CSPRFS - 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 CSPRFS( UPLO, N, NRHS, AP, AFP, IPIV,  B,  LDB,
                          X, LDX, FERR, BERR, WORK, RWORK, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           INTEGER        IPIV( * )

           REAL           BERR( * ), FERR( * ), RWORK( * )

           COMPLEX        AFP( * ), AP( * ), B( LDB, * ), WORK( *
                          ), X( LDX, * )


PURPOSE

       CSPRFS  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) COMPLEX 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) COMPLEX 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 CSPTRF, stored as a packed triangular matrix.

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

       B       (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS)
               On  entry,  the  solution matrix X, as computed by
               CSPTRS.  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) COMPLEX array, dimension (2*N)

       RWORK   (workspace) REAL 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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