DPTRFS - improve the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  symmetric
       positive  definite  and  tridiagonal,  and  provides error
       bounds and backward error estimates for the solution


SYNOPSIS

       SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X,  LDX,
                          FERR, BERR, WORK, INFO )

           INTEGER        INFO, LDB, LDX, N, NRHS

           DOUBLE         PRECISION  B( LDB, * ), BERR( * ), D( *
                          ), DF( * ), E( * ), EF( * ), FERR( * ),
                          WORK( * ), X( LDX, * )


PURPOSE

       DPTRFS  improves the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  symmetric
       positive  definite  and  tridiagonal,  and  provides error
       bounds and backward error estimates for the solution.


ARGUMENTS

       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 matrix B.  NRHS >= 0.

       D       (input) DOUBLE PRECISION array, dimension (N)
               The  n diagonal elements of the tridiagonal matrix
               A.

       E       (input) DOUBLE PRECISION array, dimension (N-1)
               The (n-1) subdiagonal elements of the  tridiagonal
               matrix A.

       DF      (input) DOUBLE PRECISION array, dimension (N)
               The  n  diagonal elements of the diagonal matrix D
               from the factorization computed by DPTTRF.

       EF      (input) DOUBLE PRECISION array, dimension (N-1)
               The (n-1) subdiagonal elements of the unit bidiag­
               onal  factor  L from the factorization computed by
               DPTTRF.

       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).

               dimension (LDX,NRHS)
               On entry, the solution matrix X,  as  computed  by
               DPTTRS.   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  forward  error bound for each solution vector
               X(j) (the j-th column of the solution  matrix  X).
               If  XTRUE  is  the  true solution corresponding 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).

       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
               (2*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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