CGERFS - improve the computed solution to a system of lin­
       ear equations and provides error bounds and backward error
       estimates for the solution


SYNOPSIS

       SUBROUTINE CGERFS( 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( * )

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

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


PURPOSE

       CGERFS improves the computed solution to a system of  lin­
       ear 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 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 array, dimension (LDAF,N)
               The factors L and U from  the  factorization  A  =
               P*L*U as computed by CGETRF.

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

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

       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
               CGETRS.  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  solu­
               tion  vector X(j) (the j-th column of the solution
               matrix X).  If XTRUE is the true  solution  corre­
               sponding  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 reli­
               able  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 ille­
               gal value


PARAMETERS

       ITMAX is the maximum number of steps of iterative  refine­
       ment.


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