DSPRFS - 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 DSPRFS( 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( * )

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


PURPOSE

       DSPRFS 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) 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)*(2*n-
               j)/2) = A(i,j) for j<=i<=n.

       AFP     (input) DOUBLE PRECISION 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 DSPTRF, stored as a packed triangular matrix.

               Details  of  the interchanges and the block struc­
               ture of D as determined by DSPTRF.

       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
               DSPTRS.   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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