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dtgsja(3P)		    Sun Performance Library		    dtgsja(3P)

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 * 1 JOBU, JOBV, JOBQ
       INTEGER M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
       DOUBLE PRECISION TOLA, TOLB
       DOUBLE  PRECISION  A(LDA,*),  B(LDB,*),	ALPHA(*),  BETA(*),  U(LDU,*),
       V(LDV,*), Q(LDQ,*), WORK(*)

       SUBROUTINE DTGSJA_64(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 * 1 JOBU, JOBV, JOBQ
       INTEGER*8 M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
       DOUBLE PRECISION TOLA, TOLB
       DOUBLE  PRECISION  A(LDA,*),  B(LDB,*),	ALPHA(*),  BETA(*),  U(LDU,*),
       V(LDV,*), Q(LDQ,*), WORK(*)

   F95 INTERFACE
       SUBROUTINE TGSJA(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(LEN=1) :: JOBU, JOBV, JOBQ
       INTEGER :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
       REAL(8) :: TOLA, TOLB
       REAL(8), DIMENSION(:) :: ALPHA, BETA, WORK
       REAL(8), DIMENSION(:,:) :: A, B, U, V, Q

       SUBROUTINE TGSJA_64(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(LEN=1) :: JOBU, JOBV, JOBQ
       INTEGER(8) :: M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
       REAL(8) :: TOLA, TOLB
       REAL(8), DIMENSION(:) :: ALPHA, BETA, WORK
       REAL(8), DIMENSION(:,:) :: A, B, U, V, Q

   C INTERFACE
       #include <sunperf.h>

       void dtgsja(char jobu, char jobv, char jobq, int m, int p, int  n,  int
		 k,  int  l,  double  *a,  int lda, double *b, int ldb, double
		 tola, double tolb, double *alpha, double  *beta,  double  *u,
		 int ldu, double *v, int ldv, double *q, int ldq, int *ncycle,
		 int *info);

       void dtgsja_64(char jobu, char jobv, char jobq, long m, long p, long n,
		 long  k,  long	 l,  double *a, long lda, double *b, long ldb,
		 double tola, double tolb, double *alpha, double *beta, double
		 *u,  long ldu, double *v, long ldv, double *q, long ldq, long
		 *ncycle, long *info);

PURPOSE
       dtgsja 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  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 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

		 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.
       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.

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

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

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

       K (input) 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.

       L (input) See the description of K.

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

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

       TOLA (input)
		 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.

       TOLB (input)
		 See the description of TOLA.

       ALPHA (output)
		 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.

       BETA (output)
		 See the description of ALPHA.

       U (input) 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	refer‐
		 enced.

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

       V (input) 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 refer‐
		 enced.

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

       Q (input) 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	refer‐
		 enced.

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

       WORK (workspace)
		 dimension(2*N)

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

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

				  6 Mar 2009			    dtgsja(3P)
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