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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.
```