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stgsja.f(3)				      LAPACK				      stgsja.f(3)

NAME
       stgsja.f -

SYNOPSIS
   Functions/Subroutines
       subroutine stgsja (JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA,
	   BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
	   STGSJA

Function/Subroutine Documentation
   subroutine stgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN,
       integerK, integerL, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B,
       integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA, real,
       dimension( ldu, * )U, integerLDU, real, dimension( ldv, * )V, integerLDV, real, dimension(
       ldq, * )Q, integerLDQ, real, dimension( * )WORK, integerNCYCLE, integerINFO)
       STGSJA

       Purpose:

	    STGSJA computes the generalized singular value decomposition (GSVD)
	    of two real upper triangular (or trapezoidal) matrices A and B.

	    On entry, it is assumed that matrices A and B have the following
	    forms, which may be obtained by the preprocessing subroutine SGGSVP
	    from a general M-by-N matrix A and P-by-N matrix B:

			 N-K-L	K    L
	       A =    K ( 0    A12  A13 ) if M-K-L >= 0;
		      L ( 0	0   A23 )
		  M-K-L ( 0	0    0	)

		       N-K-L  K    L
	       A =  K ( 0    A12  A13 ) if M-K-L < 0;
		  M-K ( 0     0   A23 )

		       N-K-L  K    L
	       B =  L ( 0     0   B13 )
		  P-L ( 0     0    0  )

	    where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
	    upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
	    otherwise A23 is (M-K)-by-L upper trapezoidal.

	    On exit,

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

	    where U, V and Q are orthogonal matrices.
	    R is a nonsingular upper triangular matrix, and D1 and D2 are
	    ``diagonal'' matrices, which are of the following structures:

	    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 ) K
			L (  0	  0   R22 ) L

	    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.

	    R = ( 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 computation of the orthogonal transformation matrices U, V or Q
	    is optional.  These matrices may either be formed explicitly, or they
	    may be postmultiplied into input matrices U1, V1, or Q1.

       Parameters:
	   JOBU

		     JOBU is CHARACTER*1
		     = 'U':  U must contain an orthogonal matrix U1 on entry, and
			     the product U1*U is returned;
		     = 'I':  U is initialized to the unit matrix, and the
			     orthogonal matrix U is returned;
		     = 'N':  U is not computed.

	   JOBV

		     JOBV is CHARACTER*1
		     = 'V':  V must contain an orthogonal matrix V1 on entry, and
			     the product V1*V is returned;
		     = 'I':  V is initialized to the unit matrix, and the
			     orthogonal matrix V is returned;
		     = 'N':  V is not computed.

	   JOBQ

		     JOBQ is CHARACTER*1
		     = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
			     the product Q1*Q is returned;
		     = 'I':  Q is initialized to the unit matrix, and the
			     orthogonal matrix Q is returned;
		     = 'N':  Q is not computed.

	   M

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

	   P

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

	   N

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

	   K

		     K is INTEGER

	   L

		     L is INTEGER

		     K and L specify the subblocks in the input matrices A and B:
		     A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
		     of A and B, whose GSVD is going to be computed by STGSJA.
		     See Further Details.

	   A

		     A is REAL array, dimension (LDA,N)
		     On entry, the M-by-N matrix A.
		     On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
		     a part of R.  See Purpose for details.

	   LDB

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

	   TOLA

		     TOLA is REAL

	   TOLB

		     TOLB is REAL

		     TOLA and TOLB are the convergence criteria for the Jacobi-
		     Kogbetliantz iteration procedure. Generally, they are the
		     same as used in the preprocessing step, say
			 TOLA = max(M,N)*norm(A)*MACHEPS,
			 TOLB = max(P,N)*norm(B)*MACHEPS.

	   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) = diag(C),
		       BETA(K+1:K+L)  = diag(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.
		     Furthermore, if K+L < N,
		       ALPHA(K+L+1:N) = 0 and
		       BETA(K+L+1:N)  = 0.

	   U

		     U is REAL array, dimension (LDU,M)
		     On entry, if JOBU = 'U', U must contain a matrix U1 (usually
		     the orthogonal matrix returned by SGGSVP).
		     On exit,
		     if JOBU = 'I', U contains the orthogonal matrix U;
		     if JOBU = 'U', U contains the product U1*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)
		     On entry, if JOBV = 'V', V must contain a matrix V1 (usually
		     the orthogonal matrix returned by SGGSVP).
		     On exit,
		     if JOBV = 'I', V contains the orthogonal matrix V;
		     if JOBV = 'V', V contains the product V1*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)
		     On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
		     the orthogonal matrix returned by SGGSVP).
		     On exit,
		     if JOBQ = 'I', Q contains the orthogonal matrix Q;
		     if JOBQ = 'Q', Q contains the product Q1*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 (2*N)

	   NCYCLE

		     NCYCLE is INTEGER
		     The number of cycles required for convergence.

	   INFO

		     INFO is INTEGER
		     = 0:  successful exit
		     < 0:  if INFO = -i, the i-th argument had an illegal value.
		     = 1:  the procedure does not converge after MAXIT cycles.

	 Internal Parameters
	 ===================

	 MAXIT	 INTEGER
		 MAXIT specifies the total loops that the iterative procedure
		 may take. If after MAXIT cycles, the routine fails to
		 converge, we return INFO = 1..fi

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Further Details:

	     STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
	     min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
	     matrix B13 to the form:

		      U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

	     where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
	     of Z.  C1 and S1 are diagonal matrices satisfying

			   C1**2 + S1**2 = I,

	     and R1 is an L-by-L nonsingular upper triangular matrix.

       Definition at line 377 of file stgsja.f.

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

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