University of Oulu

Huhtanen, M. & Kotila, V. Bit Numer Math (2019) 59: 125.

Optimal quotients for solving large eigenvalue problems

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Author: Huhtanen, Marko1; Kotila, Vesa1
Organizations: 1Faculty of Information Technology and Electrical Engineering, University of Oulu
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
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Language: English
Published: Springer Nature, 2019
Publish Date: 2019-09-07


Quotients for eigenvalue problems (generalized or not) are considered. To have a quotient optimally approximating an eigenvalue, conditions are formulated to maximize the one-dimensional projection of the eigenvalue problem. Respective optimal quotient iterations are derived under the assumption that applying the inverse is affordable. Inexact methods are also considered if applying the inverse is not affordable. Then, to approximate an eigenvector, optimality conditions are formulated to minimize linear independency over a subspace. Equivalence transformations are performed for preconditioning iterations and steering the convergence. These ideas extend to subspaces in a natural way. For the standard eigenvalue problem, a new Arnoldi method arises as an alternative to the classical Arnoldi method.

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Series: BIT numerical mathematics
ISSN: 0006-3835
ISSN-E: 1572-9125
ISSN-L: 0006-3835
Volume: 59
Issue: 1
Pages: 125 - 154
DOI: 10.1007/s10543-018-0725-x
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: The research of both authors was supported by the Academy of Finland (project 288641).
Academy of Finland Grant Number: 288641
Detailed Information: 288641 (Academy of Finland Funding decision)
Copyright information: © Springer Nature B.V. 2018. This is a post-peer-review, pre-copyedit version of an article published in Bit Numer Math. The final authenticated version is available online at: