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

           INTEGER        INFO, LDB, LDX, N, NRHS

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


PURPOSE

       SPTRFS  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) REAL array, dimension (N)
               The  n diagonal elements of the tridiagonal matrix
               A.

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

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

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

       B       (input) REAL array, dimension (LDB,NRHS)
               The right hand side matrix B.

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

               On  entry,  the  solution matrix X, as computed by
               SPTTRS.  On exit, the improved solution matrix  X.

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

       FERR    (output) REAL 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) REAL 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) REAL 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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