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sggsvd.f(3)				      LAPACK				      sggsvd.f(3)

NAME
       sggsvd.f -

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

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

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

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

       Parameters:
	   JOBU

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

	   JOBV

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

	   JOBQ

		     JOBQ is CHARACTER*1
		     = 'Q':  Orthogonal 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**T,B**T)**T.

	   A

		     A is 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

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

	   B

		     B is 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

		     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 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

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

	   V

		     V is 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

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

	   Q

		     Q is 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

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

	   WORK

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

       Internal Parameters:

	     TOLA    REAL
	     TOLB    REAL
		     TOLA and TOLB are the thresholds to determine the effective
		     rank of (A**T,B**T)**T. 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 331 of file sggsvd.f.

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

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