ZGGSVD(l)								 )								 ZGGSVD(l)

NAME

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

SYNOPSIS

       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( * )

PURPOSE

       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) )

ARGUMENTS

       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.

PARAMETERS

       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

LAPACK version 3.0						   15 June 2000 							 ZGGSVD(l)