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

           CHARACTER      UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IWORK( * )

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


PURPOSE

       SPPRFS 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) REAL 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) REAL 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  SPPTRF/CPPTRF,  packed  columnwise in a
               linear array in the same format as A (see AP).

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

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

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