ZGGSVP(l)								 )								 ZGGSVP(l)

NAME

       ZGGSVP - compute unitary 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 ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, INFO )

	   CHARACTER	  JOBQ, JOBU, JOBV

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

	   DOUBLE	  PRECISION TOLA, TOLB

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION RWORK( * )

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

PURPOSE

       ZGGSVP computes unitary matrices U, V and Q such that N-K-L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;		     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  conjugate
       transpose of Z.

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

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.

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

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

       A       (input/output) COMPLEX*16 array, dimension (LDA,N)
	       On entry, the M-by-N matrix A.  On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section.

       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 the triangular matrix described in the Purpose section.

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

       TOLA    (input) DOUBLE PRECISION
	       TOLB	(input) DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective numerical rank of matrix B and a sub-
	       block of A. 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.

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

       U       (output) COMPLEX*16 array, dimension (LDU,M)
	       If JOBU = 'U', U contains the 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,M)
	       If JOBV = 'V', V contains the 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 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.

       IWORK   (workspace) INTEGER array, dimension (N)

       RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)

       TAU     (workspace) COMPLEX*16 array, dimension (N)

       WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P))

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

       The subroutine uses LAPACK subroutine ZGEQPF 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.

LAPACK version 3.0						   15 June 2000 							 ZGGSVP(l)