NAME

stgsja - compute the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B

SYNOPSIS

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

  SUBROUTINE STGSJA_64( 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)
  CHARACTER * 1 JOBU, JOBV, JOBQ
  INTEGER*8 M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
  REAL TOLA, TOLB
  REAL A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(*)

F95 INTERFACE

  SUBROUTINE TGSJA( 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])
  CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ
  INTEGER :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
  REAL :: TOLA, TOLB
  REAL, DIMENSION(:) :: ALPHA, BETA, WORK
  REAL, DIMENSION(:,:) :: A, B, U, V, Q

  SUBROUTINE TGSJA_64( 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])
  CHARACTER(LEN=1) :: JOBU, JOBV, JOBQ
  INTEGER(8) :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
  REAL :: TOLA, TOLB
  REAL, DIMENSION(:) :: ALPHA, BETA, WORK
  REAL, DIMENSION(:,:) :: A, B, U, V, Q

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

void stgsja_64(char jobu, char jobv, char jobq, long m, long p, long n, long k, long l, float *a, long lda, float *b, long ldb, float tola, float tolb, float *alpha, float *beta, float *u, long ldu, float *v, long ldv, float *q, long ldq, long *ncycle, long *info);

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'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),

where U, V and Q are orthogonal matrices, Z' denotes the transpose of Z, 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

          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.

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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,

where U1, V1 and Q1 are orthogonal matrix, and Z' 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.

ARGUMENTS

JOBU (input)

 = '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 (input)

 = '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 (input)

 = '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 (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.
K (input)
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.
L (input)
See the description of K.
A (input/output)
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 (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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains a part of R. See Purpose for details.
LDB (input)
The leading dimension of the array B. LDB > = max(1,P).
TOLA (input)
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.
TOLB (input)
See the description of TOLA.
ALPHA (output)
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.
BETA (output)
See the description of ALPHA.
U (input/output)
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 (input)
The leading dimension of the array U. LDU > = max(1,M) if JOBU = 'U'; LDU > = 1 otherwise.
V (input/output)
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 (input)
The leading dimension of the array V. LDV > = max(1,P) if JOBV = 'V'; LDV > = 1 otherwise.
Q (input/output)
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 (input)
The leading dimension of the array Q. LDQ > = max(1,N) if JOBQ = 'Q'; LDQ > = 1 otherwise.
WORK (workspace)
dimension(2*N)
NCYCLE (output)
The number of cycles required for convergence.

INFO (output)

 = 0:  successful exit

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

 = 1:  the procedure does not converge after MAXIT cycles.

NAME

SYNOPSIS

F95 INTERFACE

C INTERFACE

PURPOSE

ARGUMENTS