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CGGSVD(1)	      LAPACK driver routine (version 3.2)	     CGGSVD(1)

NAME
       CGGSVD  -  computes 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.
	 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 array, dimension (LDA,N)
	       On  entry, the M-by-N matrix A.	On exit, A contains the trian‐
	       gular 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.

       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/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 con‐
	       verge.  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.  Further Details =============== 2-96	 Based
	       on  modifications  by  Ming  Gu	and Huan Ren, Computer Science
	       Division, University of California at Berkeley, USA

 LAPACK driver routine (version 3November 2008			     CGGSVD(1)
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