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RedHat 9 (Linux i386) - man page for sggsvd (redhat section l)

SGGSVD(l)					)					SGGSVD(l)

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

SYNOPSIS
       SUBROUTINE SGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA,  U,  LDU,
			  V, LDV, Q, LDQ, WORK, IWORK, INFO )

	   CHARACTER	  JOBQ, JOBU, JOBV

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

	   INTEGER	  IWORK( * )

	   REAL 	  A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), U( LDU, *
			  ), V( LDV, * ), WORK( * )

PURPOSE
       SGGSVD computes the generalized singular value decomposition  (GSVD)  of  an  M-by-N  real
       matrix A and P-by-N real matrix B:
	   U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )

       where  U,  V  and  Q are orthogonal matrices, and Z' is the 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 orthogonal 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 orthonormal 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, 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':  Orthogonal matrix U is computed;
	       = 'N':  U is not computed.

       JOBV    (input) CHARACTER*1
	       = 'V':  Orthogonal matrix V is computed;
	       = 'N':  V is not computed.

       JOBQ    (input) CHARACTER*1
	       = 'Q':  Orthogonal 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 the Purpose section.  K + L = effective numerical rank of (A',B')'.

       A       (input/output) REAL 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) REAL array, dimension (LDB,N)
	       On  entry, the P-by-N matrix B.	On exit, B contains the triangular matrix R if M-
	       K-L < 0.  See Purpose for details.

       LDB     (input) INTEGER
	       The leading dimension of the array B. LDA >= 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) REAL array, dimension (LDU,M)
	       If JOBU = 'U', U contains the M-by-M orthogonal 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 oth-
	       erwise.

       V       (output) REAL array, dimension (LDV,P)
	       If JOBV = 'V', V contains the P-by-P orthogonal 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 oth-
	       erwise.

       Q       (output) REAL array, dimension (LDQ,N)
	       If JOBQ = 'Q', Q contains the N-by-N orthogonal 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 oth-
	       erwise.

       WORK    (workspace) REAL array,
	       dimension (max(3*N,M,P)+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 STGSJA.

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  back-
	       ward 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 				SGGSVD(l)


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