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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, * )



     #include <sunperf.h>

     void sporfs(char uplo, int n, int nrhs, float *sa, int  lda,
               float  *af,  int  ldaf,  float *sb, int ldb, float
               *sx,  int  ldx,  float  *ferr,  float  *berr,  int
               *info);

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  lead-
               ing 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  tri-
               angular 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 com-
               puted 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  solu-
               tion  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  ele-
               ment 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 ille-
               gal value

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