N 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 NB-sized column blocks from the right to left

starting in the bottom row block and continues to the top row block

(hence _ROW in the routine name). This sweep is in reverse order of

the order in which 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

lower-trapezoidal 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 M-by-N orthonormal matrix Q_out,

i.e the columns of A are orthogonal unit vectors.

*LDA*

LDA is INTEGER

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

*T*

T is COMPLEX*16 array,

dimension (LDT, N * NIRB)

where NIRB = Number_of_input_row_blocks

= MAX( 1, CEIL((M-N)/(MB-N)) )

Let NICB = Number_of_input_col_blocks

= CEIL(N/NB)

The upper-triangular block reflectors used to define the

input matrices Q_in(k), k=(1:NIRB*NICB). The block

reflectors are stored in compact form in NIRB block

reflector sequences. Each of the NIRB block reflector

sequences is stored in a larger NB-by-N column block of T

and consists of NICB smaller NB-by-NB upper-triangular

column blocks. See 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,(N-NBLOCAL)),

where NBLOCAL=MIN(NB,N).

If LWORK = -1, then a workspace query is assumed.

The routine only calculates the optimal size of the WORK

array, returns this value as the first entry of the WORK

array, and no error message related to LWORK is issued

by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2020, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

## 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_COL**

**Purpose:**

ZUNHR_COL takes an M-by-N 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 M-by-N matrix Q_out, with the property that Q_in = Q_out*S,

where S is an N-by-N diagonal matrix with diagonal entries

equal to +1 or -1. The Householder vectors (columns V(i) of V) are

stored in A on output, and the diagonal entries of S are stored in D.

Block reflectors are also returned in T

(same output format as 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 M-by-N orthonormal matrix Q_in,

i.e the columns of A are orthogonal unit vectors.

On exit:

The elements below the diagonal of A represent the unit

lower-trapezoidal matrix V of Householder column vectors

V(i). The unit diagonal entries of V are not stored

(same format as the output below the diagonal in A from

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" LU-decomposition:

Q_in - ( S ) = V * U

( 0 )

where 0 is a (M-N)-by-(M-N) zero matrix.

*LDA*

LDA is INTEGER

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

*T*

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 upper-triangular

block reflectors used to define Q_out stored in compact

form as a sequence of upper-triangular NB-by-NB column

blocks (same format as the output T in ZGEQRT).

The matrix T and the matrix V stored on output in A

implicitly define Q_out. NOTE: The lower triangles

below the upper-triangular blocks will be filled with

zeros. See Further Details.

*LDT*

LDT is INTEGER

The leading dimension of the array T.

LDT >= max(1,min(NB,N)).

*D*

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

i-1 steps of “modified” Gaussian elimination.

See Further Details.

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Further Details:**

The computed M-by-M 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 WY-representation format in the corresponding blocks of

matrices V (stored in A) and T.

The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N

matrix A contains the column vectors V(i) in NB-size column

blocks VB(j). For example, VB(1) contains the columns

V(1), V(2), ... V(NB). NOTE: The unit entries on

the diagonal of Y are not stored in A.

The number of column blocks is

NOCB = Number_of_output_col_blocks = CEIL(N/NB)

where each block is of order NB except for the last block, which

is of order LAST_NB = N - (NOCB-1)*NB.

For example, if M=6, N=5 and NB=2, the matrix V is

V = ( VB(1), VB(2), VB(3) ) =

= ( 1 )

( v21 1 )

( v31 v32 1 )

( v41 v42 v43 1 )

( v51 v52 v53 v54 1 )

( v61 v62 v63 v54 v65 )

For each of the column blocks VB(i), an upper-triangular block

reflector TB(i) is computed. These blocks are stored as

a sequence of upper-triangular column blocks in the NB-by-N

matrix T. The size of each TB(i) block is NB-by-NB, except

for the last block, whose size is LAST_NB-by-LAST_NB.

For example, if M=6, N=5 and NB=2, the matrix T is

T = ( TB(1), TB(2), TB(3) ) =

= ( t11 t12 t13 t14 t15 )

( t22 t24 )

The M-by-M factor Q_out is given as a product of NOCB

unitary M-by-M matrices Q_out(i).

Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),

where each matrix Q_out(i) is given by the WY-representation

using corresponding blocks from the matrices V and T:

Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,

where I is the identity matrix. Here is the formula with matrix

dimensions:

Q(i){M-by-M} = I{M-by-M} -

VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},

where INB = NB, except for the last block NOCB

for which INB=LAST_NB.

=====

NOTE:

=====

If Q_in is the result of doing a QR factorization

B = Q_in * R_in, then:

B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.

So if one wants to interpret Q_out as the result

of the QR factorization of B, then the corresponding R_out

should be equal to R_out = S * R_in, i.e. some rows of R_in

should be multiplied by -1.

For the details of the algorithm, see [1].

[1] "Reconstructing Householder vectors from tall-skinny QR",

G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,

E. Solomonik, J. Parallel Distrib. Comput.,

vol. 85, pp. 3-31, 2015.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2019, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

## 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 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(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 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.

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 m-by-n matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L',

(M) if SIDE = 'R'

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 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 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 i-th 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 m-by-n matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L',

(M) if SIDE = 'R'

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNMBR**

**Purpose:**

If VECT = 'Q', ZUNMBR overwrites the general complex M-by-N 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 M-by-N 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 NQ-by-K matrix:

if nq >= k, Q = H(1) H(2) . . . H(k);

if nq < k, Q = H(1) H(2) . . . H(nq-1).

If VECT = 'P', A is assumed to have been a K-by-NQ matrix:

if k < nq, P = G(1) G(2) . . . G(k);

if k >= nq, P = G(1) G(2) . . . G(nq-1).

**Parameters**

*VECT*

VECT is CHARACTER*1

= 'Q': apply Q or Q**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 M-by-N 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNMHR**

**Purpose:**

ZUNMHR overwrites the general complex M-by-N 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

IHI-ILO elementary reflectors, as returned by ZGEHRD:

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

**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

(M-1) if SIDE = 'L'

(N-1) if SIDE = 'R'

TAU(i) must contain the scalar factor of the elementary

reflector H(i), as returned by ZGEHRD.

*C*

C is COMPLEX*16 array, dimension (LDC,N)

On entry, the M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 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(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 i-th 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 m-by-n matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L',

(M) if SIDE = 'R'

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNMLQ**

**Purpose:**

ZUNMLQ overwrites the general complex M-by-N 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 i-th 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 M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNMQL**

**Purpose:**

ZUNMQL overwrites the general complex M-by-N 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 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.

*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 M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNMQR**

**Purpose:**

ZUNMQR overwrites the general complex M-by-N 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 i-th 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 M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 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 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 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.

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 m-by-n matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L',

(M) if SIDE = 'R'

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 i-th 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 m-by-n matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L',

(M) if SIDE = 'R'

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

## 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)¶

**ZUNMRQ**

**Purpose:**

ZUNMRQ overwrites the general complex M-by-N 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 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.

*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 M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUNMRZ**

**Purpose:**

ZUNMRZ overwrites the general complex M-by-N 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 i-th 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 M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

## 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)¶

**ZUNMTR**

**Purpose:**

ZUNMTR overwrites the general complex M-by-N 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

nq-1 elementary reflectors, as returned by ZHETRD:

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

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

**Parameters**

*SIDE*

SIDE is CHARACTER*1

= 'L': apply Q or Q**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

(M-1) if SIDE = 'L'

(N-1) if SIDE = 'R'

TAU(i) must contain the scalar factor of the elementary

reflector H(i), as returned by ZHETRD.

*C*

C is COMPLEX*16 array, dimension (LDC,N)

On entry, the M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is 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 i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUPGTR**

**Purpose:**

ZUPGTR generates a complex unitary matrix Q which is defined as the

product of n-1 elementary reflectors H(i) of order n, as returned by

ZHPTRD using packed storage:

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

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

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangular packed storage used in previous

call to 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 (N-1)

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 N-by-N 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 (N-1)

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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)¶

**ZUPMTR**

**Purpose:**

ZUPMTR overwrites the general complex M-by-N 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

nq-1 elementary reflectors, as returned by ZHPTRD using packed

storage:

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

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

**Parameters**

*SIDE*

SIDE is CHARACTER*1

= 'L': apply Q or Q**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 (M-1) if SIDE = 'L'

or (N-1) if SIDE = 'R'

TAU(i) must contain the scalar factor of the elementary

reflector H(i), as returned by ZHPTRD.

*C*

C is COMPLEX*16 array, dimension (LDC,N)

On entry, the M-by-N matrix C.

On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

*LDC*

LDC is INTEGER

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

*WORK*

WORK is COMPLEX*16 array, dimension

(N) if SIDE = 'L'

(M) if SIDE = 'R'

*INFO*

INFO is INTEGER

= 0: successful exit

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

# Author¶

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