ZSPRFS - 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 ZSPRFS( 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( * )

           DOUBLE         PRECISION BERR( * ), FERR( * ),  RWORK(
                          * )

           COMPLEX*16     AFP( * ), AP( * ), B( LDB, * ), WORK( *
                          ), X( LDX, * )


PURPOSE

       ZSPRFS 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*16 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*16 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 ZSPTRF, stored as a packed triangular matrix.

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

       B       (input) COMPLEX*16 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*16 array, dimension
               (LDX,NRHS)
               On entry, the solution matrix X,  as  computed  by
               ZSPTRS.   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) COMPLEX*16 array, dimension (2*N)

       RWORK   (workspace) DOUBLE PRECISION 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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