### Abstract

Original language | English |
---|---|

Pages (from-to) | 27-44 |

Number of pages | 18 |

Journal | BIT Numerical Mathematics |

Volume | 47 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2007 |

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### Keywords

- Preconditioned iterative methods
- Newton’s method
- Inverse iteration

### Cite this

**Convergence of inexact inverse iteration with application to preconditioned iterative solves.** / Freitag, MA; Spence, A.

Research output: Contribution to journal › Article

*BIT Numerical Mathematics*, vol. 47, no. 1, pp. 27-44. https://doi.org/10.1007/s10543-006-0100-1

}

TY - JOUR

T1 - Convergence of inexact inverse iteration with application to preconditioned iterative solves

AU - Freitag, MA

AU - Spence, A

PY - 2007/3

Y1 - 2007/3

N2 - In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results.

AB - In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem A x=λM x. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results.

KW - Preconditioned iterative methods

KW - Newton’s method

KW - Inverse iteration

UR - http://www.springerlink.com/content/eg27mk213gr75061/

U2 - 10.1007/s10543-006-0100-1

DO - 10.1007/s10543-006-0100-1

M3 - Article

VL - 47

SP - 27

EP - 44

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 1

ER -