## table of contents

realOTHERcomputational(3) | LAPACK | realOTHERcomputational(3) |

# 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 vector. 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 vector. 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:**

**Author**

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 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 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 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 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

**Contributors:**

**Bugs, Examples and Comments:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**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 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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 = ( 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 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
vector.

**Purpose:**

SLARSCL2 performs a reciprocal diagonal scaling on an vector:

x <-- inv(D) * x

where the diagonal matrix D is stored as a vector.

Eventually to be replaced by BLAS_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 vector X to be scaled by D.

On exit, the scaled vector.

*LDX*

LDX is INTEGER

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

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

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

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

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

array is not used.

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

______V_____

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

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

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

( v1 v2 v3 )

. . .

. . .

1 . .

1 .

1

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

______V_____

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

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

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

. . .

( v1 v2 v3 )

( v1 v2 v3 )

V = ( v1 v2 v3 )

( v1 v2 v3 )

( v1 v2 v3 )

## subroutine slascl2 (integer M, integer N, real, dimension( * ) D, real, dimension( ldx, * ) X, integer LDX)¶

**SLASCL2** performs diagonal scaling on a vector.

**Purpose:**

SLASCL2 performs a diagonal scaling on a vector:

x <-- D * x

where the diagonal matrix D is stored as a vector.

Eventually to be replaced by BLAS_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 vector X to be scaled by D.

On exit, the scaled vector.

*LDX*

LDX is INTEGER

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

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

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

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

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

Z( k ) = ( I 0 ),

( 0 T( k ) )

where

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

## 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 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:**

## 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)

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 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:**

## 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 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:**

## 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 (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.

*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 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:**

## 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 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 columns of Q must be orthonormal.

If the projection is zero according to Kahan's "twice is enough"

criterion, then the zero vector is returned.

**Parameters**

*M1*

M1 is INTEGER

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

*M2*

M2 is INTEGER

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

*N*

N is INTEGER

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

*X1*

X1 is 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 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:**

**Author**

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:**

**Author**

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**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 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 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 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 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 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

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

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

On entry: On exit:

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

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

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

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

On entry: On exit:

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

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

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

Array elements marked * are not used by the routine.

**Contributors:**

## subroutine 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Jim Demmel, University of California, Berkeley, USA

Inderjit Dhillon, University of Texas, Austin, USA

Osni Marques, LBNL/NERSC, USA

Christof Voemel, University of California, Berkeley, USA

## subroutine 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 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 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 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 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 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 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 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 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:**

**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 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:**

**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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

The N-by-N matrix Z can be computed by

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

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**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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