• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS
• FURTHER DETAILS

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

SYNOPSIS

  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)
  CHARACTER * 1 JOBU, JOBV, JOBQ
  INTEGER M, P, N, LDA, LDB, K, L, LDU, LDV, LDQ, INFO
  INTEGER IWORK(*)
  REAL TOLA, TOLB
  REAL A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), TAU(*), WORK(*)

  SUBROUTINE SGGSVP_64( 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 * 1 JOBU, JOBV, JOBQ
  INTEGER*8 M, P, N, LDA, LDB, K, L, LDU, LDV, LDQ, INFO
  INTEGER*8 IWORK(*)
  REAL TOLA, TOLB
  REAL A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), TAU(*), WORK(*)

F95 INTERFACE

  SUBROUTINE GGSVP( 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(LEN=1) :: JOBU, JOBV, JOBQ
  INTEGER :: M, P, N, LDA, LDB, K, L, LDU, LDV, LDQ, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL :: TOLA, TOLB
  REAL, DIMENSION(:) :: TAU, WORK
  REAL, DIMENSION(:,:) :: A, B, U, V, Q

  SUBROUTINE GGSVP_64( 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(LEN=1) :: JOBU, JOBV, JOBQ
  INTEGER(8) :: M, P, N, LDA, LDB, K, L, LDU, LDV, LDQ, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL :: TOLA, TOLB
  REAL, DIMENSION(:) :: TAU, WORK
  REAL, DIMENSION(:,:) :: A, B, U, V, Q

C INTERFACE

#include <sunperf.h>

void sggsvp(char jobu, char jobv, char jobq, int m, int p, int n, float *a, int lda, float *b, int ldb, float tola, float tolb, int *k, int *l, float *u, int ldu, float *v, int ldv, float *q, int ldq, int *info);

void sggsvp_64(char jobu, char jobv, char jobq, long m, long p, long n, float *a, long lda, float *b, long ldb, float tola, float tolb, long *k, long *l, float *u, long ldu, float *v, long ldv, float *q, long ldq, long *info);

PURPOSE

sggsvp 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

 V'*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',B')'. Z' denotes the transpose of Z.

This decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see subroutine SGGSVD.

ARGUMENTS

• JOBU (input)
  

   = 'U':  Orthogonal matrix U is computed;

   = 'N':  U is not computed.

• JOBV (input)
  

   = 'V':  Orthogonal matrix V is computed;

   = 'N':  V is not computed.

• JOBQ (input)
  

   = 'Q':  Orthogonal matrix Q is computed;

   = 'N':  Q is not computed.

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

• P (input)
  The number of rows of the matrix B. P > = 0.

• N (input)
  The number of columns of the matrices A and B. N > = 0.

• A (input/output)
  On entry, the M-by-N matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section.

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

• B (input/output)
  On entry, the P-by-N matrix B. On exit, B contains the triangular matrix described in the Purpose section.

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

• TOLA (input)
  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.

• TOLB (input)
  See the description of TOLA.

• K (output)
  On exit, K and L specify the dimension of the subblocks described in Purpose. K + L = effective numerical rank of (A',B')'.

• L (output)
  See the description of K.

• U (output)
  If JOBU = 'U', U contains the orthogonal matrix U. If JOBU = 'N', U is not referenced.

• LDU (input)
  The leading dimension of the array U. LDU > = max(1,M) if JOBU = 'U'; LDU > = 1 otherwise.

• V (output)
  If JOBV = 'V', V contains the orthogonal matrix V. If JOBV = 'N', V is not referenced.

• LDV (input)
  The leading dimension of the array V. LDV > = max(1,P) if JOBV = 'V'; LDV > = 1 otherwise.

• Q (output)
  If JOBQ = 'Q', Q contains the orthogonal matrix Q. If JOBQ = 'N', Q is not referenced.

• LDQ (input)
  The leading dimension of the array Q. LDQ > = max(1,N) if JOBQ = 'Q'; LDQ > = 1 otherwise.

• IWORK (workspace)
  dimension(N)

• TAU (workspace)
  dimension(N)

• WORK (workspace)
  dimension(MAX(3*N,M,P))

• INFO (output)
  

   = 0:  successful exit

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

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.