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.TH SGGSVP l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
SGGSVP - compute orthogonal matrices U, V and Q such that N-K-L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0
.SH SYNOPSIS
.TP 19
SUBROUTINE SGGSVP(
JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
IWORK, TAU, WORK, INFO )
.TP 19
.ti +4
CHARACTER
JOBQ, JOBU, JOBV
.TP 19
.ti +4
INTEGER
INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
.TP 19
.ti +4
REAL
TOLA, TOLB
.TP 19
.ti +4
INTEGER
IWORK( * )
.TP 19
.ti +4
REAL
A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
.SH PURPOSE
SGGSVP computes orthogonal matrices U, V and Q such that N-K-L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; L ( 0 0 A23 )
.br
M-K-L ( 0 0 0 )
.br
N-K-L K L
.br
= K ( 0 A12 A13 ) if M-K-L < 0;
.br
M-K ( 0 0 A23 )
.br
N-K-L K L
.br
V'*B*Q = L ( 0 0 B13 )
.br
P-L ( 0 0 0 )
.br
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',B')'. Z' denotes the
transpose of Z.
.br
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
SGGSVD.
.br
.SH ARGUMENTS
.TP 8
JOBU (input) CHARACTER*1
= 'U': Orthogonal matrix U is computed;
.br
= 'N': U is not computed.
.TP 8
JOBV (input) CHARACTER*1
.br
= 'V': Orthogonal matrix V is computed;
.br
= 'N': V is not computed.
.TP 8
JOBQ (input) CHARACTER*1
.br
= 'Q': Orthogonal matrix Q is computed;
.br
= 'N': Q is not computed.
.TP 8
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
.TP 8
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
.TP 8
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
.TP 8
A (input/output) 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.
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
.TP 8
B (input/output) 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.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
.TP 8
TOLA (input) REAL
TOLB (input) 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.
.TP 8
K (output) INTEGER
L (output) INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A',B')'.
.TP 8
U (output) REAL array, dimension (LDU,M)
If JOBU = 'U', U contains the orthogonal matrix U.
If JOBU = 'N', U is not referenced.
.TP 8
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
.TP 8
V (output) REAL array, dimension (LDV,M)
If JOBV = 'V', V contains the orthogonal matrix V.
If JOBV = 'N', V is not referenced.
.TP 8
LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
.TP 8
Q (output) REAL array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.
.TP 8
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
.TP 8
IWORK (workspace) INTEGER array, dimension (N)
.TP 8
TAU (workspace) REAL array, dimension (N)
.TP 8
WORK (workspace) REAL array, dimension (max(3*N,M,P))
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.SH FURTHER DETAILS
The subroutine uses LAPACK subroutine SGEQPF 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.
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