ZGGSVD - compute the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B

SUBROUTINE ZGGSVD(

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

CHARACTER JOBQ, JOBU, JOBV

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

INTEGER IWORK( * )

DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )

COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

ZGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )

where U, V and Q are unitary matrices, and Z' means the conjugate 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 unitary

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 orthnormal 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, and 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) )

JOBU (input) CHARACTER*1

= 'U': Unitary matrix U is computed;

= 'N': U is not computed.

JOBV (input) CHARACTER*1

= 'V': Unitary matrix V is computed;

= 'N': V is not computed.

JOBQ (input) CHARACTER*1

= 'Q': Unitary 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 Purpose. K + L = effective numerical rank of (A',B')'.

A (input/output) COMPLEX*16 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) COMPLEX*16 array, dimension (LDB,N)

On entry, the P-by-N matrix B. On exit, B contains part of 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) DOUBLE PRECISION array, dimension (N)

BETA (output) DOUBLE PRECISION 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) COMPLEX*16 array, dimension (LDU,M)

If JOBU = 'U', U contains the M-by-M unitary 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) COMPLEX*16 array, dimension (LDV,P)

If JOBV = 'V', V contains the P-by-P unitary 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) COMPLEX*16 array, dimension (LDQ,N)

If JOBQ = 'Q', Q contains the N-by-N unitary 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) COMPLEX*16 array, dimension (max(3*N,M,P)+N)

RWORK (workspace) DOUBLE PRECISION array, dimension (2*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 ZTGSJA.

TOLA DOUBLE PRECISION

TOLB DOUBLE PRECISION 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)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. 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