CXML
        	    
LAPACK version 3.0
sporfs(3)

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

SYNTAX
  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, * )

DESCRIPTION
  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
	  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) 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 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 estimate is
	  as reliable as the estimate for RCOND, and is almost always a
	  slight overestimate of the true error.

  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 element 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.