On polynomial preconditioning and asymptotic convergence factors for indefinite Hermitian matrices

  • Freund R
  • 5


    Mendeley users who have this article in their library.
  • 10


    Citations of this article.


We are concerned with the minimal residual method combined with polynomial preconditioning for solving large linear systems Ax=b with indefinite Hermitian coefficient matrices A. The standard approach for choosing the polynomial preconditioner leads to preconditioned systems which are positive definite. Here, we investigate a different strategy which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive (negative) eigenvalues of A around 1 (around some negative constant). In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two-parameter family of Chebyshev approximation problems. The problem of selecting the parameters so that the resulting indefinite polynomial preconditioner speeds up the convergence of the minimal residual method optimally is also addressed. For this task, we propose an approach based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given. © 1991.

Get free article suggestions today

Mendeley saves you time finding and organizing research

Sign up here
Already have an account ?Sign in

Find this document


  • Roland Freund

Cite this document

Choose a citation style from the tabs below

Save time finding and organizing research with Mendeley

Sign up for free