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


SYNOPSIS

       SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX,
                          FERR, BERR, WORK, IWORK, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IWORK( * )

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


PURPOSE

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


ARGUMENTS

       UPLO    (input) CHARACTER*1
               = 'U':  Upper triangle of A is stored;
               = 'L':  Lower triangle of A is stored.

       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  triangle  of  the  symmetric
               matrix  A,  packed  columnwise  in a linear array.
               The j-th column of 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)*(2n-j)/2)
               = A(i,j) for j<=i<=n.

       AFP     (input) DOUBLE PRECISION array, dimension
               (N*(N+1)/2)
               The triangular factor U or  L  from  the  Cholesky
               factorization  A  =  U**T*U or A = L*L**T, as com­
               puted by DPPTRF/ZPPTRF,  packed  columnwise  in  a
               linear array in the same format as A (see AP).

               (LDB,NRHS)
               The right hand side matrix B.

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

       X         (input/output)   DOUBLE   PRECISION   array,
               dimension (LDX,NRHS)
               On  entry,  the  solution matrix X, as computed by
               DPPTRS.  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 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


PARAMETERS

       ITMAX  is the maximum number of steps of iterative refine­
       ment.


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