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# csyrfs

```
NAME
CSYRFS - improve the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefin-
ite, and provides error bounds and backward error estimates
for the solution

SYNOPSIS
SUBROUTINE CSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
LDB, X, LDX, FERR, BERR, WORK, RWORK,
INFO )

CHARACTER      UPLO

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

INTEGER        IPIV( * )

REAL           BERR( * ), FERR( * ), RWORK( * )

COMPLEX        A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
WORK( * ), X( LDX, * )

PURPOSE
CSYRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefin-
ite, 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) COMPLEX 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) COMPLEX array, dimension (LDAF,N)
The factored form of the matrix A.  AF contains the
block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A =
U*D*U**T or A = L*D*L**T as computed by CSYTRF.

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

IPIV    (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure
of D as determined by CSYTRF.

B       (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by
CSYTRS.  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) COMPLEX array, dimension (2*N)

RWORK   (workspace) REAL 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 refine-
ment.
```