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Iterative Eigenvalue Algorithms for Large Symmetric Matrices

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Numerische Behandlung von Eigenwertaufgaben

Summary

We compare several iterative eigenvalue algorithms applicable to large sparse symmetric matrices, of so high an order that the matrix A cannot be stored in the memory of the computer, while it is easy to compute y = Ax for a given vector x.

Our main interest is focused to the Lanczos algorithm, and algorithms that apply an optimization strategy to the Rayleigh quotient, mainly steepest descent, multiple step steepest descent and conjugate gradients. We study the rates of convergence, nd find that the theoretical bounds for all the algorithms is determined by the separation of the extreme eigenvalue from the rest of the spectrum of the matrix.

We discuss computer implementation and report several numerical tests. These show a marked superiority for the Lanczos algorithm compared to the others. Of the other algorithms, the c-g algorithm also works well in wellconditioned cases, and it can be realized by a very simple program.

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  1. Umea, Schweden

    A. Ruhe

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  1. A. Ruhe
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© 1974 Springer Basel AG

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Ruhe, A. (1974). Iterative Eigenvalue Algorithms for Large Symmetric Matrices. In: Numerische Behandlung von Eigenwertaufgaben. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 24. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5518-1_9

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  • DOI: https://doi.org/10.1007/978-3-0348-5518-1_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5519-8

  • Online ISBN: 978-3-0348-5518-1

  • eBook Packages: Springer Book Archive

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