CXML
        	    
LAPACK version 3.0
dtgsja(3)

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

SYNTAX
  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( * )

DESCRIPTION
  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.

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) contains 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 otherwise.

  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 otherwise.

  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 otherwise.

  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.