SGERFS - improve the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution

SUBROUTINE SGERFS(

TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )

CHARACTER TRANS

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

INTEGER IPIV( * ), IWORK( * )

REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR( * ), FERR( * ), WORK( * ), X( LDX, * )

SGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution.

TRANS (input) CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate transpose = Transpose)

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 original N-by-N matrix A.

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,N).

AF (input) REAL array, dimension (LDAF,N)

The factors L and U from the factorization A = P*L*U as computed by SGETRF.

LDAF (input) INTEGER

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

IPIV (input) INTEGER array, dimension (N)

The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).

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 SGETRS. 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) 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

ITMAX is the maximum number of steps of iterative refinement.