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CGGSVD(3F)							    CGGSVD(3F)

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

SYNOPSIS
     SUBROUTINE CGGSVD( 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( * )

	 REAL		ALPHA( * ), BETA( * ), RWORK( * )

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

PURPOSE
     CGGSVD 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.

									Page 1

CGGSVD(3F)							    CGGSVD(3F)

       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.

									Page 2

CGGSVD(3F)							    CGGSVD(3F)

     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 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 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) 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) COMPLEX array, dimension (LDU,M)
	     If JOBU = 'U', U contains the M-by-M unitary matrix U.  If JOBU =
	     'N', U is not referenced.

									Page 3

CGGSVD(3F)							    CGGSVD(3F)

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

     V	     (output) COMPLEX 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 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 array, dimension (max(3*N,M,P)+N)

     RWORK   (workspace) REAL array, dimension (2*N)

     IWORK   (workspace) INTEGER array, dimension (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 CTGSJA.

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.

									Page 4

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