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


SYNOPSIS

       SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB,
                          X, LDX, FERR, BERR, WORK, IWORK, INFO )

           CHARACTER      DIAG, TRANS, UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IWORK( * )

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


PURPOSE

       DTPRFS provides error bounds and backward error  estimates
       for  the  solution  to a system of linear equations with a
       triangular packed coefficient matrix.  The solution matrix
       X  must  be  computed by DTPTRS or some other means before
       entering this  routine.   DTPRFS  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.

       AP      (input) DOUBLE PRECISION array, dimension
               (N*(N+1)/2)
               The  upper  or  lower  triangular matrix A, packed

               A  is stored in the array AP as follows: if UPLO =
               'U', AP(i + (j-1)*j/2) = A(i,j)  for  1<=i<=j;  if
               UPLO  =  'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for
               j<=i<=n.  If DIAG = 'U', the diagonal elements  of
               A are not referenced and are assumed to be 1.

       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
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value


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