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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: improves the computed solution to a system of linear equations and provides error bounds and backward error estimates

```NAME
LAPACK-3  -  improves  the  computed  solution  to  a  system of linear
equations and provides error bounds and backward  error  estimates  for
the solution

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

CHARACTER      TRANS

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

INTEGER        IPIV( * )

DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * )

COMPLEX*16     A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( *  ),
X( LDX, * )

PURPOSE
ZGERFS  improves  the computed solution to a system of linear equations
and  provides  error  bounds  and  backward  error  estimates  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)

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*16 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) COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by ZGETRF.

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

IPIV    (input) INTEGER array, dimension (N)
The pivot indices from ZGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).

B       (input) COMPLEX*16 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*16 array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by ZGETRS.
On exit, the improved solution matrix X.

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

FERR    (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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*16 array, dimension (2*N)

RWORK   (workspace) DOUBLE PRECISION 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.

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
```