Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

The Chebyshev iteration revisited

by M. H. Gutknecht and S. Röllin

(Report number 2000-16)

Abstract
Compared to Krylov space methods based on orthogonal or oblique projection, the Chebyshev iteration does not require inner products and is therefore particularly suited for massively parallel computers with high communication cost. We compare six different algorithms that implement this methods and compare them with respect to roundoff effects, in particular, the ultimately achievable accuracy. Two of these algorithms replace the three-term recurrences by more accurate coupled two-term recurrences and seem to be new. But we also show that, for real data, the classical three-term Chebyshev iteration is never seriously affected by roundoff, in contrast to the corresponding version of the conjugate gradient method. Even for complex data, strong roundoff effects are seen to be limited to very special situations where convergence is anyway slow. The Chebyshev iteration is applicable to symmetric definite linear systems and to nonsymmetric matrices whose eigenvalues are known to be confined to an elliptic domain that does not include the origin. We also consider a corresponding stationary 2-step method, which has the same asymptotic convergence behavior and is additionally suitable for mildly nonlinear problems.

Keywords: sparse linear systems, Chebyshev iteration, second order Richarson iteration, coupled two-term recurrences, roundoff error analysis

@Techreport{GR00_275,
  author = {M. H. Gutknecht and S. R\"ollin},
  title = {The Chebyshev iteration revisited},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2000-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2000/2000-16.pdf },
  year = {2000}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).