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dgerfs

 NAME
      DGERFS - improve the computed solution to a system of linear
      equations and provides error bounds and backward error esti-
      mates 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 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) 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).

      IPIV    (input) INTEGER array, dimension (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).

 (LDX,NRHS)
      X       (input/output) DOUBLE PRECISION array, dimension
              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 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) DOUBLE PRECISION 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) 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 illegal
              value

 PARAMETERS
      ITMAX is the maximum number of steps of iterative

      refinement.