 
Summary: (Revised) RALTR2002021
A Chebyshevbased twostage iterative method as an
alternative to the direct solution of linear systems
Mario Arioli 1 and Daniel Ruiz 2
ABSTRACT
We consider the solution of illconditioned symmetric and positive definite large sparse
linear systems of equations. These arise, for instance, when using some symmetrising pre
conditioning technique for solving a general (possibly unsymmetric) illconditioned linear
system, or in domain decomposition of a numerically di#cult elliptic problem. We are
also concerned with the consecutive solution of several linear systems with the same ma
trix and di#erent righthand sides. In such cases, the consecutive runs of some iterative
methods like the conjugate gradient or the block conjugate gradient algorithms might be
computationally prohibitive, and it might be preferable to use direct methods which are
very well suited for this type of situation.
The purpose of our study is to analyse a twophase approach: analogously to direct meth
ods, it includes a preliminary phase which we call a ``partial spectral factorization phase'',
followed by a ``cheap'' solution phase, both only based on numerical tools that are usu
ally exploited in iterative methods. Therefore, we need not store the given illconditioned
matrix explicitly but we need only use it to compute matrixvector products. This is
of particular interest in the case of very large sparse systems and permits an e#cient
