BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//NA-M-87-04
       ENTRY:: January 28, 1996
ORGANIZATION:: Stanford University, Department of Computer Science, Numerical
               Analysis Project
       TITLE:: The convergence rate of inexact preconditioned steepest
               descent algorithms for solving linear systems
        TYPE:: Manuscript
      AUTHOR:: Munthe-Kaas, Hans
        DATE:: March 1987
       PAGES:: 20
    ABSTRACT:: The steepest descent algorithm is a classical iterative
               method for solving a linear system Ax=b, where A is a
               positive definite symmetric matrix. A common way to
               accelerate an iterative scheme is to precondition the method,
               i.e. to solve a simpler system Mz=r in each stage of the
               iteration. We analyze the effect of solving the
               preconditioner inexactly. A lower bound for the convergence
               rate is derived, and we show under what conditions this lower
               bound is obtained. Finally we describe some numerical
               experiments which show that in practical situations the lower
               bound may be too pessimistic. An amusing result is that in
               some cases small errors may lead to $\underline{higher}$
               convergence rates than if the preconditioner is solved
               exactly!
       NOTES:: [Adminitrivia V1/Prg/19960128]
         END:: STAN//NA-M-87-04