### Abstract

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

Pages (from-to) | 1069-1082 |

Number of pages | 14 |

Journal | SIAM Journal On Matrix Analysis and Applications (SIMAX) |

Volume | 28 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2006 |

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### Keywords

- Eigenvalue approximation
- Inverse iteration
- Iterative methods

### Cite this

*SIAM Journal On Matrix Analysis and Applications (SIMAX)*,

*28*(4), 1069-1082. https://doi.org/10.1137/050623255

**Shift for nonsymmetric generalised eigenvalue problems.** / Berns-Muller, J; Spence, A.

Research output: Contribution to journal › Article

*SIAM Journal On Matrix Analysis and Applications (SIMAX)*, vol. 28, no. 4, pp. 1069-1082. https://doi.org/10.1137/050623255

}

TY - JOUR

T1 - Shift for nonsymmetric generalised eigenvalue problems

AU - Berns-Muller, J

AU - Spence, A

N1 - This is the author's final, peer-reviewed version. ©Society for Industrial and Applied Mathematics

PY - 2006/12

Y1 - 2006/12

N2 - In this paper we analyze inexact inverse iteration for the nonsymmetric generalized eigenvalue problem Ax = λMx, where M is symmetric positive definite and the problem is diagonalizable. Our analysis is designed to apply to the case when A and M are large and sparse and preconditioned iterative methods are used to solve shifted linear systems with coefficient matrix A − σM. We prove a convergence result for the variable shift case (for example, where the shift is the Rayleigh quotient) which extends current results for the case of a fixed shift. Additionally, we consider the approach from [V. Simoncini and L. Elden, BIT, 42 (2002), pp. 159–182] to modify the right-hand side when using preconditioned solves. Several numerical experiments are presented thatillustrate the theory and provide a basis for the discussion of practical issues.

AB - In this paper we analyze inexact inverse iteration for the nonsymmetric generalized eigenvalue problem Ax = λMx, where M is symmetric positive definite and the problem is diagonalizable. Our analysis is designed to apply to the case when A and M are large and sparse and preconditioned iterative methods are used to solve shifted linear systems with coefficient matrix A − σM. We prove a convergence result for the variable shift case (for example, where the shift is the Rayleigh quotient) which extends current results for the case of a fixed shift. Additionally, we consider the approach from [V. Simoncini and L. Elden, BIT, 42 (2002), pp. 159–182] to modify the right-hand side when using preconditioned solves. Several numerical experiments are presented thatillustrate the theory and provide a basis for the discussion of practical issues.

KW - Eigenvalue approximation

KW - Inverse iteration

KW - Iterative methods

UR - http://scitation.aip.org/journals/doc/SIAMDL-home/jrnls/top.jsp?key=SJMAEL

U2 - 10.1137/050623255

DO - 10.1137/050623255

M3 - Article

VL - 28

SP - 1069

EP - 1082

JO - SIAM Journal On Matrix Analysis and Applications (SIMAX)

JF - SIAM Journal On Matrix Analysis and Applications (SIMAX)

SN - 0895-4798

IS - 4

ER -