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On the Vector ε-Algorithm for Solving Linear Systems of Equations

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Abstract

The four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation and the topological epsilon algorithm, when applied to linearly generated vector sequences are Krylov subspace methods and it is known that they are equivalent to some well-known conjugate gradient type methods. However, the vector ε-algorithm is an extrapolation method, older than the four extrapolation methods above, and no similar results are known for it. In this paper, a determinantal formula for the vector ε-algorithm is given. Then it is shown that, when applied to a linearly generated vector sequence, the algorithm is also a Krylov subspace method and for a class of matrices the method is equivalent to a preconditioned Lanczos method. A new determinantal formula for the CGS is given, and an algebraic comparison between the vector ε-algorithm for linear systems and CGS is also given.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Pures et Appliquées J. Liouville, Université du Littoral, Centre Universitaire de la Mi-Voix, Bâtiment Poincaré, 50 rue F. Buisson, B.P. 699, 62228, Calais Cedex, France

    A. Salam

  2. Department of Computing, University of Bradford, West Yorkshire, BD7 1DP, England

    P.R. Graves-Morris

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  1. A. Salam
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  2. P.R. Graves-Morris
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Salam, A., Graves-Morris, P. On the Vector ε-Algorithm for Solving Linear Systems of Equations. Numerical Algorithms 29, 229–247 (2002). https://doi.org/10.1023/A:1014832627766

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