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


SYNOPSIS

       SUBROUTINE DGERFS( 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( * )

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


PURPOSE

       DGERFS  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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDAF,N)
               The  factors  L  and  U from the factorization A =
               P*L*U as computed by DGETRF.

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

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

       B       (input) DOUBLE PRECISION 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)   DOUBLE   PRECISION   array,
               dimension (LDX,NRHS)
               On entry, the solution matrix X,  as  computed  by
               DGETRS.   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 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) 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) DOUBLE PRECISION 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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