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# csprfs

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

SYNOPSIS
SUBROUTINE CSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
LDX, FERR, BERR, WORK, RWORK, INFO )

CHARACTER      UPLO

INTEGER        INFO, LDB, LDX, N, NRHS

INTEGER        IPIV( * )

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

COMPLEX        AFP( * ), AP( * ), B( LDB, * ), WORK( *
), X( LDX, * )

PURPOSE
CSPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefin-
ite and packed, 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.

AP      (input) COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array.  The j-th
column of A is stored in the array AP as follows: if
UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for
j<=i<=n.

AFP     (input) COMPLEX array, dimension (N*(N+1)/2)
The factored form of the matrix A.  AFP 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 CSPTRF,
stored as a packed triangular matrix.

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

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
CSPTRS.  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.
```