DPPRFS(3)                  MathKeisan LAPACK routine                 DPPRFS(3)



NAME
       DPPRFS - the computed solution to a system of linear equations when the
       coefficient matrix is symmetric positive definite and packed, and  pro-
       vides 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 linear equations
       when the coefficient matrix is symmetric positive definite and  packed,
       and  provides  error  bounds and backward error estimates for the solu-
       tion.


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 computed  by  DPPTRF/ZPPTRF,  packed
               columnwise  in a linear array in the same format as A (see AP).

       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/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 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).
               The estimate is as reliable 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 vec-
               tor 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 illegal value

PARAMETERS
       ITMAX is the maximum number of steps of iterative refinement.



 LAPACK routine (version 3.1)    November 2006                       DPPRFS(3)