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# sgerfs

```
NAME
SGERFS - improve the computed solution to a system of linear
equations and provides error bounds and backward error esti-
mates for the solution

SYNOPSIS
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,
* )

PURPOSE
SGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error esti-
mates for the solution.

ARGUMENTS
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 = Tran-
spose)

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 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 refine-
ment.
```