BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-87-01
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: The convergence of inexact Chebyshev and Richardson iterative
               methods for solving linear systems
        TYPE:: Manuscript
      AUTHOR:: Golub, Gene H.
      AUTHOR:: Overton, Michael L.
        DATE:: February 1987
       PAGES:: 38
    ABSTRACT:: The Chebyshev and second-order Richardson methods are
               classical iterative schemes for solving linear systems. We
               consider the convergence analysis of these methods when each
               step of the iteration is carried out inexactly. This has many
               applications, since a preconditioned iteration requires, at
               each step, the solution of linear system which may be solved
               inexactly using an "inner" iteration. We derive an error
               bound which applies to the general nonsymmetric inexact
               Chebyshev iteration. We show how this simplifies slightly in
               the case of a symmetric or skew-symmetric iteration, and we
               consider both the cases of underestimating and overestimating
               the spectrum. We show that in the symmetric case, it is
               actually advantageous to underestimate the spectrum when the
               spectral radius and the degree of inexactness are both large.
               This is not true in the case of the skew-symmetric iteration.
               We show how similar results apply to the Richardson
               iteration. Finally, we describe numerical experiments which
               illustrate the results and suggest that the Chevyshev and
               Richardson methods, with reasonable param eter choices, may
               be more effective than the conjugate gradient method in the
               presence of inexactness.
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-87-01