From Wed Mar 6 14:13 MST 1996 Received: from ( []) by newton.Colorado.EDU (8.6.13/8.6.11/Unixops/Hesiod/(SDM)) with SMTP id OAA05278 for ; Wed, 6 Mar 1996 14:13:18 -0700 Received: by (5.65/DEC-OSF/1.2) id AA00433; Wed, 6 Mar 1996 14:14:24 -0700 Date: Wed, 6 Mar 1996 14:14:24 -0700 From: (Catherine H. Rachwalski) Message-Id: <> To: Subject: popov Content-Type: text Content-Length: 1980 Status: R


              S.S.Bielawski, S.G.Mulyarchik, and A.V.Popov*

    Radio Physics and Electronics Faculty, Belarusian State University,
                4 F.Scoriny Ave., Minsk, 220080, Belarus
   * - present address: The Economics Institute, University of Colorado,
                1005 12th. St., Boulder, CO, 80302, USA

        We show how auxiliary subspaces and related projectors may be used
for preconditioning nonsymmetric system of linear equations. It is shown
that preconditioned in such a way (or projected) system is better
conditioned than original system (at least if the coefficient matrix of the
system to be solved is symmetrizable). Two approaches for solving the
projected system are outlined. The first one implies straightforward
computation of the projected matrix and consequent using some direct or
iterative method. The second pproach is projection preconditioning of
conjugate gradient-type solver. The latter approach is developed here in
context with biconjugate gradient iteration and some other related
Lanczos-type algorithms. The developed technique is of no practical value
unless the auxiliary subspaces can be determined. It is natural to define a
subspace by choosing its basis. Some possible choices of basis vectors are
discussed, and one of them is presented in details. This particular choice
reduces projected system to Schur complement system if unit basis vectors
are determined according to a graph coloring. The developed technique may be
applied by itself or in conjunction with one of commonly used
preconditioners. We present the results of some numerical experiments that
have been carried out for some Lanczos-type methods using basis vectors
which are determined by red-black and some linear orderings. The other kinds
of projection preconditioning are the subjects of current research.