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Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations

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Parallel Numerical Algorithms

Part of the book series: ICASE/LaRC Interdisciplinary Series in Science and Engineering ((ICAS,volume 4))

Abstract

Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly.

The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method (Sorensen, 1992) is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

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Author information

Authors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, Houston, TX, 77251-1892, USA

    Danny C. Sorensen

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  1. Danny C. Sorensen
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Editor information

Editors and Affiliations

  1. Old Dominion University, Norfolk, Virginia, USA

    David E. Keyes

  2. University of Minnesota, Minneapolis, Minnesota, USA

    Ahmed Sameh

  3. ICASE, Hampton, Virginia, USA

    David E. Keyes  & V. Venkatakrishnan  & 

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© 1997 Springer Science+Business Media Dordrecht

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Sorensen, D.C. (1997). Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations. In: Keyes, D.E., Sameh, A., Venkatakrishnan, V. (eds) Parallel Numerical Algorithms. ICASE/LaRC Interdisciplinary Series in Science and Engineering, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5412-3_5

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  • DOI: https://doi.org/10.1007/978-94-011-5412-3_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6277-0

  • Online ISBN: 978-94-011-5412-3

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