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cggsvd.f(3)				      LAPACK				      cggsvd.f(3)

NAME
       cggsvd.f -

SYNOPSIS
   Functions/Subroutines
       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)
	    CGGSVD computes the singular value decomposition (SVD) for OTHER matrices

Function/Subroutine Documentation
   subroutine cggsvd (characterJOBU, characterJOBV, characterJOBQ, integerM, integerN, integerP,
       integerK, integerL, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, *
       )B, integerLDB, real, dimension( * )ALPHA, real, dimension( * )BETA, complex, dimension(
       ldu, * )U, integerLDU, complex, dimension( ldv, * )V, integerLDV, complex, dimension( ldq,
       * )Q, integerLDQ, complex, dimension( * )WORK, real, dimension( * )RWORK, integer,
       dimension( * )IWORK, integerINFO)
	CGGSVD computes the singular value decomposition (SVD) for OTHER matrices

       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**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )

	    where U, V and Q are unitary matrices.
	    Let K+L = the effective numerical rank of the
	    matrix (A**H,B**H)**H, 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**H.
	    If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x.
	    In some literature, the GSVD of A and B is presented in the form
			     U**H*A*X = ( 0 D1 ),   V**H*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) )

       Parameters:
	   JOBU

		     JOBU is CHARACTER*1
		     = 'U':  Unitary matrix U is computed;
		     = 'N':  U is not computed.

	   JOBV

		     JOBV is CHARACTER*1
		     = 'V':  Unitary matrix V is computed;
		     = 'N':  V is not computed.

	   JOBQ

		     JOBQ is CHARACTER*1
		     = 'Q':  Unitary matrix Q is computed;
		     = 'N':  Q is not computed.

	   M

		     M is INTEGER
		     The number of rows of the matrix A.  M >= 0.

	   N

		     N is INTEGER
		     The number of columns of the matrices A and B.  N >= 0.

	   P

		     P is INTEGER
		     The number of rows of the matrix B.  P >= 0.

	   K

		     K is INTEGER

	   L

		     L is INTEGER

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

	   A

		     A is 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

		     LDA is INTEGER
		     The leading dimension of the array A. LDA >= max(1,M).

	   B

		     B is 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

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

	   ALPHA

		     ALPHA is REAL array, dimension (N)

	   BETA

		     BETA is 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

		     U is 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

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

	   V

		     V is 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

		     LDV is INTEGER
		     The leading dimension of the array V. LDV >= max(1,P) if
		     JOBV = 'V'; LDV >= 1 otherwise.

	   Q

		     Q is 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

		     LDQ is INTEGER
		     The leading dimension of the array Q. LDQ >= max(1,N) if
		     JOBQ = 'Q'; LDQ >= 1 otherwise.

	   WORK

		     WORK is COMPLEX array, dimension (max(3*N,M,P)+N)

	   RWORK

		     RWORK is REAL array, dimension (2*N)

	   IWORK

		     IWORK is 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

		     INFO is 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.

       Internal Parameters:

	     TOLA    REAL
	     TOLB    REAL
		     TOLA and TOLB are the thresholds to determine the effective
		     rank of (A**H,B**H)**H. 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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Contributors:
	   Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley,
	   USA

       Definition at line 335 of file cggsvd.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      cggsvd.f(3)
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