SSPRFS - improve the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  symmetric
       indefinite and packed, and provides error bounds and back­
       ward error estimates for the solution


SYNOPSIS

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

           CHARACTER      UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IPIV( * ), IWORK( * )

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


PURPOSE

       SSPRFS  improves the computed solution to a system of lin­
       ear equations when the  coefficient  matrix  is  symmetric
       indefinite and packed, and provides error bounds and back­
       ward 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)*(2*n-
               j)/2) = A(i,j) for j<=i<=n.

       AFP     (input) REAL array, dimension (N*(N+1)/2)
               The factored form of the matrix A.   AFP  contains
               the  block  diagonal  matrix D and the multipliers
               used to obtain the factor U or L from the  factor­
               ization  A  = U*D*U**T or A = L*D*L**T as computed
               by SSPTRF, stored as a packed triangular matrix.

       IPIV    (input) INTEGER array, dimension (N)
               Details  of  the  interchanges   and   the   block


       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).

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