SYNOPSIS

SUBROUTINE SGGSVD(

JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO )

    

CHARACTER JOBQ, JOBU, JOBV

    

INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

    

INTEGER IWORK( * )

    

REAL A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

PURPOSE

SGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B:
    U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the effective numerical rank of the matrix (A',B')', then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respectively:
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 )

            L (  0    0   R22 )
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.

  (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 routine computes C, S, R, and optionally the orthogonal transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B):

                     A*inv(B) = U*(D1*inv(D2))*V'.
If ( A',B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
                     A'*A x = lambda* B'*B x.
In some literature, the GSVD of A and B is presented in the form
                 U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, D1 and D2 are ``diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as

                     X = Q*( I   0    )

                           ( 0 inv(R) ).

ARGUMENTS

JOBU (input) CHARACTER*1

= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.

JOBV (input) CHARACTER*1

= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.

JOBQ (input) CHARACTER*1

= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.

M (input) INTEGER

The number of rows of the matrix A. M >= 0.

N (input) INTEGER

The number of columns of the matrices A and B. N >= 0.

P (input) INTEGER

The number of rows of the matrix B. P >= 0.

K (output) INTEGER

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

A (input/output) REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, A contains the triangular matrix R, or part of R. See Purpose for details.

LDA (input) INTEGER

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

B (input/output) REAL array, dimension (LDB,N)

On entry, the P-by-N matrix B. On exit, B contains the triangular matrix R if M-K-L < 0. See Purpose for details.

LDB (input) INTEGER

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

ALPHA (output) REAL array, dimension (N)

BETA (output) 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) = C,
BETA(K+1:K+L) = 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 and ALPHA(K+L+1:N) = 0
BETA(K+L+1:N) = 0

U (output) REAL array, dimension (LDU,M)

If JOBU = 'U', U contains the M-by-M orthogonal matrix U. If JOBU = 'N', U is not referenced.

LDU (input) INTEGER

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

V (output) REAL array, dimension (LDV,P)

If JOBV = 'V', V contains the P-by-P orthogonal matrix V. If JOBV = 'N', V is not referenced.

LDV (input) INTEGER

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

Q (output) REAL array, dimension (LDQ,N)

If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. If JOBQ = 'N', Q is not referenced.

LDQ (input) INTEGER

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

WORK (workspace) REAL array,

dimension (max(3*N,M,P)+N)

IWORK (workspace/output) INTEGER array, dimension (N)

On exit, IWORK stores the sorting information. More precisely, the following loop will sort ALPHA for I = K+1, min(M,K+L) swap ALPHA(I) and ALPHA(IWORK(I)) endfor such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine STGSJA.

PARAMETERS

TOLA REAL

TOLB REAL TOLA and TOLB are the thresholds to determine the effective rank of (A',B')'. 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. Further Details =============== 2-96 Based on modifications by Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA