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


SYNOPSIS

       SUBROUTINE ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF,  IPIV,
                          B,  LDB,  X,  LDX,  FERR,  BERR,  WORK,
                          RWORK, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDA, LDAF, LDB, LDX, N, NRHS

           INTEGER        IPIV( * )

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

           COMPLEX*16     A(  LDA,  * ), AF( LDAF, * ), B( LDB, *
                          ), WORK( * ), X( LDX, * )


PURPOSE

       ZSYRFS improves the computed solution to a system of  lin­
       ear  equations  when  the  coefficient matrix is symmetric
       indefinite, 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.

       A       (input) COMPLEX*16 array, dimension (LDA,N)
               The symmetric matrix A.  If UPLO = 'U', the  lead­
               ing N-by-N upper triangular part of A contains the
               upper triangular part of the  matrix  A,  and  the
               strictly  lower triangular part of A is not refer­
               enced.  If UPLO = 'L', the  leading  N-by-N  lower
               triangular part of A contains the lower triangular
               part of the matrix A, and the strictly upper  tri­
               angular part of A is not referenced.

       LDA     (input) INTEGER
               The  leading  dimension  of  the  array A.  LDA >=
               max(1,N).

               The factored form of the matrix  A.   AF  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 ZSYTRF.

       LDAF    (input) INTEGER
               The leading dimension of the array  AF.   LDAF  >=
               max(1,N).

       IPIV    (input) INTEGER array, dimension (N)
               Details  of  the interchanges and the block struc­
               ture of D as determined by ZSYTRF.

       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
               ZSYTRS.   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

               gal value


PARAMETERS

       ITMAX  is the maximum number of steps of iterative refine­
       ment.


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