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Geometric interpretation of some Cauchy related methods

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Abstract

This paper shows that a class of methods for solving linear equations, including the Cauchy–Barzilai–Borwein method, can be interpreted by means of a simple geometric object, the Bézier parabola. This curve is built from the current iterate using a transformation characterizing the system to be solved. The localization of the next iterates in the plane of the parabola sheds some light on the behavior of the methods and provides some new understanding of their relative efficiency.

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Authors and Affiliations

  1. Institut de Mathématiques et de Modélisation de Montpellier, UMR CNRS 5149, Université Montpellier II, Cc 051, Place Eugène Bataillon, 34095, Montpellier Cedex 5, France

    Alain F. Berlinet & Christophe Roland

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  1. Alain F. Berlinet
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  2. Christophe Roland
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Correspondence to Alain F. Berlinet.

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Berlinet, A.F., Roland, C. Geometric interpretation of some Cauchy related methods. Numer. Math. 119, 437–464 (2011). https://doi.org/10.1007/s00211-011-0383-2

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  • DOI: https://doi.org/10.1007/s00211-011-0383-2

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