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

ZSYSVX(l)					)					ZSYSVX(l)

NAME
       ZSYSVX - use the diagonal pivoting factorization to compute the solution to a complex sys-
       tem of linear equations A * X = B,

SYNOPSIS
       SUBROUTINE ZSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,  LDB,  X,  LDX,  RCOND,
			  FERR, BERR, WORK, LWORK, RWORK, INFO )

	   CHARACTER	  FACT, UPLO

	   INTEGER	  INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * )

	   DOUBLE	  PRECISION BERR( * ), FERR( * ), RWORK( * )

	   COMPLEX*16	  A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE
       ZSYSVX  uses the diagonal pivoting factorization to compute the solution to a complex sys-
       tem of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and  B  are
       N-by-NRHS matrices.

       Error bounds on the solution and a condition estimate are also provided.

DESCRIPTION
       The following steps are performed:

       1. If FACT = 'N', the diagonal pivoting method is used to factor A.
	  The form of the factorization is
	     A = U * D * U**T,	if UPLO = 'U', or
	     A = L * D * L**T,	if UPLO = 'L',
	  where U (or L) is a product of permutation and unit upper (lower)
	  triangular matrices, and D is symmetric and block diagonal with
	  1-by-1 and 2-by-2 diagonal blocks.

       2. If some D(i,i)=0, so that D is exactly singular, then the routine
	  returns with INFO = i. Otherwise, the factored form of A is used
	  to estimate the condition number of the matrix A.  If the
	  reciprocal of the condition number is less than machine precision,
	  INFO = N+1 is returned as a warning, but the routine still goes on
	  to solve for X and compute error bounds as described below.

       3. The system of equations is solved for X using the factored form
	  of A.

       4. Iterative refinement is applied to improve the computed solution
	  matrix and calculate error bounds and backward error estimates
	  for it.

ARGUMENTS
       FACT    (input) CHARACTER*1
	       Specifies  whether  or  not  the factored form of A has been supplied on entry.	=
	       'F':  On entry, AF and IPIV contain the factored form of A.  A, AF and  IPIV  will
	       not be modified.  = 'N':  The matrix A will be copied to AF and factored.

       UPLO    (input) CHARACTER*1
	       = 'U':  Upper triangle of A is stored;
	       = 'L':  Lower triangle of A is stored.

       N       (input) INTEGER
	       The number of linear equations, i.e., the order of the matrix A.  N >= 0.

       NRHS    (input) INTEGER
	       The  number of right hand sides, i.e., the number of columns of the matrices B and
	       X.  NRHS >= 0.

       A       (input) COMPLEX*16 array, dimension (LDA,N)
	       The symmetric matrix A.	If UPLO = 'U', the leading N-by-N upper  triangular  part
	       of  A  contains	the upper triangular part of the matrix A, and the strictly lower
	       triangular part of A is not referenced.	If UPLO = 'L', the leading  N-by-N  lower
	       triangular  part  of A contains the lower triangular part of the matrix A, and the
	       strictly upper triangular part of A is not referenced.

       LDA     (input) INTEGER
	       The leading dimension of the array A.  LDA >= max(1,N).

       AF      (input or output) COMPLEX*16 array, dimension (LDAF,N)
	       If FACT = 'F', then AF is an input argument and on entry contains the block diago-
	       nal matrix D and the multipliers used to obtain the factor U or L from the factor-
	       ization A = U*D*U**T or A = L*D*L**T as computed by ZSYTRF.

	       If FACT = 'N', then AF is an output argument and on exit returns the block  diago-
	       nal matrix D and the multipliers used to obtain the factor U or L from the factor-
	       ization A = U*D*U**T or A = L*D*L**T.

       LDAF    (input) INTEGER
	       The leading dimension of the array AF.  LDAF >= max(1,N).

       IPIV    (input or output) INTEGER array, dimension (N)
	       If FACT = 'F', then IPIV is an input argument and on entry contains details of the
	       interchanges  and the block structure of D, as determined by ZSYTRF.  If IPIV(k) >
	       0, then rows and columns k and IPIV(k) were interchanged and D(k,k)  is	a  1-by-1
	       diagonal  block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns
	       k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal  block.
	       If  UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k)
	       were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

	       If FACT = 'N', then IPIV is an output argument and on exit contains details of the
	       interchanges and the block structure of D, as determined by ZSYTRF.

       B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
	       The N-by-NRHS right hand side matrix B.

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

       X       (output) COMPLEX*16 array, dimension (LDX,NRHS)
	       If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

       LDX     (input) INTEGER
	       The leading dimension of the array X.  LDX >= max(1,N).

       RCOND   (output) DOUBLE PRECISION
	       The estimate of the reciprocal condition number of the matrix A.  If RCOND is less
	       than the machine precision (in particular, if RCOND = 0), the matrix  is  singular
	       to working precision.  This condition is indicated by a return code of INFO > 0.

       FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
	       The  estimated  forward error bound for each solution vector X(j) (the j-th column
	       of the solution matrix X).  If XTRUE is the true solution corresponding	to  X(j),
	       FERR(j)	is  an	estimated upper bound for the magnitude of the largest element in
	       (X(j) - XTRUE) divided by the magnitude of the largest element in X(j).	The esti-
	       mate is as reliable as the estimate for RCOND, and is almost always a slight over-
	       estimate of the true error.

       BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
	       The componentwise relative backward error of each solution vector X(j) (i.e.,  the
	       smallest  relative  change in any element of A or B that makes X(j) an exact solu-
	       tion).

       WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The length of WORK.  LWORK >= 2*N, and for best performance LWORK >=  N*NB,  where
	       NB is the optimal blocksize for ZSYTRF.

	       If  LWORK = -1, then a workspace query is assumed; the routine only calculates the
	       optimal size of the WORK array, returns this value as the first entry of the  WORK
	       array, and no error message related to LWORK is issued by XERBLA.

       RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

       INFO    (output) INTEGER
	       = 0: successful exit
	       < 0: if INFO = -i, the i-th argument had an illegal value
	       > 0: if INFO = i, and i is
	       <=  N:  D(i,i) is exactly zero.	The factorization has been completed but the fac-
	       tor D is exactly singular, so the solution and error bounds could not be computed.
	       RCOND  =  0  is returned.  = N+1: D is nonsingular, but RCOND is less than machine
	       precision, meaning that the matrix is singular to  working  precision.	Neverthe-
	       less,  the  solution  and  error bounds are computed because there are a number of
	       situations where the computed solution can be more  accurate  than  the	value  of
	       RCOND would suggest.

LAPACK version 3.0			   15 June 2000 				ZSYSVX(l)


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