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

DSYSVX(l)					)					DSYSVX(l)

NAME
       DSYSVX  - use the diagonal pivoting factorization to compute the solution to a real system
       of linear equations A * X = B,

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

	   CHARACTER	  FACT, UPLO

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

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

	   DOUBLE	  PRECISION  A(  LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR( * ), FERR( *
			  ), WORK( * ), X( LDX, * )

PURPOSE
       DSYSVX uses the diagonal pivoting factorization to compute the solution to a  real  system
       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.  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DSYTRF.

	       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 DSYTRF.  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 DSYTRF.

       B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The  length  of WORK.  LWORK >= 3*N, and for best performance LWORK >= N*NB, where
	       NB is the optimal blocksize for DSYTRF.

	       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.

       IWORK   (workspace) INTEGER 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 				DSYSVX(l)


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