DTRRFS - provide error bounds and backward error estimates
       for the solution to a system of linear  equations  with  a
       triangular coefficient matrix


SYNOPSIS

       SUBROUTINE DTRRFS( UPLO,  TRANS, DIAG, N, NRHS, A, LDA, B,
                          LDB, X, LDX, FERR, BERR,  WORK,  IWORK,
                          INFO )

           CHARACTER      DIAG, TRANS, UPLO

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

           INTEGER        IWORK( * )

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


PURPOSE

       DTRRFS  provides error bounds and backward error estimates
       for the solution to a system of linear  equations  with  a
       triangular coefficient matrix.  The solution matrix X must
       be computed by DTRTRS or some other means before  entering
       this  routine.   DTRRFS  does  not do iterative refinement
       because doing so cannot improve the backward error.


ARGUMENTS

       UPLO    (input) CHARACTER*1
               = 'U':  A is upper triangular;
               = 'L':  A is lower triangular.

       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)

       DIAG    (input) CHARACTER*1
               = 'N':  A is non-unit triangular;
               = 'U':  A is unit triangular.

       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 triangular matrix  A.   If  UPLO  =  'U',  the

               A contains the upper triangular  matrix,  and  the
               strictly  lower triangular part of A is not refer­
               enced.  If UPLO = 'L', the  leading  N-by-N  lower
               triangular  part of the array A contains the lower
               triangular matrix, and the strictly upper triangu­
               lar  part  of A is not referenced.  If DIAG = 'U',
               the diagonal elements of A are also not referenced
               and are assumed to be 1.

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

       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) DOUBLE PRECISION array, dimension
               (LDX,NRHS)
               The 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

               gal value


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