SPORFS(l)		LAPACK routine (version	1.1)		    SPORFS(l)

NAME
  SPORFS - improve the computed	solution to a system of	linear equations when
  the coefficient matrix is symmetric positive definite,

SYNOPSIS

  SUBROUTINE SPORFS( UPLO, N, NRHS, A, LDA, AF,	LDAF, B, LDB, X, LDX, FERR,
		     BERR, WORK, IWORK,	INFO )

      CHARACTER	     UPLO

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

      INTEGER	     IWORK( * )

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

PURPOSE
  SPORFS improves the computed solution	to a system of linear equations	when
  the coefficient matrix is symmetric positive definite, and provides error
  bounds and backward error estimates for the solution.

ARGUMENTS

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

  N	  (input) INTEGER
	  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) REAL 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 refer-
	  enced.  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) REAL array, dimension	(LDAF,N)
	  The triangular factor	U or L from the	Cholesky factorization A =
	  U**T*U or A =	L*L**T,	as computed by SPOTRF.

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

  B	  (input) REAL array, dimension	(LDB,NRHS)
	  The right hand side matrix B.

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

  X	  (input/output) REAL array, dimension (LDX,NRHS)
	  On entry, the	solution matrix	X, as computed by SPOTRS.  On exit,
	  the improved solution	matrix X.

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

  FERR	  (output) REAL	array, dimension (NRHS)
	  The estimated	forward	error bounds for each solution vector X(j)
	  (the j-th column of the solution matrix X).  If XTRUE	is the true
	  solution, FERR(j) bounds the magnitude of the	largest	entry in
	  (X(j)	- XTRUE) divided by the	magnitude of the largest entry in
	  X(j).	 The quality of	the error bound	depends	on the quality of the
	  estimate of norm(inv(A)) computed in the code; if the	estimate of
	  norm(inv(A)) is accurate, the	error bound is guaranteed.

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

  WORK	  (workspace) REAL array, dimension (3*N)

  IWORK	  (workspace) INTEGER array, dimension (N)

  INFO	  (output) INTEGER
	  = 0:	successful exit
	  < 0:	if INFO	= -i, the i-th argument	had an illegal value

PARAMETERS

  ITMAX	is the maximum number of steps of iterative refinement.