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DTGSJA(l)					)					DTGSJA(l)

NAME
       DTGSJA  -  compute  the	generalized singular value decomposition (GSVD) of two real upper
       triangular (or trapezoidal) matrices A and B

SYNOPSIS
       SUBROUTINE DTGSJA( 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 )

	   CHARACTER	  JOBQ, JOBU, JOBV

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

	   DOUBLE	  PRECISION TOLA, TOLB

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

PURPOSE
       DTGSJA computes the generalized singular value decomposition (GSVD) of two real upper tri-
       angular	(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  DGGSVP
       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'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),

       where U, V and Q are orthogonal matrices, Z' denotes the transpose of Z, R is a	nonsingu-
       lar  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

		 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 matri-
       ces U1, V1, or Q1.

ARGUMENTS
       JOBU    (input) 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    (input) 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    (input) 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       (input) INTEGER
	       The number of rows of the matrix A.  M >= 0.

       P       (input) INTEGER
	       The number of rows of the matrix B.  P >= 0.

       N       (input) INTEGER
	       The number of columns of the matrices A and B.  N >= 0.

       K       (input) INTEGER
	       L       (input) 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 DTGSJA.	See Further details.

       A       (input/output) DOUBLE PRECISION 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     (input) INTEGER
	       The leading dimension of the array A. LDA >= max(1,M).

       B       (input/output) DOUBLE PRECISION 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) con-
	       tains a part of R.  See Purpose for details.

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

       TOLA    (input) DOUBLE PRECISION
	       TOLB    (input) DOUBLE PRECISION 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)*MAZHEPS,	 TOLB	=
	       max(P,N)*norm(B)*MAZHEPS.

       ALPHA   (output) DOUBLE PRECISION array, dimension (N)
	       BETA	(output)  DOUBLE  PRECISION  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       (input/output) DOUBLE PRECISION array, dimension (LDU,M)
	       On entry, if JOBU = 'U', U must contain a matrix U1 (usually the orthogonal matrix
	       returned by DGGSVP).  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     (input) INTEGER
	       The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 oth-
	       erwise.

       V       (input/output) DOUBLE PRECISION array, dimension (LDV,P)
	       On entry, if JOBV = 'V', V must contain a matrix V1 (usually the orthogonal matrix
	       returned  by DGGSVP).  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     (input) INTEGER
	       The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 oth-
	       erwise.

       Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
	       On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the orthogonal matrix
	       returned by DGGSVP).  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     (input) INTEGER
	       The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 oth-
	       erwise.

       WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)

       NCYCLE  (output) INTEGER
	       The number of cycles required for convergence.

       INFO    (output) 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.

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.

	       Further Details ===============

	       DTGSJA  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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,

	       where U1, V1 and Q1 are orthogonal matrix, and Z' 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.

LAPACK version 3.0			   15 June 2000 				DTGSJA(l)
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