SGERFS - improve the computed solution to a system of lin­
       ear equations and provides error bounds and backward error
       estimates 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 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 =
               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).

               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 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) 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 ille­
               gal value


PARAMETERS

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


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