Provided by: liblapack-doc_3.11.0-2build1_all 
NAME
realOTHERcomputational - real
SYNOPSIS
Functions
subroutine sbbcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2,
LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, WORK,
LWORK, INFO)
SBBCSD
subroutine sgghd3 (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK,
INFO)
SGGHD3
subroutine sgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
SGGHRD
subroutine sggqrf (N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
SGGQRF
subroutine sggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
SGGRQF
subroutine sggsvp3 (JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU,
V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
SGGSVP3
subroutine sgsvj0 (JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP, WORK,
LWORK, INFO)
SGSVJ0 pre-processor for the routine sgesvj.
subroutine sgsvj1 (JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV, EPS, SFMIN, TOL, NSWEEP,
WORK, LWORK, INFO)
SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only
particular pivots.
subroutine shsein (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL, LDVL, VR, LDVR, MM,
M, WORK, IFAILL, IFAILR, INFO)
SHSEIN
subroutine shseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)
SHSEQR
subroutine sla_lin_berr (N, NZ, NRHS, RES, AYB, BERR)
SLA_LIN_BERR computes a component-wise relative backward error.
subroutine sla_wwaddw (N, X, Y, W)
SLA_WWADDW adds a vector into a doubled-single vector.
subroutine slals0 (ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, PERM, GIVPTR, GIVCOL,
LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO)
SLALS0 applies back multiplying factors in solving the least squares problem using
divide and conquer SVD approach. Used by sgelsd.
subroutine slalsa (ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, LDU, VT, K, DIFL, DIFR,
Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine slalsd (UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, IWORK, INFO)
SLALSD uses the singular value decomposition of A to solve the least squares problem.
real function slansf (NORM, TRANSR, UPLO, N, A, WORK)
SLANSF
subroutine slarscl2 (M, N, D, X, LDX)
SLARSCL2 performs reciprocal diagonal scaling on a matrix.
subroutine slarz (SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK)
SLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
subroutine slarzb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, C, LDC, WORK,
LDWORK)
SLARZB applies a block reflector or its transpose to a general matrix.
subroutine slarzt (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine slascl2 (M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a matrix.
subroutine slatrz (M, N, L, A, LDA, TAU, WORK)
SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine sopgtr (UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
SOPGTR
subroutine sopmtr (SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
SOPMTR
subroutine sorbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22,
THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
SORBDB
subroutine sorbdb1 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1,
WORK, LWORK, INFO)
SORBDB1
subroutine sorbdb2 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1,
WORK, LWORK, INFO)
SORBDB2
subroutine sorbdb3 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1,
WORK, LWORK, INFO)
SORBDB3
subroutine sorbdb4 (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1,
PHANTOM, WORK, LWORK, INFO)
SORBDB4
subroutine sorbdb5 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK,
INFO)
SORBDB5
subroutine sorbdb6 (M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK,
INFO)
SORBDB6
recursive subroutine sorcsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11,
LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T,
LDV2T, WORK, LWORK, IWORK, INFO)
SORCSD
subroutine sorcsd2by1 (JOBU1, JOBU2, JOBV1T, M, P, Q, X11, LDX11, X21, LDX21, THETA, U1,
LDU1, U2, LDU2, V1T, LDV1T, WORK, LWORK, IWORK, INFO)
SORCSD2BY1
subroutine sorg2l (M, N, K, A, LDA, TAU, WORK, INFO)
SORG2L generates all or part of the orthogonal matrix Q from a QL factorization
determined by sgeqlf (unblocked algorithm).
subroutine sorg2r (M, N, K, A, LDA, TAU, WORK, INFO)
SORG2R generates all or part of the orthogonal matrix Q from a QR factorization
determined by sgeqrf (unblocked algorithm).
subroutine sorghr (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
SORGHR
subroutine sorgl2 (M, N, K, A, LDA, TAU, WORK, INFO)
SORGL2
subroutine sorglq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGLQ
subroutine sorgql (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQL
subroutine sorgqr (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
subroutine sorgr2 (M, N, K, A, LDA, TAU, WORK, INFO)
SORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization
determined by sgerqf (unblocked algorithm).
subroutine sorgrq (M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGRQ
subroutine sorgtr (UPLO, N, A, LDA, TAU, WORK, LWORK, INFO)
SORGTR
subroutine sorm2l (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization
determined by sgeqlf (unblocked algorithm).
subroutine sorm2r (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization
determined by sgeqrf (unblocked algorithm).
subroutine sormbr (VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMBR
subroutine sormhr (SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMHR
subroutine sorml2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization
determined by sgelqf (unblocked algorithm).
subroutine sormlq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMLQ
subroutine sormql (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQL
subroutine sormqr (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
subroutine sormr2 (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization
determined by sgerqf (unblocked algorithm).
subroutine sormr3 (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, INFO)
SORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization
determined by stzrzf (unblocked algorithm).
subroutine sormrq (SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMRQ
subroutine sormrz (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMRZ
subroutine sormtr (SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
subroutine spbcon (UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, IWORK, INFO)
SPBCON
subroutine spbequ (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
SPBEQU
subroutine spbrfs (UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, LDX, FERR, BERR,
WORK, IWORK, INFO)
SPBRFS
subroutine spbstf (UPLO, N, KD, AB, LDAB, INFO)
SPBSTF
subroutine spbtf2 (UPLO, N, KD, AB, LDAB, INFO)
SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite
band matrix (unblocked algorithm).
subroutine spbtrf (UPLO, N, KD, AB, LDAB, INFO)
SPBTRF
subroutine spbtrs (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
SPBTRS
subroutine spftrf (TRANSR, UPLO, N, A, INFO)
SPFTRF
subroutine spftri (TRANSR, UPLO, N, A, INFO)
SPFTRI
subroutine spftrs (TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
SPFTRS
subroutine sppcon (UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO)
SPPCON
subroutine sppequ (UPLO, N, AP, S, SCOND, AMAX, INFO)
SPPEQU
subroutine spprfs (UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SPPRFS
subroutine spptrf (UPLO, N, AP, INFO)
SPPTRF
subroutine spptri (UPLO, N, AP, INFO)
SPPTRI
subroutine spptrs (UPLO, N, NRHS, AP, B, LDB, INFO)
SPPTRS
subroutine spstf2 (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric
positive semidefinite matrix.
subroutine spstrf (UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric
positive semidefinite matrix.
subroutine ssbgst (VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO)
SSBGST
subroutine ssbtrd (VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO)
SSBTRD
subroutine ssfrk (TRANSR, UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C)
SSFRK performs a symmetric rank-k operation for matrix in RFP format.
subroutine sspcon (UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SSPCON
subroutine sspgst (ITYPE, UPLO, N, AP, BP, INFO)
SSPGST
subroutine ssprfs (UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
INFO)
SSPRFS
subroutine ssptrd (UPLO, N, AP, D, E, TAU, INFO)
SSPTRD
subroutine ssptrf (UPLO, N, AP, IPIV, INFO)
SSPTRF
subroutine ssptri (UPLO, N, AP, IPIV, WORK, INFO)
SSPTRI
subroutine ssptrs (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
SSPTRS
subroutine sstegr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
WORK, LWORK, IWORK, LIWORK, INFO)
SSTEGR
subroutine sstein (N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
subroutine sstemr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ,
TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
SSTEMR
subroutine stbcon (NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, IWORK, INFO)
STBCON
subroutine stbrfs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, FERR, BERR,
WORK, IWORK, INFO)
STBRFS
subroutine stbtrs (UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
STBTRS
subroutine stfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine stftri (TRANSR, UPLO, DIAG, N, A, INFO)
STFTRI
subroutine stfttp (TRANSR, UPLO, N, ARF, AP, INFO)
STFTTP copies a triangular matrix from the rectangular full packed format (TF) to the
standard packed format (TP).
subroutine stfttr (TRANSR, UPLO, N, ARF, A, LDA, INFO)
STFTTR copies a triangular matrix from the rectangular full packed format (TF) to the
standard full format (TR).
subroutine stgsen (IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q,
LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
STGSEN
subroutine stgsja (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)
STGSJA
subroutine stgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM,
M, WORK, LWORK, IWORK, INFO)
STGSNA
subroutine stpcon (NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK, INFO)
STPCON
subroutine stpmqrt (SIDE, TRANS, M, N, K, L, NB, V, LDV, T, LDT, A, LDA, B, LDB, WORK,
INFO)
STPMQRT
subroutine stpqrt (M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
STPQRT
subroutine stpqrt2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)
STPQRT2 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 stprfs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, FERR, BERR, WORK,
IWORK, INFO)
STPRFS
subroutine stptri (UPLO, DIAG, N, AP, INFO)
STPTRI
subroutine stptrs (UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO)
STPTRS
subroutine stpttf (TRANSR, UPLO, N, AP, ARF, INFO)
STPTTF copies a triangular matrix from the standard packed format (TP) to the
rectangular full packed format (TF).
subroutine stpttr (UPLO, N, AP, A, LDA, INFO)
STPTTR copies a triangular matrix from the standard packed format (TP) to the standard
full format (TR).
subroutine strcon (NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, IWORK, INFO)
STRCON
subroutine strevc (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
STREVC
subroutine strevc3 (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK,
LWORK, INFO)
STREVC3
subroutine strexc (COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
STREXC
subroutine strrfs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, LDX, FERR, BERR, WORK,
IWORK, INFO)
STRRFS
subroutine strsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK,
IWORK, LIWORK, INFO)
STRSEN
subroutine strsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M,
WORK, LDWORK, IWORK, INFO)
STRSNA
subroutine strti2 (UPLO, DIAG, N, A, LDA, INFO)
STRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
subroutine strtri (UPLO, DIAG, N, A, LDA, INFO)
STRTRI
subroutine strtrs (UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, INFO)
STRTRS
subroutine strttf (TRANSR, UPLO, N, A, LDA, ARF, INFO)
STRTTF copies a triangular matrix from the standard full format (TR) to the
rectangular full packed format (TF).
subroutine strttp (UPLO, N, A, LDA, AP, INFO)
STRTTP copies a triangular matrix from the standard full format (TR) to the standard
packed format (TP).
subroutine stzrzf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
STZRZF
Detailed Description
This is the group of real other Computational routines
Function Documentation
subroutine sbbcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T,
character TRANS, integer M, integer P, integer Q, real, dimension( * ) THETA, real,
dimension( * ) PHI, real, dimension( ldu1, * ) U1, integer LDU1, real, dimension( ldu2, *
) U2, integer LDU2, real, dimension( ldv1t, * ) V1T, integer LDV1T, real, dimension(
ldv2t, * ) V2T, integer LDV2T, real, dimension( * ) B11D, real, dimension( * ) B11E, real,
dimension( * ) B12D, real, dimension( * ) B12E, real, dimension( * ) B21D, real,
dimension( * ) B21E, real, dimension( * ) B22D, real, dimension( * ) B22E, real,
dimension( * ) WORK, integer LWORK, integer INFO)
SBBCSD
Purpose:
SBBCSD computes the CS decomposition of an orthogonal matrix in
bidiagonal-block form,
[ B11 | B12 0 0 ]
[ 0 | 0 -I 0 ]
X = [----------------]
[ B21 | B22 0 0 ]
[ 0 | 0 0 I ]
[ C | -S 0 0 ]
[ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T
= [---------] [---------------] [---------] .
[ | U2 ] [ S | C 0 0 ] [ | V2 ]
[ 0 | 0 0 I ]
X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
transposed and/or permuted. This can be done in constant time using
the TRANS and SIGNS options. See SORCSD for details.)
The bidiagonal matrices B11, B12, B21, and B22 are represented
implicitly by angles THETA(1:Q) and PHI(1:Q-1).
The orthogonal matrices U1, U2, V1T, and V2T are input/output.
The input matrices are pre- or post-multiplied by the appropriate
singular vector matrices.
Parameters
JOBU1
JOBU1 is CHARACTER
= 'Y': U1 is updated;
otherwise: U1 is not updated.
JOBU2
JOBU2 is CHARACTER
= 'Y': U2 is updated;
otherwise: U2 is not updated.
JOBV1T
JOBV1T is CHARACTER
= 'Y': V1T is updated;
otherwise: V1T is not updated.
JOBV2T
JOBV2T is CHARACTER
= 'Y': V2T is updated;
otherwise: V2T is not updated.
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
M
M is INTEGER
The number of rows and columns in X, the orthogonal matrix in
bidiagonal-block form.
P
P is INTEGER
The number of rows in the top-left block of X. 0 <= P <= M.
Q
Q is INTEGER
The number of columns in the top-left block of X.
0 <= Q <= MIN(P,M-P,M-Q).
THETA
THETA is REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (LDV1T,Q)
On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
by the transpose of the right singular vector
matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
LDV1T
LDV1T is INTEGER
The leading dimension of the array V1T, LDV1T >= MAX(1,Q).
V2T
V2T is REAL array, dimension (LDV2T,M-Q)
On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
premultiplied by the transpose of the right
singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
[ B22 0 0 ; 0 0 I ].
LDV2T
LDV2T is INTEGER
The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).
B11D
B11D is REAL array, dimension (Q)
When SBBCSD converges, B11D contains the cosines of THETA(1),
..., THETA(Q). If SBBCSD fails to converge, then B11D
contains the diagonal of the partially reduced top-left
block.
B11E
B11E is REAL array, dimension (Q-1)
When SBBCSD converges, B11E contains zeros. If SBBCSD fails
to converge, then B11E contains the superdiagonal of the
partially reduced top-left block.
B12D
B12D is REAL array, dimension (Q)
When SBBCSD converges, B12D contains the negative sines of
THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
B12D contains the diagonal of the partially reduced top-right
block.
B12E
B12E is REAL array, dimension (Q-1)
When SBBCSD converges, B12E contains zeros. If SBBCSD fails
to converge, then B12E contains the subdiagonal of the
partially reduced top-right block.
B21D
B21D is REAL array, dimension (Q)
When SBBCSD converges, B21D contains the negative sines of
THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
B21D contains the diagonal of the partially reduced bottom-left
block.
B21E
B21E is REAL array, dimension (Q-1)
When SBBCSD converges, B21E contains zeros. If SBBCSD fails
to converge, then B21E contains the subdiagonal of the
partially reduced bottom-left block.
B22D
B22D is REAL array, dimension (Q)
When SBBCSD converges, B22D contains the negative sines of
THETA(1), ..., THETA(Q). If SBBCSD fails to converge, then
B22D contains the diagonal of the partially reduced bottom-right
block.
B22E
B22E is REAL array, dimension (Q-1)
When SBBCSD converges, B22E contains zeros. If SBBCSD fails
to converge, then B22E contains the subdiagonal of the
partially reduced bottom-right block.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= MAX(1,8*Q).
If LWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the WORK array,
returns this value as the first entry of the work array, and
no error message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if SBBCSD did not converge, INFO specifies the number
of nonzero entries in PHI, and B11D, B11E, etc.,
contain the partially reduced matrix.
Internal Parameters:
TOLMUL REAL, default = MAX(10,MIN(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop.
Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
are within TOLMUL*EPS of either bound.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgghd3 (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI,
real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real,
dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, real,
dimension( * ) WORK, integer LWORK, integer INFO)
SGGHD3
Purpose:
SGGHD3 reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then SGGHD3 reduces the original
problem to generalized Hessenberg form.
This is a blocked variant of SGGHRD, using matrix-matrix
multiplications for parts of the computation to enhance performance.
Parameters
COMPQ
COMPQ is CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of A which are to be
reduced. It is assumed that A is already upper triangular
in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
normally set by a previous call to SGGBAL; 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 REAL 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 REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q**T B Z. The
elements below the diagonal are set to zero.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q
Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1,
typically from the QR factorization of B.
On exit, if COMPQ='I', the orthogonal matrix Q, and if
COMPQ = 'V', the product Q1*Q.
Not referenced if COMPQ='N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
Z
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1.
On exit, if COMPZ='I', the orthogonal matrix Z, and if
COMPZ = 'V', the product Z1*Z.
Not referenced if COMPZ='N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
WORK
WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= 1.
For optimum performance LWORK >= 6*N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
This routine reduces A to Hessenberg form and maintains B in triangular form
using a blocked variant of Moler and Stewart's original algorithm,
as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
(BIT 2008).
subroutine sgghrd (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI,
real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real,
dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer
INFO)
SGGHRD
Purpose:
SGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then SGGHRD reduces the original
problem to generalized Hessenberg form.
Parameters
COMPQ
COMPQ is CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of A which are to be
reduced. It is assumed that A is already upper triangular
in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
normally set by a previous call to SGGBAL; 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 REAL 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 REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q**T B Z. The
elements below the diagonal are set to zero.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q
Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1,
typically from the QR factorization of B.
On exit, if COMPQ='I', the orthogonal matrix Q, and if
COMPQ = 'V', the product Q1*Q.
Not referenced if COMPQ='N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
Z
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1.
On exit, if COMPZ='I', the orthogonal matrix Z, and if
COMPZ = 'V', the product Z1*Z.
Not referenced if COMPZ='N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
This routine reduces A to Hessenberg and B to triangular form by
an unblocked reduction, as described in _Matrix_Computations_,
by Golub and Van Loan (Johns Hopkins Press.)
subroutine sggqrf (integer N, integer M, integer P, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAUA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * )
TAUB, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGGQRF
Purpose:
SGGQRF computes a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B:
A = Q*R, B = Q*T*Z,
where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:
if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
( 0 ) N-M N M-N
M
where R11 is upper triangular, and
if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
P-N N ( T21 ) P
P
where T12 or T21 is upper triangular.
In particular, if B is square and nonsingular, the GQR factorization
of A and B implicitly gives the QR factorization of inv(B)*A:
inv(B)*A = Z**T*(inv(T)*R)
where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.
Parameters
N
N is INTEGER
The number of rows of the matrices A and B. N >= 0.
M
M is INTEGER
The number of columns of the matrix A. M >= 0.
P
P is INTEGER
The number of columns of the matrix B. P >= 0.
A
A is REAL array, dimension (LDA,M)
On entry, the N-by-M matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(N,M)-by-M upper trapezoidal matrix R (R is
upper triangular if N >= M); the elements below the diagonal,
with the array TAUA, represent the orthogonal matrix Q as a
product of min(N,M) elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAUA
TAUA is REAL array, dimension (min(N,M))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q (see Further Details).
B
B is REAL array, dimension (LDB,P)
On entry, the N-by-P matrix B.
On exit, if N <= P, the upper triangle of the subarray
B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
if N > P, the elements on and above the (N-P)-th subdiagonal
contain the N-by-P upper trapezoidal matrix T; the remaining
elements, with the array TAUB, represent the orthogonal
matrix Z as a product of elementary reflectors (see Further
Details).
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
TAUB
TAUB is REAL array, dimension (min(N,P))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Z (see Further Details).
WORK
WORK is REAL 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 SORMQR.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(n,m).
Each H(i) has the form
H(i) = I - taua * v * v**T
where taua is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine SORGQR.
To use Q to update another matrix, use LAPACK subroutine SORMQR.
The matrix Z is represented as a product of elementary reflectors
Z = H(1) H(2) . . . H(k), where k = min(n,p).
Each H(i) has the form
H(i) = I - taub * v * v**T
where taub is a real scalar, and v is a real vector with
v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGRQ.
To use Z to update another matrix, use LAPACK subroutine SORMRQ.
subroutine sggrqf (integer M, integer P, integer N, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAUA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * )
TAUB, real, dimension( * ) WORK, integer LWORK, integer INFO)
SGGRQF
Purpose:
SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:
A = R*Q, B = Z*T*Q,
where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:
if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N
where R12 or R21 is upper triangular, and
if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N
where T11 is upper triangular.
In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):
A*inv(B) = (R*inv(T))*Z**T
where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
transpose of the matrix Z.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
if M > N, the elements on and above the (M-N)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R; the remaining
elements, with the array TAUA, represent the orthogonal
matrix Q as a product of elementary reflectors (see Further
Details).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAUA
TAUA is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q (see Further Details).
B
B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, the elements on and above the diagonal of the array
contain the min(P,N)-by-N upper trapezoidal matrix T (T is
upper triangular if P >= N); the elements below the diagonal,
with the array TAUB, represent the orthogonal matrix Z as a
product of elementary reflectors (see Further Details).
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TAUB
TAUB is REAL array, dimension (min(P,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Z (see Further Details).
WORK
WORK is REAL 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 SORMRQ.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - taua * v * v**T
where taua is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine SORGRQ.
To use Q to update another matrix, use LAPACK subroutine SORMRQ.
The matrix Z is represented as a product of elementary reflectors
Z = H(1) H(2) . . . H(k), where k = min(p,n).
Each H(i) has the form
H(i) = I - taub * v * v**T
where taub is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGQR.
To use Z to update another matrix, use LAPACK subroutine SORMQR.
subroutine sggsvp3 (character JOBU, character JOBV, character JOBQ, integer M, integer P,
integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
LDB, real TOLA, real TOLB, integer K, integer L, real, dimension( ldu, * ) U, integer LDU,
real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ,
integer, dimension( * ) IWORK, real, dimension( * ) TAU, real, dimension( * ) WORK,
integer LWORK, integer INFO)
SGGSVP3
Purpose:
SGGSVP3 computes orthogonal matrices U, V and Q such that
N-K-L K L
U**T*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V**T*B*Q = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
SGGSVD3.
Parameters
JOBU
JOBU is CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
JOBV
JOBV is CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.
JOBQ
JOBQ is CHARACTER*1
= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A
A is REAL 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 REAL 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 REAL
TOLB
TOLB is REAL
TOLA and TOLB are the thresholds to determine the effective
numerical rank of matrix B and a subblock of A. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
K
K is INTEGER
L
L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section.
K + L = effective numerical rank of (A**T,B**T)**T.
U
U is REAL array, dimension (LDU,M)
If JOBU = 'U', U contains the orthogonal matrix U.
If JOBU = 'N', U is not referenced.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V
V is REAL array, dimension (LDV,P)
If JOBV = 'V', V contains the orthogonal matrix V.
If JOBV = 'N', V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q
Q is REAL array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
IWORK
IWORK is INTEGER array, dimension (N)
TAU
TAU is REAL array, dimension (N)
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The subroutine uses LAPACK subroutine SGEQP3 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.
SGGSVP3 replaces the deprecated subroutine SGGSVP.
subroutine sgsvj0 (character*1 JOBV, integer M, integer N, real, dimension( lda, * ) A,
integer LDA, real, dimension( n ) D, real, dimension( n ) SVA, integer MV, real,
dimension( ldv, * ) V, integer LDV, real EPS, real SFMIN, real TOL, integer NSWEEP, real,
dimension( lwork ) WORK, integer LWORK, integer INFO)
SGSVJ0 pre-processor for the routine sgesvj.
Purpose:
SGSVJ0 is called from SGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as SGESVJ 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 REAL array, dimension (LDA,N)
On entry, M-by-N matrix A, such that A*diag(D) represents
the input matrix.
On exit,
A_onexit * D_onexit represents the input matrix A*diag(D)
post-multiplied by a sequence of Jacobi rotations, where the
rotation threshold and the total number of sweeps are given in
TOL and NSWEEP, respectively.
(See the descriptions of D, TOL and NSWEEP.)
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is REAL array, dimension (N)
The array D accumulates the scaling factors from the fast scaled
Jacobi rotations.
On entry, A*diag(D) represents the input matrix.
On exit, A_onexit*diag(D_onexit) represents the input matrix
post-multiplied by a sequence of Jacobi rotations, where the
rotation threshold and the total number of sweeps are given in
TOL and NSWEEP, respectively.
(See the descriptions of A, TOL and NSWEEP.)
SVA
SVA is REAL 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 REAL 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 REAL
EPS = SLAMCH('Epsilon')
SFMIN
SFMIN is REAL
SFMIN = SLAMCH('Safe Minimum')
TOL
TOL is REAL
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 REAL array, dimension (LWORK)
LWORK
LWORK is INTEGER
LWORK is the dimension of WORK. LWORK >= M.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, then the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
SGSVJ0 is used just to enable SGESVJ to call a simplified version of itself to work on
a submatrix of the original matrix.
Contributors:
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
Bugs, Examples and Comments:
Please report all bugs and send interesting test examples and comments to
drmac@math.hr. Thank you.
subroutine sgsvj1 (character*1 JOBV, integer M, integer N, integer N1, real, dimension( lda, *
) A, integer LDA, real, dimension( n ) D, real, dimension( n ) SVA, integer MV, real,
dimension( ldv, * ) V, integer LDV, real EPS, real SFMIN, real TOL, integer NSWEEP, real,
dimension( lwork ) WORK, integer LWORK, integer INFO)
SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only
particular pivots.
Purpose:
SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as SGESVJ 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
~~~~~~~~~~~~~~~
SGSVJ1 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
EPS = SLAMCH('Epsilon')
SFMIN
SFMIN is REAL
SFMIN = SLAMCH('Safe Minimum')
TOL
TOL is REAL
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 REAL array, dimension (LWORK)
LWORK
LWORK is INTEGER
LWORK is the dimension of WORK. LWORK >= M.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, then the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
subroutine shsein (character SIDE, character EIGSRC, character INITV, logical, dimension( * )
SELECT, integer N, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR,
real, dimension( * ) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension(
ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * ) WORK, integer,
dimension( * ) IFAILL, integer, dimension( * ) IFAILR, integer INFO)
SHSEIN
Purpose:
SHSEIN uses inverse iteration to find specified right and/or left
eigenvectors of a real upper Hessenberg matrix H.
The right eigenvector x and the left eigenvector y of the matrix H
corresponding to an eigenvalue w are defined by:
H * x = w * x, y**h * H = w * y**h
where y**h denotes the conjugate transpose of the vector y.
Parameters
SIDE
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
EIGSRC
EIGSRC is CHARACTER*1
Specifies the source of eigenvalues supplied in (WR,WI):
= 'Q': the eigenvalues were found using SHSEQR; 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 SHSEIN 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, SHSEIN must always perform inverse iteration
using the whole matrix H.
INITV
INITV is CHARACTER*1
= 'N': no initial vectors are supplied;
= 'U': user-supplied initial vectors are stored in the arrays
VL and/or VR.
SELECT
SELECT is LOGICAL array, dimension (N)
Specifies the eigenvectors to be computed. To select the
real eigenvector corresponding to a real eigenvalue WR(j),
SELECT(j) must be set to .TRUE.. To select the complex
eigenvector corresponding to a complex eigenvalue
(WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
.FALSE..
N
N is INTEGER
The order of the matrix H. N >= 0.
H
H is REAL array, dimension (LDH,N)
The upper Hessenberg matrix H.
If a NaN is detected in H, the routine will return with INFO=-6.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
WR
WR is REAL array, dimension (N)
WI
WI is REAL array, dimension (N)
On entry, the real and imaginary parts of the eigenvalues of
H; a complex conjugate pair of eigenvalues must be stored in
consecutive elements of WR and WI.
On exit, WR may have been altered since close eigenvalues
are perturbed slightly in searching for independent
eigenvectors.
VL
VL is REAL array, dimension (LDVL,MM)
On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
contain starting vectors for the inverse iteration for the
left eigenvectors; the starting vector for each eigenvector
must be in the same column(s) in which the eigenvector will
be stored.
On exit, if SIDE = 'L' or 'B', the left eigenvectors
specified by SELECT will be stored consecutively in the
columns of VL, in the same order as their eigenvalues. A
complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = 'R', VL is not referenced.
LDVL
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
VR
VR is REAL array, dimension (LDVR,MM)
On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
contain starting vectors for the inverse iteration for the
right eigenvectors; the starting vector for each eigenvector
must be in the same column(s) in which the eigenvector will
be stored.
On exit, if SIDE = 'R' or 'B', the right eigenvectors
specified by SELECT will be stored consecutively in the
columns of VR, in the same order as their eigenvalues. A
complex eigenvector corresponding to a complex eigenvalue is
stored in two consecutive columns, the first holding the real
part and the second the imaginary part.
If SIDE = 'L', VR is not referenced.
LDVR
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
MM
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M
M is INTEGER
The number of columns in the arrays VL and/or VR required to
store the eigenvectors; each selected real eigenvector
occupies one column and each selected complex eigenvector
occupies two columns.
WORK
WORK is REAL array, dimension ((N+2)*N)
IFAILL
IFAILL is INTEGER array, dimension (MM)
If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
eigenvector in the i-th column of VL (corresponding to the
eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
eigenvector converged satisfactorily. If the i-th and (i+1)th
columns of VL hold a complex eigenvector, then IFAILL(i) and
IFAILL(i+1) are set to the same value.
If SIDE = 'R', IFAILL is not referenced.
IFAILR
IFAILR is INTEGER array, dimension (MM)
If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
eigenvector in the i-th column of VR (corresponding to the
eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
eigenvector converged satisfactorily. If the i-th and (i+1)th
columns of VR hold a complex eigenvector, then IFAILR(i) and
IFAILR(i+1) are set to the same value.
If SIDE = 'L', IFAILR is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, i is the number of eigenvectors which
failed to converge; see IFAILL and IFAILR for further
details.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x|+|y|.
subroutine shseqr (character JOB, character COMPZ, integer N, integer ILO, integer IHI, real,
dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI,
real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK,
integer INFO)
SHSEQR
Purpose:
SHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.
Optionally Z may be postmultiplied into an input orthogonal
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
Parameters
JOB
JOB is CHARACTER*1
= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T.
COMPZ
COMPZ is CHARACTER*1
= 'N': no Schur vectors are computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= 'V': Z must contain an orthogonal matrix Q on entry, and
the product Q*Z is returned.
N
N is INTEGER
The order of the matrix H. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL, and then passed to ZGEHRD
when the matrix output by SGEBAL 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 REAL array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and JOB = 'S', then H contains the
upper quasi-triangular matrix T from the Schur decomposition
(the Schur form); 2-by-2 diagonal blocks (corresponding to
complex conjugate pairs of eigenvalues) are returned in
standard form, with H(i,i) = H(i+1,i+1) and
H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
contents of H are unspecified on exit. (The output value of
H when INFO > 0 is given under the description of INFO
below.)
Unlike earlier versions of SHSEQR, this subroutine may
explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
WR
WR is REAL array, dimension (N)
WI
WI is REAL array, dimension (N)
The real and imaginary parts, respectively, of the computed
eigenvalues. If two eigenvalues are computed as a complex
conjugate pair, they are stored in consecutive elements of
WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and
WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in
the same order as on the diagonal of the Schur form returned
in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
WI(i+1) = -WI(i).
Z
Z is REAL array, dimension (LDZ,N)
If COMPZ = 'N', Z is not referenced.
If COMPZ = 'I', on entry Z need not be set and on exit,
if INFO = 0, Z contains the orthogonal matrix Z of the Schur
vectors of H. If COMPZ = 'V', on entry Z must contain an
N-by-N matrix Q, which is assumed to be equal to the unit
matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
if INFO = 0, Z contains Q*Z.
Normally Q is the orthogonal matrix generated by SORGHR
after the call to SGEHRD 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 REAL 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 SHSEQR does a workspace query.
In this case, SHSEQR 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, SHSEQR failed to compute all of
the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed. (Failures are rare.)
If INFO > 0 and JOB = 'E', then on exit, the
remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.
If INFO > 0 and JOB = 'S', then on exit
(*) (initial value of H)*U = U*(final value of H)
where U is an orthogonal matrix. The final
value of H is upper Hessenberg and quasi-triangular
in rows and columns INFO+1 through IHI.
If INFO > 0 and COMPZ = 'V', then on exit
(final value of Z) = (initial value of Z)*U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)
If INFO > 0 and COMPZ = 'I', then on exit
(final value of Z) = U
where U is the orthogonal matrix in (*) (regard-
less of the value of JOB.)
If INFO > 0 and COMPZ = 'N', then Z is not
accessed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
Further Details:
Default values supplied by
ILAENV(ISPEC,'SHSEQR',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 SLAHQR vs SLAQR0 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
SLAHQR 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 sla_lin_berr (integer N, integer NZ, integer NRHS, real, dimension( n, nrhs ) RES,
real, dimension( n, nrhs ) AYB, real, dimension( nrhs ) BERR)
SLA_LIN_BERR computes a component-wise relative backward error.
Purpose:
SLA_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 REAL array, dimension (N,NRHS)
The residual matrix, i.e., the matrix R in the relative backward
error formula above.
AYB
AYB is REAL 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 sla_gerfsx_extended.f).
BERR
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error from the formula above.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sla_wwaddw (integer N, real, dimension( * ) X, real, dimension( * ) Y, real,
dimension( * ) W)
SLA_WWADDW adds a vector into a doubled-single vector.
Purpose:
SLA_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 REAL array, dimension (N)
The first part of the doubled-single accumulation vector.
Y
Y is REAL array, dimension (N)
The second part of the doubled-single accumulation vector.
W
W is REAL array, dimension (N)
The vector to be added.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine slals0 (integer ICOMPQ, integer NL, integer NR, integer SQRE, integer NRHS, real,
dimension( ldb, * ) B, integer LDB, real, dimension( ldbx, * ) BX, integer LDBX, integer,
dimension( * ) PERM, integer GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer
LDGCOL, real, dimension( ldgnum, * ) GIVNUM, integer LDGNUM, real, dimension( ldgnum, * )
POLES, real, dimension( * ) DIFL, real, dimension( ldgnum, * ) DIFR, real, dimension( * )
Z, integer K, real C, real S, real, dimension( * ) WORK, integer INFO)
SLALS0 applies back multiplying factors in solving the least squares problem using divide
and conquer SVD approach. Used by sgelsd.
Purpose:
SLALS0 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
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 REAL
S contains garbage if SQRE =0 and the S-value of a Givens
rotation related to the right null space if SQRE = 1.
WORK
WORK is REAL array, dimension ( K )
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at
Berkeley, USA
Osni Marques, LBNL/NERSC, USA
subroutine slalsa (integer ICOMPQ, integer SMLSIZ, integer N, integer NRHS, real, dimension(
ldb, * ) B, integer LDB, real, dimension( ldbx, * ) BX, integer LDBX, real, dimension(
ldu, * ) U, integer LDU, real, dimension( ldu, * ) VT, integer, dimension( * ) K, real,
dimension( ldu, * ) DIFL, real, dimension( ldu, * ) DIFR, real, dimension( ldu, * ) Z,
real, dimension( ldu, * ) POLES, integer, dimension( * ) GIVPTR, integer, dimension(
ldgcol, * ) GIVCOL, integer LDGCOL, integer, dimension( ldgcol, * ) PERM, real, dimension(
ldu, * ) GIVNUM, real, dimension( * ) C, real, dimension( * ) S, real, dimension( * )
WORK, integer, dimension( * ) IWORK, integer INFO)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
Purpose:
SLALSA 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, SLALSA applies the inverse of the left singular vector
matrix of an upper bidiagonal matrix to the right hand side; and if
ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
right hand side. The singular vector matrices were generated in
compact form by SLALSA.
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 REAL 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 REAL 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 REAL 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 REAL array, dimension ( LDU, SMLSIZ+1 ).
On entry, VT**T contains the right singular vector matrices of
all subproblems at the bottom level.
K
K is INTEGER array, dimension ( N ).
DIFL
DIFL is REAL array, dimension ( LDU, NLVL ).
where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
DIFR
DIFR is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension ( N ).
On entry, if the I-th subproblem is not square,
S( I ) contains the S-value of a Givens rotation related to
the right null space of the I-th subproblem.
WORK
WORK is REAL array, dimension (N)
IWORK
IWORK is INTEGER array, dimension (3*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at
Berkeley, USA
Osni Marques, LBNL/NERSC, USA
subroutine slalsd (character UPLO, integer SMLSIZ, integer N, integer NRHS, real, dimension( *
) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, real RCOND, integer
RANK, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SLALSD uses the singular value decomposition of A to solve the least squares problem.
Purpose:
SLALSD 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 REAL 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 REAL array, dimension (N-1)
Contains the super-diagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
B
B is REAL 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 REAL
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 REAL array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
IWORK
IWORK is INTEGER array, dimension at least
(3*N*NLVL + 11*N)
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute a singular value while
working on the submatrix lying in rows and columns
INFO/(N+1) through MOD(INFO,N+1).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at
Berkeley, USA
Osni Marques, LBNL/NERSC, USA
real function slansf (character NORM, character TRANSR, character UPLO, integer N, real,
dimension( 0: * ) A, real, dimension( 0: * ) WORK)
SLANSF
Purpose:
SLANSF returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
real symmetric matrix A in RFP format.
Returns
SLANSF
SLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in SLANSF as described
above.
TRANSR
TRANSR is CHARACTER*1
Specifies whether the RFP format of A is normal or
transposed format.
= 'N': RFP format is Normal;
= 'T': RFP format is Transpose.
UPLO
UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
= 'U': RFP A came from an upper triangular matrix;
= 'L': RFP A came from a lower triangular matrix.
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, SLANSF is
set to zero.
A
A is REAL array, dimension ( N*(N+1)/2 );
On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
part of the symmetric matrix A stored in RFP format. See the
'Notes' below for more details.
Unchanged on exit.
WORK
WORK is REAL array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine slarscl2 (integer M, integer N, real, dimension( * ) D, real, dimension( ldx, * )
X, integer LDX)
SLARSCL2 performs reciprocal diagonal scaling on a matrix.
Purpose:
SLARSCL2 performs a reciprocal diagonal scaling on a matrix:
x <-- inv(D) * x
where the diagonal matrix D is stored as a vector.
Eventually to be replaced by BLAS_sge_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 REAL array, length M
Diagonal matrix D, stored as a vector of length M.
X
X is REAL array, dimension (LDX,N)
On entry, the matrix X to be scaled by D.
On exit, the scaled matrix.
LDX
LDX is INTEGER
The leading dimension of the matrix X. LDX >= M.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine slarz (character SIDE, integer M, integer N, integer L, real, dimension( * ) V,
integer INCV, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * )
WORK)
SLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Purpose:
SLARZ applies a real elementary reflector H to a real M-by-N
matrix C, from either the left or the right. H is represented in the
form
H = I - tau * v * v**T
where tau is a real scalar and v is a real vector.
If tau = 0, then H is taken to be the unit matrix.
H is a product of k elementary reflectors as returned by STZRZF.
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 REAL array, dimension (1+(L-1)*abs(INCV))
The vector v in the representation of H as returned by
STZRZF. V is not used if TAU = 0.
INCV
INCV is INTEGER
The increment between elements of v. INCV <> 0.
TAU
TAU is REAL
The value tau in the representation of H.
C
C is REAL 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 REAL array, dimension
(N) if SIDE = 'L'
or (M) if SIDE = 'R'
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
subroutine slarzb (character SIDE, character TRANS, character DIRECT, character STOREV,
integer M, integer N, integer K, integer L, real, dimension( ldv, * ) V, integer LDV,
real, dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real,
dimension( ldwork, * ) WORK, integer LDWORK)
SLARZB applies a block reflector or its transpose to a general matrix.
Purpose:
SLARZB applies a real block reflector H or its transpose H**T to
a real distributed M-by-N C from the left or the right.
Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply H or H**T from the Left
= 'R': apply H or H**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'C': apply H**T (Transpose)
DIRECT
DIRECT is CHARACTER*1
Indicates how H is formed from a product of elementary
reflectors
= 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV
STOREV is CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= 'C': Columnwise (not supported yet)
= 'R': Rowwise
M
M is INTEGER
The number of rows of the matrix C.
N
N is INTEGER
The number of columns of the matrix C.
K
K is INTEGER
The order of the matrix T (= the number of elementary
reflectors whose product defines the block reflector).
L
L is INTEGER
The number of columns of the matrix V containing the
meaningful part of the Householder reflectors.
If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
V
V is REAL 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 REAL 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 REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension (LDWORK,K)
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
If SIDE = 'L', LDWORK >= max(1,N);
if SIDE = 'R', LDWORK >= max(1,M).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
subroutine slarzt (character DIRECT, character STOREV, integer N, integer K, real, dimension(
ldv, * ) V, integer LDV, real, dimension( * ) TAU, real, dimension( ldt, * ) T, integer
LDT)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
Purpose:
SLARZT forms the triangular factor T of a real block reflector
H of order > n, which is defined as a product of k elementary
reflectors.
If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
If STOREV = 'C', the vector which defines the elementary reflector
H(i) is stored in the i-th column of the array V, and
H = I - V * T * V**T
If STOREV = 'R', the vector which defines the elementary reflector
H(i) is stored in the i-th row of the array V, and
H = I - V**T * T * V
Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
Parameters
DIRECT
DIRECT is CHARACTER*1
Specifies the order in which the elementary reflectors are
multiplied to form the block reflector:
= 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV
STOREV is CHARACTER*1
Specifies how the vectors which define the elementary
reflectors are stored (see also Further Details):
= 'C': columnwise (not supported yet)
= 'R': rowwise
N
N is INTEGER
The order of the block reflector H. N >= 0.
K
K is INTEGER
The order of the triangular factor T (= the number of
elementary reflectors). K >= 1.
V
V is REAL 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).
T
T is REAL array, dimension (LDT,K)
The k by k triangular factor T of the block reflector.
If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
lower triangular. The rest of the array is not used.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= K.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored; the corresponding
array elements are modified but restored on exit. The rest of the
array is not used.
DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
______V_____
( v1 v2 v3 ) / \
( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
( v1 v2 v3 )
. . .
. . .
1 . .
1 .
1
DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
______V_____
1 / \
. 1 ( 1 . . . . v1 v1 v1 v1 v1 )
. . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
. . . ( . . 1 . . v3 v3 v3 v3 v3 )
. . .
( v1 v2 v3 )
( v1 v2 v3 )
V = ( v1 v2 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
subroutine slascl2 (integer M, integer N, real, dimension( * ) D, real, dimension( ldx, * ) X,
integer LDX)
SLASCL2 performs diagonal scaling on a matrix.
Purpose:
SLASCL2 performs a diagonal scaling on a matrix:
x <-- D * x
where the diagonal matrix D is stored as a vector.
Eventually to be replaced by BLAS_sge_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 REAL array, length M
Diagonal matrix D, stored as a vector of length M.
X
X is REAL array, dimension (LDX,N)
On entry, the matrix X to be scaled by D.
On exit, the scaled matrix.
LDX
LDX is INTEGER
The leading dimension of the matrix X. LDX >= M.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine slatrz (integer M, integer N, integer L, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK)
SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
Purpose:
SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
[ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
matrix and, R and A1 are M-by-M upper triangular matrices.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
L
L is INTEGER
The number of columns of the matrix A containing the
meaningful part of the Householder vectors. N-M >= L >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the leading M-by-N upper trapezoidal part of the
array A must contain the matrix to be factorized.
On exit, the leading M-by-M upper triangular part of A
contains the upper triangular matrix R, and elements N-L+1 to
N of the first M rows of A, with the array TAU, represent the
orthogonal matrix Z as a product of M elementary reflectors.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is REAL array, dimension (M)
The scalar factors of the elementary reflectors.
WORK
WORK is REAL array, dimension (M)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
The factorization is obtained by Householder's method. The kth
transformation matrix, Z( k ), which is used to introduce zeros into
the ( m - k + 1 )th row of A, is given in the form
Z( k ) = ( I 0 ),
( 0 T( k ) )
where
T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
( 0 )
( z( k ) )
tau is a scalar and z( k ) is an l element vector. tau and z( k )
are chosen to annihilate the elements of the kth row of A2.
The scalar tau is returned in the kth element of TAU and the vector
u( k ) in the kth row of A2, such that the elements of z( k ) are
in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
the upper triangular part of A1.
Z is given by
Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
subroutine sopgtr (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * )
TAU, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK, integer INFO)
SOPGTR
Purpose:
SOPGTR generates a real orthogonal matrix Q which is defined as the
product of n-1 elementary reflectors H(i) of order n, as returned by
SSPTRD 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 SSPTRD;
= 'L': Lower triangular packed storage used in previous
call to SSPTRD.
N
N is INTEGER
The order of the matrix Q. N >= 0.
AP
AP is REAL array, dimension (N*(N+1)/2)
The vectors which define the elementary reflectors, as
returned by SSPTRD.
TAU
TAU is REAL array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SSPTRD.
Q
Q is REAL array, dimension (LDQ,N)
The N-by-N orthogonal matrix Q.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
WORK
WORK is REAL array, dimension (N-1)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sopmtr (character SIDE, character UPLO, character TRANS, integer M, integer N,
real, dimension( * ) AP, real, dimension( * ) TAU, real, dimension( ldc, * ) C, integer
LDC, real, dimension( * ) WORK, integer INFO)
SOPMTR
Purpose:
SOPMTR overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
nq-1 elementary reflectors, as returned by SSPTRD using packed
storage:
if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangular packed storage used in previous
call to SSPTRD;
= 'L': Lower triangular packed storage used in previous
call to SSPTRD.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
AP
AP is REAL 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 SSPTRD. AP is modified by the routine but
restored on exit.
TAU
TAU is REAL 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 SSPTRD.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension
(N) if SIDE = 'L'
(M) if SIDE = 'R'
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorbdb (character TRANS, character SIGNS, integer M, integer P, integer Q, real,
dimension( ldx11, * ) X11, integer LDX11, real, dimension( ldx12, * ) X12, integer LDX12,
real, dimension( ldx21, * ) X21, integer LDX21, real, dimension( ldx22, * ) X22, integer
LDX22, real, dimension( * ) THETA, real, dimension( * ) PHI, real, dimension( * ) TAUP1,
real, dimension( * ) TAUP2, real, dimension( * ) TAUQ1, real, dimension( * ) TAUQ2, real,
dimension( * ) WORK, integer LWORK, integer INFO)
SORBDB
Purpose:
SORBDB simultaneously bidiagonalizes the blocks of an M-by-M
partitioned orthogonal matrix X:
[ B11 | B12 0 0 ]
[ X11 | X12 ] [ P1 | ] [ 0 | 0 -I 0 ] [ Q1 | ]**T
X = [-----------] = [---------] [----------------] [---------] .
[ X21 | X22 ] [ | P2 ] [ B21 | B22 0 0 ] [ | Q2 ]
[ 0 | 0 0 I ]
X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
not the case, then X must be transposed and/or permuted. This can be
done in constant time using the TRANS and SIGNS options. See SORCSD
for details.)
The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
represented implicitly by Householder vectors.
B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS
SIGNS is CHARACTER
= 'O': The lower-left block is made nonpositive (the
'other' convention);
otherwise: The upper-right block is made nonpositive (the
'default' convention).
M
M is INTEGER
The number of rows and columns in X.
P
P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q).
X11
X11 is REAL array, dimension (LDX11,Q)
On entry, the top-left block of the orthogonal matrix to be
reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X11) specify reflectors for P1,
the rows of triu(X11,1) specify reflectors for Q1;
else TRANS = 'T', and
the rows of triu(X11) specify reflectors for P1,
the columns of tril(X11,-1) specify reflectors for Q1.
LDX11
LDX11 is INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q.
X12
X12 is REAL array, dimension (LDX12,M-Q)
On entry, the top-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X12) specify the first P reflectors for
Q2;
else TRANS = 'T', and
the columns of tril(X12) specify the first P reflectors
for Q2.
LDX12
LDX12 is INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q.
X21
X21 is REAL array, dimension (LDX21,Q)
On entry, the bottom-left block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the columns of tril(X21) specify reflectors for P2;
else TRANS = 'T', and
the rows of triu(X21) specify reflectors for P2.
LDX21
LDX21 is INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q.
X22
X22 is REAL array, dimension (LDX22,M-Q)
On entry, the bottom-right block of the orthogonal matrix to
be reduced. On exit, the form depends on TRANS:
If TRANS = 'N', then
the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
M-P-Q reflectors for Q2,
else TRANS = 'T', and
the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
M-P-Q reflectors for P2.
LDX22
LDX22 is INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA
THETA is REAL 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 REAL 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 REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.
TAUP2
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
TAUQ2
TAUQ2 is REAL array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The bidiagonal blocks B11, B12, B21, and B22 are represented
implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
lower bidiagonal. Every entry in each bidiagonal band is a product
of a sine or cosine of a THETA with a sine or cosine of a PHI. See
[1] or SORCSD for details.
P1, P2, Q1, and Q2 are represented as products of elementary
reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2
using SORGQR and SORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
subroutine sorbdb1 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer
LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real,
dimension(*) PHI, real, dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*)
TAUQ1, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB1
Purpose:
SORBDB1 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 SORBDB2, SORBDB3, and SORBDB4 handle cases in
which Q is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
angles THETA, PHI.
Parameters
M
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P
P is INTEGER
The number of rows in X11. 0 <= P <= M.
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q).
X11
X11 is REAL 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 REAL 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 REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.
TAUP2
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
subroutine sorbdb2 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer
LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real,
dimension(*) PHI, real, dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*)
TAUQ1, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB2
Purpose:
SORBDB2 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 SORBDB1, SORBDB3, and SORBDB4 handle cases in
which P is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
angles THETA, PHI.
Parameters
M
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P
P is INTEGER
The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11
X11 is REAL 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 REAL 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 REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1
TAUP1 is REAL array, dimension (P-1)
The scalar factors of the elementary reflectors that define
P1.
TAUP2
TAUP2 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
subroutine sorbdb3 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer
LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real,
dimension(*) PHI, real, dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*)
TAUQ1, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB3
Purpose:
SORBDB3 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 SORBDB1, SORBDB2, and SORBDB4 handle cases in
which M-P is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
M
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P
P is INTEGER
The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11
X11 is REAL 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 REAL 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 REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.
TAUP2
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
subroutine sorbdb4 (integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer
LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real,
dimension(*) PHI, real, dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*)
TAUQ1, real, dimension(*) PHANTOM, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB4
Purpose:
SORBDB4 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 SORBDB1, SORBDB2, and SORBDB3 handle cases in
which M-Q is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.
B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
implicitly by angles THETA, PHI.
Parameters
M
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P
P is INTEGER
The number of rows in X11. 0 <= P <= M.
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M and
M-Q <= min(P,M-P,Q).
X11
X11 is REAL 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 REAL 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 REAL array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI
PHI is REAL array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1
TAUP1 is REAL array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
P1.
TAUP2
TAUP2 is REAL array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
PHANTOM
PHANTOM is REAL 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 REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or SORCSD for details.
P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
subroutine sorbdb5 (integer M1, integer M2, integer N, real, dimension(*) X1, integer INCX1,
real, dimension(*) X2, integer INCX2, real, dimension(ldq1,*) Q1, integer LDQ1, real,
dimension(ldq2,*) Q2, integer LDQ2, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB5
Purpose:
SORBDB5 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= N.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorbdb6 (integer M1, integer M2, integer N, real, dimension(*) X1, integer INCX1,
real, dimension(*) X2, integer INCX2, real, dimension(ldq1,*) Q1, integer LDQ1, real,
dimension(ldq2,*) Q2, integer LDQ2, real, dimension(*) WORK, integer LWORK, integer INFO)
SORBDB6
Purpose:
SORBDB6 orthogonalizes the column vector
X = [ X1 ]
[ X2 ]
with respect to the columns of
Q = [ Q1 ] .
[ Q2 ]
The Euclidean norm of X must be one and the columns of Q must be
orthonormal. The orthogonalized vector will be zero if and only if it
lies entirely in the range of Q.
The projection is computed with at most two iterations of the
classical Gram-Schmidt algorithm, see
* L. Giraud, J. Langou, M. RozložnÃk. 'On the round-off error
analysis of the Gram-Schmidt algorithm with reorthogonalization.'
2002. CERFACS Technical Report No. TR/PA/02/33. URL:
https://www.cerfacs.fr/algor/reports/2002/TR_PA_02_33.pdf
Parameters
M1
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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= N.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
recursive subroutine sorcsd (character JOBU1, character JOBU2, character JOBV1T, character
JOBV2T, character TRANS, character SIGNS, integer M, integer P, integer Q, real,
dimension( ldx11, * ) X11, integer LDX11, real, dimension( ldx12, * ) X12, integer LDX12,
real, dimension( ldx21, * ) X21, integer LDX21, real, dimension( ldx22,
* ) X22, integer LDX22, real, dimension( * ) THETA, real, dimension( ldu1, * ) U1, integer
LDU1, real, dimension( ldu2, * ) U2, integer LDU2, real, dimension( ldv1t, * ) V1T,
integer LDV1T, real, dimension( ldv2t, * ) V2T, integer LDV2T, real, dimension( * ) WORK,
integer LWORK, integer, dimension( * ) IWORK, integer INFO)
SORCSD
Purpose:
SORCSD computes the CS decomposition of an M-by-M partitioned
orthogonal matrix X:
[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
X = [-----------] = [---------] [---------------------] [---------] .
[ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ]
[ 0 S 0 | 0 C 0 ]
[ 0 0 I | 0 0 0 ]
X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
(M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
which R = MIN(P,M-P,Q,M-Q).
Parameters
JOBU1
JOBU1 is CHARACTER
= 'Y': U1 is computed;
otherwise: U1 is not computed.
JOBU2
JOBU2 is CHARACTER
= 'Y': U2 is computed;
otherwise: U2 is not computed.
JOBV1T
JOBV1T is CHARACTER
= 'Y': V1T is computed;
otherwise: V1T is not computed.
JOBV2T
JOBV2T is CHARACTER
= 'Y': V2T is computed;
otherwise: V2T is not computed.
TRANS
TRANS is CHARACTER
= 'T': X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise: X, U1, U2, V1T, and V2T are stored in column-
major order.
SIGNS
SIGNS is CHARACTER
= 'O': The lower-left block is made nonpositive (the
'other' convention);
otherwise: The upper-right block is made nonpositive (the
'default' convention).
M
M is INTEGER
The number of rows and columns in X.
P
P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11
X11 is REAL array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX11
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).
X12
X12 is REAL array, dimension (LDX12,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX12
LDX12 is INTEGER
The leading dimension of X12. LDX12 >= MAX(1,P).
X21
X21 is REAL array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX21
LDX21 is INTEGER
The leading dimension of X11. LDX21 >= MAX(1,M-P).
X22
X22 is REAL array, dimension (LDX22,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX22
LDX22 is INTEGER
The leading dimension of X11. LDX22 >= MAX(1,M-P).
THETA
THETA is REAL 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 REAL array, dimension (LDU1,P)
If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
LDU1
LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
MAX(1,P).
U2
U2 is REAL array, dimension (LDU2,M-P)
If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.
LDU2
LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
MAX(1,M-P).
V1T
V1T is REAL array, dimension (LDV1T,Q)
If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.
LDV1T
LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
MAX(1,Q).
V2T
V2T is REAL array, dimension (LDV2T,M-Q)
If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
matrix V2**T.
LDV2T
LDV2T is INTEGER
The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
MAX(1,M-Q).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI's.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the work array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: SBBCSD did not converge. See the description of WORK
above for details.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorcsd2by1 (character JOBU1, character JOBU2, character JOBV1T, integer M, integer
P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21,
integer LDX21, real, dimension(*) THETA, real, dimension(ldu1,*) U1, integer LDU1, real,
dimension(ldu2,*) U2, integer LDU2, real, dimension(ldv1t,*) V1T, integer LDV1T, real,
dimension(*) WORK, integer LWORK, integer, dimension(*) IWORK, integer INFO)
SORCSD2BY1
Purpose:
SORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
orthonormal columns that has been partitioned into a 2-by-1 block
structure:
[ I1 0 0 ]
[ 0 C 0 ]
[ X11 ] [ U1 | ] [ 0 0 0 ]
X = [-----] = [---------] [----------] V1**T .
[ X21 ] [ | U2 ] [ 0 0 0 ]
[ 0 S 0 ]
[ 0 0 I2]
X11 is P-by-Q. The orthogonal matrices U1, U2, and V1 are P-by-P,
(M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R
nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which
R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a
K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).
Parameters
JOBU1
JOBU1 is CHARACTER
= 'Y': U1 is computed;
otherwise: U1 is not computed.
JOBU2
JOBU2 is CHARACTER
= 'Y': U2 is computed;
otherwise: U2 is not computed.
JOBV1T
JOBV1T is CHARACTER
= 'Y': V1T is computed;
otherwise: V1T is not computed.
M
M is INTEGER
The number of rows in X.
P
P is INTEGER
The number of rows in X11. 0 <= P <= M.
Q
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11
X11 is REAL array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX11
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).
X21
X21 is REAL array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.
LDX21
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= MAX(1,M-P).
THETA
THETA is REAL 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 REAL array, dimension (P)
If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
LDU1
LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
MAX(1,P).
U2
U2 is REAL array, dimension (M-P)
If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.
LDU2
LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
MAX(1,M-P).
V1T
V1T is REAL array, dimension (Q)
If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.
LDV1T
LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
MAX(1,Q).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI's.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the work array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: SBBCSD did not converge. See the description of WORK
above for details.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms,
50(1):33-65, 2009.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorg2l (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined
by sgeqlf (unblocked algorithm).
Purpose:
SORG2L generates an m by n real matrix Q with orthonormal columns,
which is defined as the last n columns of a product of k elementary
reflectors of order m
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF.
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 REAL 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 SGEQLF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorg2r (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined
by sgeqrf (unblocked algorithm).
Purpose:
SORG2R generates an m by n real matrix Q with orthonormal columns,
which is defined as the first n columns of a product of k elementary
reflectors of order m
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF.
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 REAL 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 SGEQRF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorghr (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer
LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGHR
Purpose:
SORGHR generates a real orthogonal matrix Q which is defined as the
product of IHI-ILO elementary reflectors of order N, as returned by
SGEHRD:
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 SGEHRD. 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 REAL array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by SGEHRD.
On exit, the N-by-N orthogonal matrix Q.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU
TAU is REAL array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEHRD.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorgl2 (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORGL2
Purpose:
SORGL2 generates an m by n real matrix Q with orthonormal rows,
which is defined as the first m rows of a product of k elementary
reflectors of order n
Q = H(k) . . . H(2) H(1)
as returned by SGELQF.
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 REAL 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 SGELQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
WORK
WORK is REAL array, dimension (M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorglq (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGLQ
Purpose:
SORGLQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the first M rows of a product of K elementary
reflectors of order N
Q = H(k) . . . H(2) H(1)
as returned by SGELQF.
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 REAL 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 SGELQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorgql (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGQL
Purpose:
SORGQL generates an M-by-N real matrix Q with orthonormal columns,
which is defined as the last N columns of a product of K elementary
reflectors of order M
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF.
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 REAL 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 SGEQLF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorgqr (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGQR
Purpose:
SORGQR generates an M-by-N real matrix Q with orthonormal columns,
which is defined as the first N columns of a product of K elementary
reflectors of order M
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF.
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 REAL 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 SGEQRF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorgr2 (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)
SORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization
determined by sgerqf (unblocked algorithm).
Purpose:
SORGR2 generates an m by n real matrix Q with orthonormal rows,
which is defined as the last m rows of a product of k elementary
reflectors of order n
Q = H(1) H(2) . . . H(k)
as returned by SGERQF.
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 REAL 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 SGERQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
WORK
WORK is REAL array, dimension (M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorgrq (integer M, integer N, integer K, real, dimension( lda, * ) A, integer LDA,
real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGRQ
Purpose:
SORGRQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the last M rows of a product of K elementary
reflectors of order N
Q = H(1) H(2) . . . H(k)
as returned by SGERQF.
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 REAL 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 SGERQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorgtr (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real,
dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORGTR
Purpose:
SORGTR generates a real orthogonal matrix Q which is defined as the
product of n-1 elementary reflectors of order N, as returned by
SSYTRD:
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 SSYTRD;
= 'L': Lower triangle of A contains elementary reflectors
from SSYTRD.
N
N is INTEGER
The order of the matrix Q. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by SSYTRD.
On exit, the N-by-N orthogonal matrix Q.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU
TAU is REAL array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SSYTRD.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N-1).
For optimum performance LWORK >= (N-1)*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorm2l (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer INFO)
SORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization
determined by sgeqlf (unblocked algorithm).
Purpose:
SORM2L overwrites the general real m by n matrix C with
Q * C if SIDE = 'L' and TRANS = 'N', or
Q**T * C if SIDE = 'L' and TRANS = 'T', or
C * Q if SIDE = 'R' and TRANS = 'N', or
C * Q**T if SIDE = 'R' and TRANS = 'T',
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF. Q is of order m if SIDE = 'L' and of order n
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left
= 'R': apply Q or Q**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply Q (No transpose)
= 'T': apply Q**T (Transpose)
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
A
A is REAL 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
SGEQLF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
C
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension
(N) if SIDE = 'L',
(M) if SIDE = 'R'
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorm2r (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization
determined by sgeqrf (unblocked algorithm).
Purpose:
SORM2R overwrites the general real m by n matrix C with
Q * C if SIDE = 'L' and TRANS = 'N', or
Q**T* C if SIDE = 'L' and TRANS = 'T', or
C * Q if SIDE = 'R' and TRANS = 'N', or
C * Q**T if SIDE = 'R' and TRANS = 'T',
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF. Q is of order m if SIDE = 'L' and of order n
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left
= 'R': apply Q or Q**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply Q (No transpose)
= 'T': apply Q**T (Transpose)
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
A
A is REAL 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
SGEQRF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
C
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension
(N) if SIDE = 'L',
(M) if SIDE = 'R'
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormbr (character VECT, character SIDE, character TRANS, integer M, integer N,
integer K, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real,
dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer LWORK, integer
INFO)
SORMBR
Purpose:
If VECT = 'Q', SORMBR overwrites the general real M-by-N matrix C
with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
If VECT = 'P', SORMBR overwrites the general real M-by-N matrix C
with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': P * C C * P
TRANS = 'T': P**T * C C * P**T
Here Q and P**T are the orthogonal matrices determined by SGEBRD when
reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
P**T are defined as products of elementary reflectors H(i) and G(i)
respectively.
Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
order of the orthogonal matrix Q or P**T that is applied.
If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
if nq >= k, Q = H(1) H(2) . . . H(k);
if nq < k, Q = H(1) H(2) . . . H(nq-1).
If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
if k < nq, P = G(1) G(2) . . . G(k);
if k >= nq, P = G(1) G(2) . . . G(nq-1).
Parameters
VECT
VECT is CHARACTER*1
= 'Q': apply Q or Q**T;
= 'P': apply P or P**T.
SIDE
SIDE is CHARACTER*1
= 'L': apply Q, Q**T, P or P**T from the Left;
= 'R': apply Q, Q**T, P or P**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q or P;
= 'T': Transpose, apply Q**T or P**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
If VECT = 'Q', the number of columns in the original
matrix reduced by SGEBRD.
If VECT = 'P', the number of rows in the original
matrix reduced by SGEBRD.
K >= 0.
A
A is REAL 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 SGEBRD.
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 REAL 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 SGEBRD in the array argument TAUQ or TAUP.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
or P*C or P**T*C or C*P or C*P**T.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormhr (character SIDE, character TRANS, integer M, integer N, integer ILO, integer
IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension(
ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMHR
Purpose:
SORMHR overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
IHI-ILO elementary reflectors, as returned by SGEHRD:
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI must have the same values as in the previous call
of SGEHRD. 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 REAL array, dimension
(LDA,M) if SIDE = 'L'
(LDA,N) if SIDE = 'R'
The vectors which define the elementary reflectors, as
returned by SGEHRD.
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 REAL 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 SGEHRD.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sorml2 (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer INFO)
SORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization
determined by sgelqf (unblocked algorithm).
Purpose:
SORML2 overwrites the general real m by n matrix C with
Q * C if SIDE = 'L' and TRANS = 'N', or
Q**T* C if SIDE = 'L' and TRANS = 'T', or
C * Q if SIDE = 'R' and TRANS = 'N', or
C * Q**T if SIDE = 'R' and TRANS = 'T',
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGELQF. Q is of order m if SIDE = 'L' and of order n
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left
= 'R': apply Q or Q**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply Q (No transpose)
= 'T': apply Q**T (Transpose)
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
A
A is REAL 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
SGELQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
C
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension
(N) if SIDE = 'L',
(M) if SIDE = 'R'
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormlq (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMLQ
Purpose:
SORMLQ overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGELQF. Q is of order M if SIDE = 'L' and of order N
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
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 REAL 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
SGELQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGELQF.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormql (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMQL
Purpose:
SORMQL overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by SGEQLF. Q is of order M if SIDE = 'L' and of order N
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
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 REAL 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
SGEQLF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQLF.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormqr (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMQR
Purpose:
SORMQR overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGEQRF. Q is of order M if SIDE = 'L' and of order N
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
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 REAL 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
SGEQRF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGEQRF.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormr2 (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer INFO)
SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization
determined by sgerqf (unblocked algorithm).
Purpose:
SORMR2 overwrites the general real m by n matrix C with
Q * C if SIDE = 'L' and TRANS = 'N', or
Q**T* C if SIDE = 'L' and TRANS = 'T', or
C * Q if SIDE = 'R' and TRANS = 'N', or
C * Q**T if SIDE = 'R' and TRANS = 'T',
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGERQF. Q is of order m if SIDE = 'L' and of order n
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left
= 'R': apply Q or Q**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply Q (No transpose)
= 'T': apply Q' (Transpose)
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
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 REAL 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
SGERQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
C
C is REAL array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension
(N) if SIDE = 'L',
(M) if SIDE = 'R'
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormr3 (character SIDE, character TRANS, integer M, integer N, integer K, integer
L, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension(
ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer INFO)
SORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization
determined by stzrzf (unblocked algorithm).
Purpose:
SORMR3 overwrites the general real m by n matrix C with
Q * C if SIDE = 'L' and TRANS = 'N', or
Q**T* C if SIDE = 'L' and TRANS = 'C', or
C * Q if SIDE = 'R' and TRANS = 'N', or
C * Q**T if SIDE = 'R' and TRANS = 'C',
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by STZRZF. Q is of order m if SIDE = 'L' and of order n
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left
= 'R': apply Q or Q**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply Q (No transpose)
= 'T': apply Q**T (Transpose)
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
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 REAL 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
STZRZF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by STZRZF.
C
C is REAL array, dimension (LDC,N)
On entry, the m-by-n matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL array, dimension
(N) if SIDE = 'L',
(M) if SIDE = 'R'
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
subroutine sormrq (character SIDE, character TRANS, integer M, integer N, integer K, real,
dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc, * ) C,
integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMRQ
Purpose:
SORMRQ overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by SGERQF. Q is of order M if SIDE = 'L' and of order N
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
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 REAL 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
SGERQF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sormrz (character SIDE, character TRANS, integer M, integer N, integer K, integer
L, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension(
ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMRZ
Purpose:
SORMRZ overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by STZRZF. Q is of order M if SIDE = 'L' and of order N
if SIDE = 'R'.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
K
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 REAL 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
STZRZF 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 REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by STZRZF.
C
C is REAL 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 REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
subroutine sormtr (character SIDE, character UPLO, character TRANS, integer M, integer N,
real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( ldc,
* ) C, integer LDC, real, dimension( * ) WORK, integer LWORK, integer INFO)
SORMTR
Purpose:
SORMTR overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix of order nq, with nq = m if
SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
nq-1 elementary reflectors, as returned by SSYTRD:
if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A contains elementary reflectors
from SSYTRD;
= 'L': Lower triangle of A contains elementary reflectors
from SSYTRD.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M
M is INTEGER
The number of rows of the matrix C. M >= 0.
N
N is INTEGER
The number of columns of the matrix C. N >= 0.
A
A is REAL array, dimension
(LDA,M) if SIDE = 'L'
(LDA,N) if SIDE = 'R'
The vectors which define the elementary reflectors, as
returned by SSYTRD.
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 REAL 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 SSYTRD.
C
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spbcon (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB,
integer LDAB, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * )
IWORK, integer INFO)
SPBCON
Purpose:
SPBCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite band matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangular factor stored in AB;
= 'L': Lower triangular factor stored in AB.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB
AB is REAL array, dimension (LDAB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A, stored in the
first KD+1 rows of the array. The j-th column of U or L is
stored in the j-th column of the array AB as follows:
if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
ANORM
ANORM is REAL
The 1-norm (or infinity-norm) of the symmetric band matrix A.
RCOND
RCOND is REAL
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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spbequ (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB,
integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)
SPBEQU
Purpose:
SPBEQU computes row and column scalings intended to equilibrate a
symmetric positive definite band matrix A and reduce its condition
number (with respect to the two-norm). S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangular of A is stored;
= 'L': Lower triangular of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB
AB is REAL array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band matrix A,
stored in the first KD+1 rows of the array. The j-th column
of A is stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array A. LDAB >= KD+1.
S
S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND
SCOND is REAL
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 REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the i-th diagonal element is nonpositive.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spbrfs (character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab,
* ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, real, dimension(
ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * )
FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK,
integer INFO)
SPBRFS
Purpose:
SPBRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite
and banded, and provides error bounds and backward error estimates
for the solution.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AB
AB is REAL array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band matrix A,
stored in the first KD+1 rows of the array. The j-th column
of A is stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
AFB
AFB is REAL array, dimension (LDAFB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A as computed by
SPBTRF, 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 REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SPBTRS.
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 REAL 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 REAL 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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spbstf (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB,
integer LDAB, integer INFO)
SPBSTF
Purpose:
SPBSTF computes a split Cholesky factorization of a real
symmetric positive definite band matrix A.
This routine is designed to be used in conjunction with SSBGST.
The factorization has the form A = S**T*S where S is a band matrix
of the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
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 REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first kd+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the factor S from the split Cholesky
factorization A = S**T*S. See Further Details.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The band storage scheme is illustrated by the following example, when
N = 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
Array elements marked * are not used by the routine.
subroutine spbtf2 (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB,
integer LDAB, integer INFO)
SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band
matrix (unblocked algorithm).
Purpose:
SPBTF2 computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
A = U**T * U , if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix, U**T is the transpose of U, and
L is lower triangular.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
AB
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, the leading minor of order k is not
positive definite, and the factorization could not be
completed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = 'L' the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
subroutine spbtrf (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB,
integer LDAB, integer INFO)
SPBTRF
Purpose:
SPBTRF computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
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 REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = 'L' the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
Contributors:
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
subroutine spbtrs (character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab,
* ) AB, integer LDAB, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SPBTRS
Purpose:
SPBTRS solves a system of linear equations A*X = B with a symmetric
positive definite band matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by SPBTRF.
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 REAL array, dimension (LDAB,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T of the band matrix A, stored in the
first KD+1 rows of the array. The j-th column of U or L is
stored in the j-th column of the array AB as follows:
if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
B
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spftrf (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) A,
integer INFO)
SPFTRF
Purpose:
SPFTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A.
The factorization has the form
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
This is the block version of the algorithm, calling Level 3 BLAS.
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': The Normal TRANSR of RFP A is stored;
= 'T': The Transpose TRANSR of RFP A is stored.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of RFP A is stored;
= 'L': Lower triangle of RFP A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension ( N*(N+1)/2 );
On entry, the symmetric matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
the transpose of RFP A as defined when
TRANSR = 'N'. The contents of RFP A are defined by UPLO as
follows: If UPLO = 'U' the RFP A contains the NT elements of
upper packed A. If UPLO = 'L' the RFP A contains the elements
of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
is odd. See the Note below for more details.
On exit, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**T*U or RFP A = L*L**T.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine spftri (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) A,
integer INFO)
SPFTRI
Purpose:
SPFTRI computes the inverse of a real (symmetric) positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by SPFTRF.
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': The Normal TRANSR of RFP A is stored;
= 'T': The Transpose TRANSR of RFP A is stored.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension ( N*(N+1)/2 )
On entry, the symmetric matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
the transpose of RFP A as defined when
TRANSR = 'N'. The contents of RFP A are defined by UPLO as
follows: If UPLO = 'U' the RFP A contains the nt elements of
upper packed A. If UPLO = 'L' the RFP A contains the elements
of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
is odd. See the Note below for more details.
On exit, the symmetric inverse of the original matrix, in the
same storage format.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the (i,i) element of the factor U or L is
zero, and the inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine spftrs (character TRANSR, character UPLO, integer N, integer NRHS, real, dimension(
0: * ) A, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SPFTRS
Purpose:
SPFTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A using the Cholesky factorization
A = U**T*U or A = L*L**T computed by SPFTRF.
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': The Normal TRANSR of RFP A is stored;
= 'T': The Transpose TRANSR of RFP A is stored.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of RFP A is stored;
= 'L': Lower triangle of RFP A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A
A is REAL 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**T, as computed by SPFTRF.
See note below for more details about RFP A.
B
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine sppcon (character UPLO, integer N, real, dimension( * ) AP, real ANORM, real RCOND,
real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SPPCON
Purpose:
SPPCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite packed matrix using
the Cholesky factorization A = U**T*U or A = L*L**T computed by
SPPTRF.
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 REAL array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, packed columnwise in a linear
array. The j-th column of U or L is stored in the array AP
as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
ANORM
ANORM is REAL
The 1-norm (or infinity-norm) of the symmetric matrix A.
RCOND
RCOND is REAL
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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sppequ (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) S,
real SCOND, real AMAX, integer INFO)
SPPEQU
Purpose:
SPPEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A in packed storage and reduce
its condition number (with respect to the two-norm). S contains the
scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
This choice of S puts the condition number of B within a factor N of
the smallest possible condition number over all possible diagonal
scalings.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
S
S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND
SCOND is REAL
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 REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spprfs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real,
dimension( * ) AFP, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X,
integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * )
WORK, integer, dimension( * ) IWORK, integer INFO)
SPPRFS
Purpose:
SPPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite
and packed, and provides error bounds and backward error estimates
for the solution.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
AP
AP is REAL array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
AFP
AFP is REAL array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by SPPTRF/CPPTRF,
packed columnwise in a linear array in the same format as A
(see AP).
B
B is REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SPPTRS.
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 REAL 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 REAL 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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spptrf (character UPLO, integer N, real, dimension( * ) AP, integer INFO)
SPPTRF
Purpose:
SPPTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A stored in packed format.
The factorization has the form
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
See below for further details.
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T, in the same
storage format as A.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
subroutine spptri (character UPLO, integer N, real, dimension( * ) AP, integer INFO)
SPPTRI
Purpose:
SPPTRI computes the inverse of a real symmetric positive definite
matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
computed by SPPTRF.
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 REAL array, dimension (N*(N+1)/2)
On entry, the triangular factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T, packed columnwise as
a linear array. The j-th column of U or L is stored in the
array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
On exit, the upper or lower triangle of the (symmetric)
inverse of A, overwriting the input factor U or L.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the (i,i) element of the factor U or L is
zero, and the inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spptrs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real,
dimension( ldb, * ) B, integer LDB, integer INFO)
SPPTRS
Purpose:
SPPTRS solves a system of linear equations A*X = B with a symmetric
positive definite matrix A in packed storage using the Cholesky
factorization A = U**T*U or A = L*L**T computed by SPPTRF.
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 REAL array, dimension (N*(N+1)/2)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, packed columnwise in a linear
array. The j-th column of U or L is stored in the array AP
as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
B
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spstf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA,
integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer
INFO)
SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric
positive semidefinite matrix.
Purpose:
SPSTF2 computes the Cholesky factorization with complete
pivoting of a real symmetric positive semidefinite matrix A.
The factorization has the form
P**T * A * P = U**T * U , if UPLO = 'U',
P**T * A * P = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular, and
P is stored as vector PIV.
This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 2 BLAS.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if 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 REAL
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 REAL array, dimension (2*N)
Work space.
INFO
INFO is INTEGER
< 0: If INFO = -K, the K-th argument had an illegal value,
= 0: algorithm completed successfully, and
> 0: the matrix A is either rank deficient with computed rank
as returned in RANK, or is not positive semidefinite. See
Section 7 of LAPACK Working Note #161 for further
information.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine spstrf (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA,
integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer
INFO)
SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric
positive semidefinite matrix.
Purpose:
SPSTRF computes the Cholesky factorization with complete
pivoting of a real symmetric positive semidefinite matrix A.
The factorization has the form
P**T * A * P = U**T * U , if UPLO = 'U',
P**T * A * P = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular, and
P is stored as vector PIV.
This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 3 BLAS.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if 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 REAL
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 REAL array, dimension (2*N)
Work space.
INFO
INFO is INTEGER
< 0: If INFO = -K, the K-th argument had an illegal value,
= 0: algorithm completed successfully, and
> 0: the matrix A is either rank deficient with computed rank
as returned in RANK, or is not positive semidefinite. See
Section 7 of LAPACK Working Note #161 for further
information.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ssbgst (character VECT, character UPLO, integer N, integer KA, integer KB, real,
dimension( ldab, * ) AB, integer LDAB, real, dimension( ldbb, * ) BB, integer LDBB, real,
dimension( ldx, * ) X, integer LDX, real, dimension( * ) WORK, integer INFO)
SSBGST
Purpose:
SSBGST reduces a real symmetric-definite banded generalized
eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y,
such that C has the same bandwidth as A.
B must have been previously factorized as S**T*S by SPBSTF, using a
split Cholesky factorization. A is overwritten by C = X**T*A*X, where
X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
bandwidth of A.
Parameters
VECT
VECT is CHARACTER*1
= 'N': do not form the transformation matrix X;
= 'V': form X.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
KA
KA is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB
KB is INTEGER
The number of superdiagonals of the matrix B if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0.
AB
AB is REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first ka+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
On exit, the transformed matrix X**T*A*X, stored in the same
format as A.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
BB
BB is REAL array, dimension (LDBB,N)
The banded factor S from the split Cholesky factorization of
B, as returned by SPBSTF, 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 REAL 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 REAL array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ssbtrd (character VECT, character UPLO, integer N, integer KD, real, dimension(
ldab, * ) AB, integer LDAB, real, dimension( * ) D, real, dimension( * ) E, real,
dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK, integer INFO)
SSBTRD
Purpose:
SSBTRD reduces a real symmetric band matrix A to symmetric
tridiagonal form T by an orthogonal similarity transformation:
Q**T * A * Q = T.
Parameters
VECT
VECT is CHARACTER*1
= 'N': do not form Q;
= 'V': form Q;
= 'U': update a matrix X, by forming X*Q.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
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 REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
D
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T.
E
E is REAL 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 REAL array, dimension (LDQ,N)
On entry, if VECT = 'U', then Q must contain an N-by-N
matrix X; if VECT = 'N' or 'V', then Q need not be set.
On exit:
if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
if VECT = 'U', Q contains the product X*Q;
if VECT = 'N', the array Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Modified by Linda Kaufman, Bell Labs.
subroutine ssfrk (character TRANSR, character UPLO, character TRANS, integer N, integer K,
real ALPHA, real, dimension( lda, * ) A, integer LDA, real BETA, real, dimension( * ) C)
SSFRK performs a symmetric rank-k operation for matrix in RFP format.
Purpose:
Level 3 BLAS like routine for C in RFP Format.
SSFRK performs one of the symmetric rank--k operations
C := alpha*A*A**T + beta*C,
or
C := alpha*A**T*A + beta*C,
where alpha and beta are real scalars, C is an n--by--n symmetric
matrix and A is an n--by--k matrix in the first case and a k--by--n
matrix in the second case.
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': The Normal Form of RFP A is stored;
= 'T': The Transpose Form of RFP A is stored.
UPLO
UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the array C is to be referenced as
follows:
UPLO = 'U' or 'u' Only the upper triangular part of C
is to be referenced.
UPLO = 'L' or 'l' Only the lower triangular part of C
is to be referenced.
Unchanged on exit.
TRANS
TRANS is CHARACTER*1
On entry, TRANS specifies the operation to be performed as
follows:
TRANS = 'N' or 'n' C := alpha*A*A**T + beta*C.
TRANS = 'T' or 't' C := alpha*A**T*A + beta*C.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the order of the matrix C. N must be
at least zero.
Unchanged on exit.
K
K is INTEGER
On entry with TRANS = 'N' or 'n', K specifies the number
of columns of the matrix A, and on entry with TRANS = 'T'
or 't', K specifies the number of rows of the matrix A. K
must be at least zero.
Unchanged on exit.
ALPHA
ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A
A is REAL 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 REAL
On entry, BETA specifies the scalar beta.
Unchanged on exit.
C
C is REAL array, dimension (NT)
NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
Format. RFP Format is described by TRANSR, UPLO and N.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sspcon (character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * )
IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK,
integer INFO)
SSPCON
Purpose:
SSPCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric packed matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by SSPTRF.
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 REAL 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 SSPTRF, 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 SSPTRF.
ANORM
ANORM is REAL
The 1-norm of the original matrix A.
RCOND
RCOND is REAL
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 REAL array, dimension (2*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sspgst (integer ITYPE, character UPLO, integer N, real, dimension( * ) AP, real,
dimension( * ) BP, integer INFO)
SSPGST
Purpose:
SSPGST reduces a real symmetric-definite generalized eigenproblem
to standard form, using packed storage.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
B must have been previously factorized as U**T*U or L*L**T by SPPTRF.
Parameters
ITYPE
ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T*A*L.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored and B is factored as
U**T*U;
= 'L': Lower triangle of A is stored and B is factored as
L*L**T.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
AP
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.
BP
BP is REAL 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 SPPTRF.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ssprfs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real,
dimension( * ) AFP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer
LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension(
* ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
SSPRFS
Purpose:
SSPRFS 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 REAL 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 REAL 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 SSPTRF, 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 SSPTRF.
B
B is REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SSPTRS.
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 REAL 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 REAL 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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ssptrd (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) D,
real, dimension( * ) E, real, dimension( * ) TAU, integer INFO)
SSPTRD
Purpose:
SSPTRD reduces a real symmetric matrix A stored in packed form to
symmetric tridiagonal form T by an orthogonal similarity
transformation: Q**T * A * Q = T.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.
D
D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E
E is REAL 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 REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).
subroutine ssptrf (character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * )
IPIV, integer INFO)
SSPTRF
Purpose:
SSPTRF computes the factorization of a real symmetric matrix A stored
in packed format using the Bunch-Kaufman diagonal pivoting method:
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is REAL 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 Tennessee
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 ssptri (character UPLO, integer N, real, dimension( * ) AP, integer, dimension( * )
IPIV, real, dimension( * ) WORK, integer INFO)
SSPTRI
Purpose:
SSPTRI computes the inverse of a real symmetric indefinite matrix
A in packed storage using the factorization A = U*D*U**T or
A = L*D*L**T computed by SSPTRF.
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 REAL 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 SSPTRF,
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 SSPTRF.
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ssptrs (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, integer,
dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)
SSPTRS
Purpose:
SSPTRS solves a system of linear equations A*X = B with a real
symmetric matrix A stored in packed format using the factorization
A = U*D*U**T or A = L*D*L**T computed by SSPTRF.
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 REAL 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 SSPTRF, 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 SSPTRF.
B
B is REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sstegr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real,
dimension( * ) E, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real,
dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * )
ISUPPZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer
LIWORK, integer INFO)
SSTEGR
Purpose:
SSTEGR 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.
SSTEGR is a compatibility wrapper around the improved SSTEMR routine.
See SSTEMR for further details.
One important change is that the ABSTOL parameter no longer provides any
benefit and hence is no longer used.
Note : SSTEGR and SSTEMR 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 REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix
T. On exit, D is overwritten.
E
E is REAL 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 REAL
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 REAL
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 REAL
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 REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z
Z is REAL 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 REAL 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 SLARRE,
if INFO = 2X, internal error in SLARRV.
Here, the digit X = ABS( IINFO ) < 10, where IINFO is
the nonzero error code returned by SLARRE or
SLARRV, respectively.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, LBNL/NERSC, USA
subroutine sstein (integer N, real, dimension( * ) D, real, dimension( * ) E, integer M, real,
dimension( * ) W, integer, dimension( * ) IBLOCK, integer, dimension( * ) ISPLIT, real,
dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * )
IWORK, integer, dimension( * ) IFAIL, integer INFO)
SSTEIN
Purpose:
SSTEIN computes the eigenvectors of a real symmetric tridiagonal
matrix T corresponding to specified eigenvalues, using inverse
iteration.
The maximum number of iterations allowed for each eigenvector is
specified by an internal parameter MAXITS (currently set to 5).
Parameters
N
N is INTEGER
The order of the matrix. N >= 0.
D
D is REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E
E is REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix
T, in elements 1 to N-1.
M
M is INTEGER
The number of eigenvectors to be found. 0 <= M <= N.
W
W is REAL 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 SSTEBZ 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 SSTEBZ 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 SSTEBZ is expected here. )
Z
Z is REAL array, dimension (LDZ, M)
The computed eigenvectors. The eigenvector associated
with the eigenvalue W(i) is stored in the i-th column of
Z. Any vector which fails to converge is set to its current
iterate after MAXITS iterations.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
WORK
WORK is REAL array, dimension (5*N)
IWORK
IWORK is INTEGER array, dimension (N)
IFAIL
IFAIL is INTEGER array, dimension (M)
On normal exit, all elements of IFAIL are zero.
If one or more eigenvectors fail to converge after
MAXITS iterations, then their indices are stored in
array IFAIL.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, then i eigenvectors failed to converge
in MAXITS iterations. Their indices are stored in
array IFAIL.
Internal Parameters:
MAXITS INTEGER, default = 5
The maximum number of iterations performed.
EXTRA INTEGER, default = 2
The number of iterations performed after norm growth
criterion is satisfied, should be at least 1.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sstemr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real,
dimension( * ) E, real VL, real VU, integer IL, integer IU, integer M, real, dimension( *
) W, real, dimension( ldz, * ) Z, integer LDZ, integer NZC, integer, dimension( * )
ISUPPZ, logical TRYRAC, real, dimension( * ) WORK, integer LWORK, integer, dimension( * )
IWORK, integer LIWORK, integer INFO)
SSTEMR
Purpose:
SSTEMR 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.SSTEMR works only on machines which follow IEEE-754
floating-point standard in their handling of infinities and NaNs.
This permits the use of efficient inner loops avoiding a check for
zero divisors.
Parameters
JOBZ
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE
RANGE is CHARACTER*1
= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU]
will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.
N
N is INTEGER
The order of the matrix. N >= 0.
D
D is REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix
T. On exit, D is overwritten.
E
E is REAL 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 REAL
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 REAL
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 REAL array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z
Z is REAL 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 REAL 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 SLARRE,
if INFO = 2X, internal error in SLARRV.
Here, the digit X = ABS( IINFO ) < 10, where IINFO is
the nonzero error code returned by SLARRE or
SLARRV, respectively.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
subroutine stbcon (character NORM, character UPLO, character DIAG, integer N, integer KD,
real, dimension( ldab, * ) AB, integer LDAB, real RCOND, real, dimension( * ) WORK,
integer, dimension( * ) IWORK, integer INFO)
STBCON
Purpose:
STBCON 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 REAL 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 REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK
WORK is REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stbrfs (character UPLO, character TRANS, character DIAG, integer N, integer KD,
integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B,
integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real,
dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer
INFO)
STBRFS
Purpose:
STBRFS 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 STBTRS or some other
means before entering this routine. STBRFS does not do iterative
refinement because doing so cannot improve the backward error.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals or subdiagonals of the
triangular band matrix A. KD >= 0.
NRHS
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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stbtrs (character UPLO, character TRANS, character DIAG, integer N, integer KD,
integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B,
integer LDB, integer INFO)
STBTRS
Purpose:
STBTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular band matrix of order N, and B is an
N-by NRHS matrix. A check is made to verify that A is nonsingular.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS
TRANS is CHARACTER*1
Specifies the form the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of superdiagonals or subdiagonals of the
triangular band matrix A. KD >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AB
AB is REAL 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 REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the
solutions X have not been computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stfsm (character TRANSR, character SIDE, character UPLO, character TRANS, character
DIAG, integer M, integer N, real ALPHA, real, dimension( 0: * ) A, real, dimension( 0:
ldb-1, 0: * ) B, integer LDB)
STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
Purpose:
Level 3 BLAS like routine for A in RFP Format.
STFSM solves the matrix equation
op( A )*X = alpha*B or X*op( A ) = alpha*B
where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit, upper or lower triangular matrix and op( A ) is one of
op( A ) = A or op( A ) = A**T.
A is in Rectangular Full Packed (RFP) Format.
The matrix X is overwritten on B.
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': The Normal Form of RFP A is stored;
= 'T': The Transpose Form of RFP A is stored.
SIDE
SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:
SIDE = 'L' or 'l' op( A )*X = alpha*B.
SIDE = 'R' or 'r' X*op( A ) = alpha*B.
Unchanged on exit.
UPLO
UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
UPLO = 'L' or 'l' RFP A came from a lower triangular matrix
Unchanged on exit.
TRANS
TRANS is CHARACTER*1
On entry, TRANS specifies the form of op( A ) to be used
in the matrix multiplication as follows:
TRANS = 'N' or 'n' op( A ) = A.
TRANS = 'T' or 't' op( A ) = A'.
Unchanged on exit.
DIAG
DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:
DIAG = 'U' or 'u' A is assumed to be unit triangular.
DIAG = 'N' or 'n' A is not assumed to be unit
triangular.
Unchanged on exit.
M
M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the number of columns of B. N must be
at least zero.
Unchanged on exit.
ALPHA
ALPHA is REAL
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 REAL array, dimension (NT)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = 'T' then RFP is the transpose of RFP A as
defined when TRANSR = 'N'. The contents of RFP A are defined
by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
elements of upper packed A either in normal or
transpose Format. If UPLO = 'L' the RFP A contains
the NT elements of lower packed A either in normal or
transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.
B
B is REAL array, dimension (LDB,N)
Before entry, the leading m by n part of the array B must
contain the right-hand side matrix B, and on exit is
overwritten by the solution matrix X.
LDB
LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in the calling (sub) program. LDB must be at least
max( 1, m ).
Unchanged on exit.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine stftri (character TRANSR, character UPLO, character DIAG, integer N, real,
dimension( 0: * ) A, integer INFO)
STFTRI
Purpose:
STFTRI computes the inverse of a triangular matrix A stored in RFP
format.
This is a Level 3 BLAS version of the algorithm.
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': The Normal TRANSR of RFP A is stored;
= 'T': The Transpose TRANSR of RFP A is stored.
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (NT);
NT=N*(N+1)/2. On entry, the triangular factor of a Hermitian
Positive Definite matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
the transpose of RFP A as defined when
TRANSR = 'N'. The contents of RFP A are defined by UPLO as
follows: If UPLO = 'U' the RFP A contains the nt elements of
upper packed A; If UPLO = 'L' the RFP A contains the nt
elements of lower packed A. The LDA of RFP A is (N+1)/2 when
TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
even and N is odd. See the Note below for more details.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, A(i,i) is exactly zero. The triangular
matrix is singular and its inverse can not be computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine stfttp (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) ARF,
real, dimension( 0: * ) AP, integer INFO)
STFTTP copies a triangular matrix from the rectangular full packed format (TF) to the
standard packed format (TP).
Purpose:
STFTTP copies a triangular matrix A from rectangular full packed
format (TF) to standard packed format (TP).
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': ARF is in Normal format;
= 'T': ARF is in Transpose format;
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
ARF
ARF is REAL 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 REAL array, dimension ( N*(N+1)/2 ),
On exit, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine stfttr (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) ARF,
real, dimension( 0: lda-1, 0: * ) A, integer LDA, integer INFO)
STFTTR copies a triangular matrix from the rectangular full packed format (TF) to the
standard full format (TR).
Purpose:
STFTTR copies a triangular matrix A from rectangular full packed
format (TF) to standard full format (TR).
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': ARF is in Normal format;
= 'T': ARF is in Transpose format.
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
N
N is INTEGER
The order of the matrices ARF and A. N >= 0.
ARF
ARF is REAL array, dimension (N*(N+1)/2).
On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
matrix A in RFP format. See the 'Notes' below for more
details.
A
A is REAL array, dimension (LDA,N)
On exit, the triangular matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine stgsen (integer IJOB, logical WANTQ, logical WANTZ, logical, dimension( * ) SELECT,
integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
LDB, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA,
real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ,
integer M, real PL, real PR, real, dimension( * ) DIF, real, dimension( * ) WORK, integer
LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
STGSEN
Purpose:
STGSEN reorders the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans-
formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
appears in the leading diagonal blocks of the upper quasi-triangular
matrix A and the upper triangular B. The leading columns of Q and
Z form orthonormal bases of the corresponding left and right eigen-
spaces (deflating subspaces). (A, B) must be in generalized real
Schur canonical form (as returned by SGGES), i.e. A is block upper
triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
triangular.
STGSEN also computes the generalized eigenvalues
w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
of the reordered matrix pair (A, B).
Optionally, STGSEN computes the estimates of reciprocal condition
numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
(A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
between the matrix pairs (A11, B11) and (A22,B22) that correspond to
the selected cluster and the eigenvalues outside the cluster, resp.,
and norms of 'projections' onto left and right eigenspaces w.r.t.
the selected cluster in the (1,1)-block.
Parameters
IJOB
IJOB is INTEGER
Specifies whether condition numbers are required for the
cluster of eigenvalues (PL and PR) or the deflating subspaces
(Difu and Difl):
=0: Only reorder w.r.t. SELECT. No extras.
=1: Reciprocal of norms of 'projections' onto left and right
eigenspaces w.r.t. the selected cluster (PL and PR).
=2: Upper bounds on Difu and Difl. F-norm-based estimate
(DIF(1:2)).
=3: Estimate of Difu and Difl. 1-norm-based estimate
(DIF(1:2)).
About 5 times as expensive as IJOB = 2.
=4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
version to get it all.
=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
WANTQ
WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ
WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
SELECT
SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster.
To select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
N
N is INTEGER
The order of the matrices A and B. N >= 0.
A
A is REAL array, dimension(LDA,N)
On entry, the upper quasi-triangular matrix A, with (A, B) in
generalized real Schur canonical form.
On exit, A is overwritten by the reordered matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is REAL array, dimension(LDB,N)
On entry, the upper triangular matrix B, with (A, B) in
generalized real Schur canonical form.
On exit, B is overwritten by the reordered matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ALPHAR
ALPHAR is REAL array, dimension (N)
ALPHAI
ALPHAI is REAL array, dimension (N)
BETA
BETA is REAL array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
and BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real generalized Schur form of (A,B) were further reduced
to triangular form using complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Q
Q is REAL array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
On exit, Q has been postmultiplied by the left orthogonal
transformation matrix which reorder (A, B); The leading M
columns of Q form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTQ = .FALSE., Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1;
and if WANTQ = .TRUE., LDQ >= N.
Z
Z is REAL array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
On exit, Z has been postmultiplied by the left orthogonal
transformation matrix which reorder (A, B); The leading M
columns of Z form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTZ = .FALSE., Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1;
If WANTZ = .TRUE., LDZ >= N.
M
M is INTEGER
The dimension of the specified pair of left and right eigen-
spaces (deflating subspaces). 0 <= M <= N.
PL
PL is REAL
PR
PR is REAL
If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
reciprocal of the norm of 'projections' onto left and right
eigenspaces with respect to the selected cluster.
0 < PL, PR <= 1.
If M = 0 or M = N, PL = PR = 1.
If IJOB = 0, 2 or 3, PL and PR are not referenced.
DIF
DIF is REAL array, dimension (2).
If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
estimates of Difu and Difl.
If M = 0 or N, DIF(1:2) = F-norm([A, B]).
If IJOB = 0 or 1, DIF is not referenced.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 4*N+16.
If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= 1.
If IJOB = 1, 2 or 4, LIWORK >= N+6.
If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
=0: Successful exit.
<0: If INFO = -i, the i-th argument had an illegal value.
=1: Reordering of (A, B) failed because the transformed
matrix pair (A, B) would be too far from generalized
Schur form; the problem is very ill-conditioned.
(A, B) may have been partially reordered.
If requested, 0 is returned in DIF(*), PL and PR.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
STGSEN first collects the selected eigenvalues by computing
orthogonal U and W that move them to the top left corner of (A, B).
In other words, the selected eigenvalues are the eigenvalues of
(A11, B11) in:
U**T*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) n2
n1 n2 n1 n2
where N = n1+n2 and U**T means the transpose of U. The first n1 columns
of U and W span the specified pair of left and right eigenspaces
(deflating subspaces) of (A, B).
If (A, B) has been obtained from the generalized real Schur
decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
reordered generalized real Schur form of (C, D) is given by
(C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
and the first n1 columns of Q*U and Z*W span the corresponding
deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
Note that if the selected eigenvalue is sufficiently ill-conditioned,
then its value may differ significantly from its value before
reordering.
The reciprocal condition numbers of the left and right eigenspaces
spanned by the first n1 columns of U and W (or Q*U and Z*W) may
be returned in DIF(1:2), corresponding to Difu and Difl, resp.
The Difu and Difl are defined as:
Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
and
Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
where sigma-min(Zu) is the smallest singular value of the
(2*n1*n2)-by-(2*n1*n2) matrix
Zu = [ kron(In2, A11) -kron(A22**T, In1) ]
[ kron(In2, B11) -kron(B22**T, In1) ].
Here, Inx is the identity matrix of size nx and A22**T is the
transpose of A22. kron(X, Y) is the Kronecker product between
the matrices X and Y.
When DIF(2) is small, small changes in (A, B) can cause large changes
in the deflating subspace. An approximate (asymptotic) bound on the
maximum angular error in the computed deflating subspaces is
EPS * norm((A, B)) / DIF(2),
where EPS is the machine precision.
The reciprocal norm of the projectors on the left and right
eigenspaces associated with (A11, B11) may be returned in PL and PR.
They are computed as follows. First we compute L and R so that
P*(A, B)*Q is block diagonal, where
P = ( I -L ) n1 Q = ( I R ) n1
( 0 I ) n2 and ( 0 I ) n2
n1 n2 n1 n2
and (L, R) is the solution to the generalized Sylvester equation
A11*R - L*A22 = -A12
B11*R - L*B22 = -B12
Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is
EPS * norm((A, B)) / PL.
There are also global error bounds which valid for perturbations up
to a certain restriction: A lower bound (x) on the smallest
F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
(i.e. (A + E, B + F), is
x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
An approximate bound on x can be computed from DIF(1:2), PL and PR.
If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
(L', R') and unperturbed (L, R) left and right deflating subspaces
associated with the selected cluster in the (1,1)-blocks can be
bounded as
max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
See LAPACK User's Guide section 4.11 or the following references
for more information.
Note that if the default method for computing the Frobenius-norm-
based estimate DIF is not wanted (see SLATDF), then the parameter
IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
(IJOB = 2 will be used)). See STGSYL for more details.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
87 Umea, Sweden.
References:
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea
University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
1996.
subroutine stgsja (character JOBU, character JOBV, character JOBQ, integer M, integer P,
integer N, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real,
dimension( ldb, * ) B, integer LDB, real TOLA, real TOLB, real, dimension( * ) ALPHA,
real, dimension( * ) BETA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv,
* ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK,
integer NCYCLE, integer INFO)
STGSJA
Purpose:
STGSJA computes the generalized singular value decomposition (GSVD)
of two real upper triangular (or trapezoidal) matrices A and B.
On entry, it is assumed that matrices A and B have the following
forms, which may be obtained by the preprocessing subroutine SGGSVP
from a general M-by-N matrix A and P-by-N matrix B:
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
A = K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
B = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal.
On exit,
U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
where U, V and Q are orthogonal matrices.
R is a nonsingular upper triangular matrix, and D1 and D2 are
``diagonal'' matrices, which are of the following structures:
If M-K-L >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 ) K
L ( 0 0 R22 ) L
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
The computation of the orthogonal transformation matrices U, V or Q
is optional. These matrices may either be formed explicitly, or they
may be postmultiplied into input matrices U1, V1, or Q1.
Parameters
JOBU
JOBU is CHARACTER*1
= 'U': U must contain an orthogonal matrix U1 on entry, and
the product U1*U is returned;
= 'I': U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= 'N': U is not computed.
JOBV
JOBV is CHARACTER*1
= 'V': V must contain an orthogonal matrix V1 on entry, and
the product V1*V is returned;
= 'I': V is initialized to the unit matrix, and the
orthogonal matrix V is returned;
= 'N': V is not computed.
JOBQ
JOBQ is CHARACTER*1
= 'Q': Q must contain an orthogonal matrix Q1 on entry, and
the product Q1*Q is returned;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'N': Q is not computed.
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
K
K is INTEGER
L
L is INTEGER
K and L specify the subblocks in the input matrices A and B:
A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
of A and B, whose GSVD is going to be computed by STGSJA.
See Further Details.
A
A is REAL 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 REAL 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 REAL
TOLB
TOLB is REAL
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)*MACHEPS,
TOLB = max(P,N)*norm(B)*MACHEPS.
ALPHA
ALPHA is REAL array, dimension (N)
BETA
BETA is REAL 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 REAL array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the orthogonal matrix returned by SGGSVP).
On exit,
if JOBU = 'I', U contains the orthogonal matrix U;
if JOBU = 'U', U contains the product U1*U.
If JOBU = 'N', U is not referenced.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V
V is REAL array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the orthogonal matrix returned by SGGSVP).
On exit,
if JOBV = 'I', V contains the orthogonal matrix V;
if JOBV = 'V', V contains the product V1*V.
If JOBV = 'N', V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q
Q is REAL array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the orthogonal matrix returned by SGGSVP).
On exit,
if JOBQ = 'I', Q contains the orthogonal matrix Q;
if JOBQ = 'Q', Q contains the product Q1*Q.
If JOBQ = 'N', Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK
WORK is REAL array, dimension (2*N)
NCYCLE
NCYCLE is INTEGER
The number of cycles required for convergence.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.
Internal Parameters
===================
MAXIT INTEGER
MAXIT specifies the total loops that the iterative procedure
may take. If after MAXIT cycles, the routine fails to
converge, we return INFO = 1..fi
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
matrix B13 to the form:
U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
of Z. C1 and S1 are diagonal matrices satisfying
C1**2 + S1**2 = I,
and R1 is an L-by-L nonsingular upper triangular matrix.
subroutine stgsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N,
real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real,
dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real,
dimension( * ) S, real, dimension( * ) DIF, integer MM, integer M, real, dimension( * )
WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)
STGSNA
Purpose:
STGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B) in
generalized real Schur canonical form (or of any matrix pair
(Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
Z**T denotes the transpose of Z.
(A, B) must be in generalized real Schur form (as returned by SGGES),
i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
blocks. B is upper triangular.
Parameters
JOB
JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (DIF):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S and DIF).
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT
SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the eigenpair corresponding to a real eigenvalue w(j),
SELECT(j) must be set to .TRUE.. To select condition numbers
corresponding to a complex conjugate pair of eigenvalues w(j)
and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N
N is INTEGER
The order of the square matrix pair (A, B). N >= 0.
A
A is REAL array, dimension (LDA,N)
The upper quasi-triangular matrix A in the pair (A,B).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is REAL 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 REAL 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 STGEVC.
If JOB = 'V', VL is not referenced.
LDVL
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1.
If JOB = 'E' or 'B', LDVL >= N.
VR
VR is REAL array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns ov VR, as returned by STGEVC.
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 REAL array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus
S(j), DIF(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = 'V', S is not referenced.
DIF
DIF is REAL array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of DIF are set to the same value. If
the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
is set to 0; this can only occur when the true value would be
very small anyway.
If JOB = 'E', DIF is not referenced.
MM
MM is INTEGER
The number of elements in the arrays S and DIF. MM >= M.
M
M is INTEGER
The number of elements of the arrays S and DIF used to store
the specified condition numbers; for each selected real
eigenvalue one element is used, and for each selected complex
conjugate pair of eigenvalues, two elements are used.
If HOWMNY = 'A', M is set to N.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (N + 6)
If JOB = 'E', IWORK is not referenced.
INFO
INFO is INTEGER
=0: Successful exit
<0: If INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The reciprocal of the condition number of a generalized eigenvalue
w = (a, b) is defined as
S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
where u and v are the left and right eigenvectors of (A, B)
corresponding to w; |z| denotes the absolute value of the complex
number, and norm(u) denotes the 2-norm of the vector u.
The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
of the matrix pair (A, B). If both a and b equal zero, then (A B) is
singular and S(I) = -1 is returned.
An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact
eigenvalue lambda is
chord(w, lambda) <= EPS * norm(A, B) / S(I)
where EPS is the machine precision.
The reciprocal of the condition number DIF(i) of right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows:
a) If the i-th eigenvalue w = (a,b) is real
Suppose U and V are orthogonal transformations such that
U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
( 0 S22 ),( 0 T22 ) n-1
1 n-1 1 n-1
Then the reciprocal condition number DIF(i) is
Difl((a, b), (S22, T22)) = sigma-min( Zl ),
where sigma-min(Zl) denotes the smallest singular value of the
2(n-1)-by-2(n-1) matrix
Zl = [ kron(a, In-1) -kron(1, S22) ]
[ kron(b, In-1) -kron(1, T22) ] .
Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
Kronecker product between the matrices X and Y.
Note that if the default method for computing DIF(i) is wanted
(see SLATDF), then the parameter DIFDRI (see below) should be
changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
See STGSYL for more details.
b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
Suppose U and V are orthogonal transformations such that
U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
( 0 S22 ),( 0 T22) n-2
2 n-2 2 n-2
and (S11, T11) corresponds to the complex conjugate eigenvalue
pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
that
U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
( 0 s22 ) ( 0 t22 )
where the generalized eigenvalues w = s11/t11 and
conjg(w) = s22/t22.
Then the reciprocal condition number DIF(i) is bounded by
min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
Z1 is the complex 2-by-2 matrix
Z1 = [ s11 -s22 ]
[ t11 -t22 ],
This is done by computing (using real arithmetic) the
roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
where Z1**T denotes the transpose of Z1 and det(X) denotes
the determinant of X.
and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ]
[ kron(T11**T, In-2) -kron(I2, T22) ]
Note that if the default method for computing DIF is wanted (see
SLATDF), then the parameter DIFDRI (see below) should be changed
from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
for more details.
For each eigenvalue/vector specified by SELECT, DIF stores a
Frobenius norm-based estimate of Difl.
An approximate error bound for the i-th computed eigenvector VL(i) or
VR(i) is given by
EPS * norm(A, B) / DIF(i).
See ref. [2-3] for more details and further references.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
87 Umea, Sweden.
References:
[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 stpcon (character NORM, character UPLO, character DIAG, integer N, real, dimension(
* ) AP, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer
INFO)
STPCON
Purpose:
STPCON 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 REAL 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 REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK
WORK is REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stpmqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer
L, integer NB, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T,
integer LDT, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
integer LDB, real, dimension( * ) WORK, integer INFO)
STPMQRT
Purpose:
STPMQRT applies a real orthogonal matrix Q obtained from a
'triangular-pentagonal' real block reflector H to a general
real matrix C, which consists of two blocks A and B.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply Q or Q^T from the Left;
= 'R': apply Q or Q^T from the Right.
TRANS
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q^T.
M
M is INTEGER
The number of rows of the matrix B. M >= 0.
N
N is INTEGER
The number of columns of the matrix B. N >= 0.
K
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
L
L is INTEGER
The order of the trapezoidal part of V.
K >= L >= 0. See Further Details.
NB
NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in CTPQRT.
V
V is REAL 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 REAL 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 REAL array, dimension
(LDA,N) if SIDE = 'L' or
(LDA,K) if SIDE = 'R'
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
Q*C or Q^T*C or C*Q or C*Q^T. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDC >= max(1,K);
If SIDE = 'R', LDC >= max(1,M).
B
B is REAL array, dimension (LDB,N)
On entry, the M-by-N matrix B.
On exit, B is overwritten by the corresponding block of
Q*C or Q^T*C or C*Q or C*Q^T. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).
WORK
WORK is REAL array. The dimension of WORK is
N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The columns of the pentagonal matrix V contain the elementary reflectors
H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
trapezoidal block V2:
V = [V1]
[V2].
The size of the trapezoidal block V2 is determined by the parameter L,
where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular;
if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
If SIDE = 'L': C = [A] where A is K-by-N, B is M-by-N and V is M-by-K.
[B]
If SIDE = 'R': C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K.
The real orthogonal matrix Q is formed from V and T.
If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
If TRANS='T' and SIDE='L', C is on exit replaced with Q^T * C.
If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
If TRANS='T' and SIDE='R', C is on exit replaced with C * Q^T.
subroutine stpqrt (integer M, integer N, integer L, integer NB, real, dimension( lda, * ) A,
integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldt, * ) T,
integer LDT, real, dimension( * ) WORK, integer INFO)
STPQRT
Purpose:
STPQRT computes a blocked QR factorization of a real
'triangular-pentagonal' matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.
Parameters
M
M is INTEGER
The number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrix B, and the order of the
triangular matrix A.
N >= 0.
L
L is INTEGER
The number of rows of the upper trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
NB
NB is INTEGER
The block size to be used in the blocked QR. N >= NB >= 1.
A
A is REAL 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 REAL 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 REAL 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 REAL array, dimension (NB*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The input matrix C is a (N+M)-by-N matrix
C = [ A ]
[ B ]
where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:
B = [ B1 ] <- (M-L)-by-N rectangular
[ B2 ] <- L-by-N upper trapezoidal.
The upper trapezoidal matrix B2 consists of the first L rows of a
N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is upper triangular.
The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C
C = [ A ] <- upper triangular N-by-N
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ] <- identity, N-by-N
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
V = [ V1 ] <- (M-L)-by-N rectangular
[ V2 ] <- L-by-N upper trapezoidal.
The columns of V represent the vectors which define the H(i)'s.
The number of blocks is B = ceiling(N/NB), where each
block is of order NB except for the last block, which is of order
IB = N - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T's are stored in the NB-by-N matrix T as
T = [T1 T2 ... TB].
subroutine stpqrt2 (integer M, integer N, integer L, real, dimension( lda, * ) A, integer LDA,
real, dimension( ldb, * ) B, integer LDB, real, dimension( ldt, * ) T, integer LDT,
integer INFO)
STPQRT2 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:
STPQRT2 computes a QR factorization of a real 'triangular-pentagonal'
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.
Parameters
M
M is INTEGER
The total number of rows of the matrix B.
M >= 0.
N
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 REAL 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 REAL 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 REAL array, dimension (LDT,N)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The input matrix C is a (N+M)-by-N matrix
C = [ A ]
[ B ]
where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:
B = [ B1 ] <- (M-L)-by-N rectangular
[ B2 ] <- L-by-N upper trapezoidal.
The upper trapezoidal matrix B2 consists of the first L rows of a
N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is upper triangular.
The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C
C = [ A ] <- upper triangular N-by-N
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ] <- identity, N-by-N
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
V = [ V1 ] <- (M-L)-by-N rectangular
[ V2 ] <- L-by-N upper trapezoidal.
The columns of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by
H = I - W * T * W^H
where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.
subroutine stprfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS,
real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, *
) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( *
) WORK, integer, dimension( * ) IWORK, integer INFO)
STPRFS
Purpose:
STPRFS 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 STPTRS or some other
means before entering this routine. STPRFS does not do iterative
refinement because doing so cannot improve the backward error.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
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 REAL array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array. The j-th column of A is stored in the array
AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
If DIAG = 'U', the diagonal elements of A are not referenced
and are assumed to be 1.
B
B is REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stptri (character UPLO, character DIAG, integer N, real, dimension( * ) AP, integer
INFO)
STPTRI
Purpose:
STPTRI computes the inverse of a real upper or lower triangular
matrix A stored in packed format.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
A triangular matrix A can be transferred to packed storage using one
of the following program segments:
UPLO = 'U': UPLO = 'L':
JC = 1 JC = 1
DO 2 J = 1, N DO 2 J = 1, N
DO 1 I = 1, J DO 1 I = J, N
AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J)
1 CONTINUE 1 CONTINUE
JC = JC + J JC = JC + N - J + 1
2 CONTINUE 2 CONTINUE
subroutine stptrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS,
real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, integer INFO)
STPTRS
Purpose:
STPTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular matrix of order N stored in packed format,
and B is an N-by-NRHS matrix. A check is made to verify that A is
nonsingular.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AP
AP is REAL 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 REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the
solutions X have not been computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stpttf (character TRANSR, character UPLO, integer N, real, dimension( 0: * ) AP,
real, dimension( 0: * ) ARF, integer INFO)
STPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular
full packed format (TF).
Purpose:
STPTTF copies a triangular matrix A from standard packed format (TP)
to rectangular full packed format (TF).
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': ARF in Normal format is wanted;
= 'T': ARF in Conjugate-transpose format is wanted.
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is REAL 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 REAL array, dimension ( N*(N+1)/2 ),
On exit, the upper or lower triangular matrix A stored in
RFP format. For a further discussion see Notes below.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine stpttr (character UPLO, integer N, real, dimension( * ) AP, real, dimension( lda, *
) A, integer LDA, integer INFO)
STPTTR copies a triangular matrix from the standard packed format (TP) to the standard
full format (TR).
Purpose:
STPTTR 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 REAL 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 REAL array, dimension ( LDA, N )
On exit, the triangular matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strcon (character NORM, character UPLO, character DIAG, integer N, real, dimension(
lda, * ) A, integer LDA, real RCOND, real, dimension( * ) WORK, integer, dimension( * )
IWORK, integer INFO)
STRCON
Purpose:
STRCON 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 REAL 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 REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
WORK
WORK is REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer
N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldvl, * ) VL, integer LDVL,
real, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * )
WORK, integer INFO)
STREVC
Purpose:
STREVC computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by SHSEQR.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, (y**H)*T = w*(y**H)
where y**H denotes the conjugate transpose of y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.
This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.
Parameters
SIDE
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.
SELECT
SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be
computed.
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
.FALSE..
Not referenced if HOWMNY = 'A' or 'B'.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = 'R'.
LDVL
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1, and if
SIDE = 'L' or 'B', LDVL >= N.
VR
VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of T;
if HOWMNY = 'B', the matrix Q*X;
if HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = 'L'.
LDVR
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N.
MM
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = 'A' or 'B', M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
WORK
WORK is REAL array, dimension (3*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.
Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.
subroutine strevc3 (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer
N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldvl, * ) VL, integer LDVL,
real, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, real, dimension( * )
WORK, integer LWORK, integer INFO)
STREVC3
Purpose:
STREVC3 computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by SHSEQR.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, (y**T)*T = w*(y**T)
where y**T denotes the transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.
This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.
This uses a Level 3 BLAS version of the back transformation.
Parameters
SIDE
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.
SELECT
SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be
computed.
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
.FALSE..
Not referenced if HOWMNY = 'A' or 'B'.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = 'R'.
LDVL
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
VR
VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of T;
if HOWMNY = 'B', the matrix Q*X;
if HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = 'L'.
LDVR
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
MM
MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = 'A' or 'B', M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
LWORK
LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,3*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.
Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.
subroutine strexc (character COMPQ, integer N, real, dimension( ldt, * ) T, integer LDT, real,
dimension( ldq, * ) Q, integer LDQ, integer IFST, integer ILST, real, dimension( * ) WORK,
integer INFO)
STREXC
Purpose:
STREXC reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
moved to row ILST.
The real Schur form T is reordered by an orthogonal similarity
transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
is updated by postmultiplying it with Z.
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.
Parameters
COMPQ
COMPQ is CHARACTER*1
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
N
N is INTEGER
The order of the matrix T. N >= 0.
If N == 0 arguments ILST and IFST may be any value.
T
T is REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
Schur canonical form.
On exit, the reordered upper quasi-triangular matrix, again
in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q
Q is REAL array, dimension (LDQ,N)
On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
On exit, if COMPQ = 'V', Q has been postmultiplied by the
orthogonal transformation matrix Z which reorders T.
If COMPQ = 'N', Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1, and if
COMPQ = 'V', LDQ >= max(1,N).
IFST
IFST is INTEGER
ILST
ILST is INTEGER
Specify the reordering of the diagonal blocks of T.
The block with row index IFST is moved to row ILST, by a
sequence of transpositions between adjacent blocks.
On exit, if IFST pointed on entry to the second row of a
2-by-2 block, it is changed to point to the first row; ILST
always points to the first row of the block in its final
position (which may differ from its input value by +1 or -1).
1 <= IFST <= N; 1 <= ILST <= N.
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: two adjacent blocks were too close to swap (the problem
is very ill-conditioned); T may have been partially
reordered, and ILST points to the first row of the
current position of the block being moved.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strrfs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS,
real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real,
dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR,
real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
STRRFS
Purpose:
STRRFS 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 STRTRS or some other
means before entering this routine. STRRFS does not do iterative
refinement because doing so cannot improve the backward error.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (3*N)
IWORK
IWORK is INTEGER array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strsen (character JOB, character COMPQ, logical, dimension( * ) SELECT, integer N,
real, dimension( ldt, * ) T, integer LDT, real, dimension( ldq, * ) Q, integer LDQ, real,
dimension( * ) WR, real, dimension( * ) WI, integer M, real S, real SEP, real, dimension(
* ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
STRSEN
Purpose:
STRSEN reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
the leading diagonal blocks of the upper quasi-triangular matrix T,
and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.
Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.
Parameters
JOB
JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= 'N': none;
= 'E': for eigenvalues only (S);
= 'V': for invariant subspace only (SEP);
= 'B': for both eigenvalues and invariant subspace (S and
SEP).
COMPQ
COMPQ is CHARACTER*1
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
SELECT
SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, T is overwritten by the reordered matrix T, again in
Schur canonical form, with the selected eigenvalues in the
leading diagonal blocks.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q
Q is REAL array, dimension (LDQ,N)
On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
On exit, if COMPQ = 'V', Q has been postmultiplied by the
orthogonal transformation matrix which reorders T; the
leading M columns of Q form an orthonormal basis for the
specified invariant subspace.
If COMPQ = 'N', Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
WR
WR is REAL array, dimension (N)
WI
WI is REAL array, dimension (N)
The real and imaginary parts, respectively, of the reordered
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T, with WR(i) = T(i,i) and, if
T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
WI(i+1) = -WI(i). Note that if a complex eigenvalue is
sufficiently ill-conditioned, then its value may differ
significantly from its value before reordering.
M
M is INTEGER
The dimension of the specified invariant subspace.
0 < = M <= N.
S
S is REAL
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 REAL
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 REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
If JOB = 'N', LWORK >= max(1,N);
if JOB = 'E', LWORK >= max(1,M*(N-M));
if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If JOB = 'N' or 'E', LIWORK >= 1;
if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very ill-conditioned);
T may have been partially reordered, and WR and WI
contain the eigenvalues in the same order as in T; S and
SEP (if requested) are set to zero.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
STRSEN first collects the selected eigenvalues by computing an
orthogonal transformation Z to move them to the top left corner of T.
In other words, the selected eigenvalues are the eigenvalues of T11
in:
Z**T * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2
where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.
If T has been obtained from the real Schur factorization of a matrix
A = Q*T*Q**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
the corresponding invariant subspace of A.
The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that
P = ( I R ) n1
( 0 0 ) n2
n1 n2
is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:
T11*R - R*T22 = T12.
Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound
(1 + F-norm(R)**2)**(-1/2)
on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).
An approximate error bound for the computed average of the
eigenvalues of T11 is
EPS * norm(T) / S
where EPS is the machine precision.
The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:
sep( T11, T22 ) = sigma-min( C )
where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix
C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is
EPS * norm(T) / SEP
subroutine strsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N,
real, dimension( ldt, * ) T, integer LDT, real, dimension( ldvl, * ) VL, integer LDVL,
real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) S, real, dimension( * )
SEP, integer MM, integer M, real, dimension( ldwork, * ) WORK, integer LDWORK, integer,
dimension( * ) IWORK, integer INFO)
STRSNA
Purpose:
STRSNA estimates reciprocal condition numbers for specified
eigenvalues and/or right eigenvectors of a real upper
quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
orthogonal).
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.
Parameters
JOB
JOB is CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (SEP):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (SEP);
= 'B': for both eigenvalues and eigenvectors (S and SEP).
HOWMNY
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT
SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the eigenpair corresponding to a real eigenvalue w(j),
SELECT(j) must be set to .TRUE.. To select condition numbers
corresponding to a complex conjugate pair of eigenvalues w(j)
and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is REAL array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvectors of T
(or of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VL, as returned by
SHSEIN or STREVC.
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 REAL array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of T
(or of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VR, as returned by
SHSEIN or STREVC.
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 REAL array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus
S(j), SEP(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = 'V', S is not referenced.
SEP
SEP is REAL array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of SEP are set to the same value. If
the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
is set to 0; this can only occur when the true value would be
very small anyway.
If JOB = 'E', SEP is not referenced.
MM
MM is INTEGER
The number of elements in the arrays S (if JOB = 'E' or 'B')
and/or SEP (if JOB = 'V' or 'B'). MM >= M.
M
M is INTEGER
The number of elements of the arrays S and/or SEP actually
used to store the estimated condition numbers.
If HOWMNY = 'A', M is set to N.
WORK
WORK is REAL array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
IWORK
IWORK is INTEGER array, dimension (2*(N-1))
If JOB = 'E', IWORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The reciprocal of the condition number of an eigenvalue lambda is
defined as
S(lambda) = |v**T*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**T denotes the transpose of v, and norm(u)
denotes the Euclidean norm. These reciprocal condition numbers always
lie between zero (very badly conditioned) and one (very well
conditioned). If n = 1, S(lambda) is defined to be 1.
An approximate error bound for a computed eigenvalue W(i) is given by
EPS * norm(T) / S(i)
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose
T = ( lambda c )
( 0 T22 )
Then the reciprocal condition number is
SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
where sigma-min denotes the smallest singular value. We approximate
the smallest singular value by the reciprocal of an estimate of the
one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
defined to be abs(T(1,1)).
An approximate error bound for a computed right eigenvector VR(i)
is given by
EPS * norm(T) / SEP(i)
subroutine strti2 (character UPLO, character DIAG, integer N, real, dimension( lda, * ) A,
integer LDA, integer INFO)
STRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
Purpose:
STRTI2 computes the inverse of a real upper or lower triangular
matrix.
This is the Level 2 BLAS version of the algorithm.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = 'U', the
leading n by n upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If DIAG = 'U', the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strtri (character UPLO, character DIAG, integer N, real, dimension( lda, * ) A,
integer LDA, integer INFO)
STRTRI
Purpose:
STRTRI computes the inverse of a real upper or lower triangular
matrix A.
This is the Level 3 BLAS version of the algorithm.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strtrs (character UPLO, character TRANS, character DIAG, integer N, integer NRHS,
real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB,
integer INFO)
STRTRS
Purpose:
STRTRS solves a triangular system of the form
A * X = B or A**T * X = B,
where A is a triangular matrix of order N, and B is an N-by-NRHS
matrix. A check is made to verify that A is nonsingular.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
DIAG
DIAG is CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N
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 REAL 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 REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, if INFO = 0, the solution matrix X.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the solutions
X have not been computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine strttf (character TRANSR, character UPLO, integer N, real, dimension( 0: lda-1, 0:
* ) A, integer LDA, real, dimension( 0: * ) ARF, integer INFO)
STRTTF copies a triangular matrix from the standard full format (TR) to the rectangular
full packed format (TF).
Purpose:
STRTTF copies a triangular matrix A from standard full format (TR)
to rectangular full packed format (TF) .
Parameters
TRANSR
TRANSR is CHARACTER*1
= 'N': ARF in Normal form is wanted;
= 'T': ARF in Transpose form is wanted.
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL 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 REAL array, dimension (NT).
NT=N*(N+1)/2. On exit, the triangular matrix A in RFP format.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.
RFP A RFP A
03 04 05 33 43 53
13 14 15 00 44 54
23 24 25 10 11 55
33 34 35 20 21 22
00 44 45 30 31 32
01 11 55 40 41 42
02 12 22 50 51 52
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
03 13 23 33 00 01 02 33 00 10 20 30 40 50
04 14 24 34 44 11 12 43 44 11 21 31 41 51
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.
RFP A RFP A
02 03 04 00 33 43
12 13 14 10 11 44
22 23 24 20 21 22
00 33 34 30 31 32
01 11 44 40 41 42
Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:
RFP A RFP A
02 12 22 00 01 00 10 20 30 40 50
03 13 23 33 11 33 11 21 31 41 51
04 14 24 34 44 43 44 22 32 42 52
subroutine strttp (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real,
dimension( * ) AP, integer INFO)
STRTTP copies a triangular matrix from the standard full format (TR) to the standard
packed format (TP).
Purpose:
STRTTP 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 REAL array, dimension (LDA,N)
On exit, the triangular matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AP
AP is REAL array, dimension (N*(N+1)/2)
On exit, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine stzrzf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real,
dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)
STZRZF
Purpose:
STZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
to upper triangular form by means of orthogonal transformations.
The upper trapezoidal matrix A is factored as
A = ( R 0 ) * Z,
where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
triangular matrix.
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= M.
A
A is REAL array, dimension (LDA,N)
On entry, the leading M-by-N upper trapezoidal part of the
array A must contain the matrix to be factorized.
On exit, the leading M-by-M upper triangular part of A
contains the upper triangular matrix R, and elements M+1 to
N of the first M rows of A, with the array TAU, represent the
orthogonal matrix Z as a product of M elementary reflectors.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is REAL array, dimension (M)
The scalar factors of the elementary reflectors.
WORK
WORK is REAL 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
The N-by-N matrix Z can be computed by
Z = Z(1)*Z(2)* ... *Z(M)
where each N-by-N Z(k) is given by
Z(k) = I - tau(k)*v(k)*v(k)**T
with v(k) is the kth row vector of the M-by-N matrix
V = ( I A(:,M+1:N) )
I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from STZRZF,
and tau(k) is the kth element of the array TAU.
Author
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