dggsvp (l) - Linux Manuals

dggsvp: computes orthogonal matrices U, V and Q such that N-K-L K L UaqAQ = K ( 0 A12 A13 ) if M-K-L >= 0

Command to display dggsvp manual in Linux: $ man l dggsvp

NAME

DGGSVP - computes orthogonal matrices U, V and Q such that N-K-L K L Uaq*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0

SYNOPSIS

SUBROUTINE DGGSVP(: JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO )
: CHARACTER JOBQ, JOBU, JOBV
: INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
: DOUBLE PRECISION TOLA, TOLB
: INTEGER IWORK( * )
: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )

PURPOSE

DGGSVP computes orthogonal matrices U, V and Q such that
        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

Vaq*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 (Aaq,Baq)aq. Zaq denotes the transpose of Z.
This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine DGGSVD.

ARGUMENTS

JOBU (input) CHARACTER*1: = aqUaq: Orthogonal matrix U is computed;
= aqNaq: U is not computed.
JOBV (input) CHARACTER*1: = aqVaq: Orthogonal matrix V is computed;
= aqNaq: V is not computed.
JOBQ (input) CHARACTER*1: = aqQaq: Orthogonal matrix Q is computed;
= aqNaq: Q is not computed.
M (input) INTEGER: The number of rows of the matrix A. M >= 0.
P (input) INTEGER: The number of rows of the matrix B. P >= 0.
N (input) INTEGER: The number of columns of the matrices A and B. N >= 0.
A (input/output) DOUBLE PRECISION 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 (input) INTEGER: The leading dimension of the array A. LDA >= max(1,M).
B (input/output) DOUBLE PRECISION 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 (input) INTEGER: The leading dimension of the array B. LDB >= max(1,P).
TOLA (input) DOUBLE PRECISION: TOLB (input) DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition.
K (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 (Aaq,Baq)aq.
U (output) DOUBLE PRECISION array, dimension (LDU,M): If JOBU = aqUaq, U contains the orthogonal matrix U. If JOBU = aqNaq, U is not referenced.
LDU (input) INTEGER: The leading dimension of the array U. LDU >= max(1,M) if JOBU = aqUaq; LDU >= 1 otherwise.
V (output) DOUBLE PRECISION array, dimension (LDV,P): If JOBV = aqVaq, V contains the orthogonal matrix V. If JOBV = aqNaq, V is not referenced.
LDV (input) INTEGER: The leading dimension of the array V. LDV >= max(1,P) if JOBV = aqVaq; LDV >= 1 otherwise.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N): If JOBQ = aqQaq, Q contains the orthogonal matrix Q. If JOBQ = aqNaq, Q is not referenced.
LDQ (input) INTEGER: The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = aqQaq; LDQ >= 1 otherwise.
IWORK (workspace) INTEGER array, dimension (N)
TAU (workspace) DOUBLE PRECISION array, dimension (N)
WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))
INFO (output) INTEGER: = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

The subroutine uses LAPACK subroutine DGEQPF 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.