# man csprfs (Fonctions bibliothèques) - improve the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution

## NAME

CSPRFS - improve the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite 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 indefinite 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)*(2*n-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 bound for each solution 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 element 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) 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 refinement.