### Abstract

The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an "inner" iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.

Original language | English (US) |
---|---|

Pages (from-to) | 571-593 |

Number of pages | 23 |

Journal | Numerische Mathematik |

Volume | 53 |

Issue number | 5 |

DOIs | |

State | Published - Aug 1988 |

### Fingerprint

### Keywords

- Subject Classifications: AMS(MOS): 65F10, CR: G1.3

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics
- Mathematics(all)

### Cite this

**The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems.** / Golub, Gene H.; Overton, Michael L.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 53, no. 5, pp. 571-593. https://doi.org/10.1007/BF01397553

}

TY - JOUR

T1 - The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems

AU - Golub, Gene H.

AU - Overton, Michael L.

PY - 1988/8

Y1 - 1988/8

N2 - The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an "inner" iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.

AB - The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an "inner" iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.

KW - Subject Classifications: AMS(MOS): 65F10, CR: G1.3

UR - http://www.scopus.com/inward/record.url?scp=0001263639&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001263639&partnerID=8YFLogxK

U2 - 10.1007/BF01397553

DO - 10.1007/BF01397553

M3 - Article

AN - SCOPUS:0001263639

VL - 53

SP - 571

EP - 593

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 5

ER -