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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}
}

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