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RedHat 9 (Linux i386) - man page for ztgsja (redhat section l)

ZTGSJA(l)					)					ZTGSJA(l)

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

SYNOPSIS
       SUBROUTINE ZTGSJA( 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 ALPHA( * ), BETA( * )

	   COMPLEX*16	  A(  LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK(
			  * )

PURPOSE
       ZTGSJA computes the generalized singular value decomposition (GSVD) of two  complex  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  ZGGSVP
       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 unitary matrices, Z' denotes the conjugate 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 unitary 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 a unitary matrix U1 on  entry,  and  the	product  U1*U  is
	       returned; = 'I':  U is initialized to the unit matrix, and the unitary matrix U is
	       returned; = 'N':  U is not computed.

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

       JOBQ    (input) CHARACTER*1
	       = 'Q':  Q must contain a unitary matrix Q1 on  entry,  and  the	product  Q1*Q  is
	       returned; = 'I':  Q is initialized to the unit matrix, and the unitary 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 ZTGSJA.  See Further details.

       A       (input/output) COMPLEX*16 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) COMPLEX*16 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
	       BETA(K+L+1:N)  = 0.

       U       (input/output) COMPLEX*16 array, dimension (LDU,M)
	       On entry, if JOBU = 'U', U must contain a matrix U1 (usually  the  unitary  matrix
	       returned  by ZGGSVP).  On exit, if JOBU = 'I', U contains the unitary 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) COMPLEX*16 array, dimension (LDV,P)
	       On  entry,  if  JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix
	       returned by ZGGSVP).  On exit, if JOBV = 'I', V contains the unitary 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) COMPLEX*16 array, dimension (LDQ,N)
	       On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually  the  unitary  matrix
	       returned  by ZGGSVP).  On exit, if JOBQ = 'I', Q contains the unitary 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) COMPLEX*16 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 ===============

	       ZTGSJA 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 unitary matrix, and Z' is the conjugate 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 				ZTGSJA(l)


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