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sporfs

 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 esti-
      mates 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 fac-
              torization 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 accu-
              rate, 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.