Ovtchinnikov, E. and Xanthis, L. 2001. Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates. *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.* 457 (2006), pp. 441-451. doi:10.1098/rspa.2000.0674

Title | Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates |
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Authors | Ovtchinnikov, E. and Xanthis, L. |

Abstract | We present a new subspace iteration method for the efficient computation of several smallest eigenvalues of the generalized eigenvalue problem Au = lambda Bu for symmetric positive definite operators A and B. We call this method successive eigenvalue relaxation, or the SER method (homoechon of the classical successive over-relaxation, or SOR method for linear systems). In particular, there are two significant features of SER which render it computationally attractive: (i) it can effectively deal with preconditioned large-scale eigenvalue problems, and (ii) its practical implementation does not require any information about the preconditioner used: it can routinely accommodate sophisticated preconditioners designed to meet more exacting requirements (e.g. three-dimensional elasticity problems with small thickness parameters). We endow SER with theoretical convergence estimates allowing for multiple and clusters of eigenvalues and illustrate their usefulness in a numerical example for a discretized partial differential equation exhibiting clusters of eigenvalues. |

Keywords | large-scale eigenvalue problems, eigensolvers with preconditioning, subspace iteration, convergence rate, multiple and clustered eigenvalues |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Journal citation | 457 (2006), pp. 441-451 |

ISSN | 1364-5021 |

Year | 08 Feb 2001 |

Digital Object Identifier (DOI) | doi:10.1098/rspa.2000.0674 |

Publication dates | |

Published | 08 Feb 2001 |

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