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Combination of the sequential secant method and Broyden's method with projected updates

Kombination des sequentiellen Sekantenverfahrens mit dem Verfahren von Broyden mit projizierten Korrekturen

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Abstract

We introduce a new algorithm for solving nonlinear simultaneous equations, which is a combination of the sequential secant method with Broyden's Quasi-Newton method with projected updates as introduced by Gay and Schnabel. The new algorithm has the order of convergence of the sequential secant method and the choice of the first increments is justified by the minimum variation principles of Quasi-Newton methods. Two versions of the method are compared numerically with some well-known test problems.

Zusammenfassung

Wir stellen einen neuen Algorithmus zum Lösen von Systemen nichtlinearer Ungleichungen vor, der eine Kombination des sequentiellen Sekantenverfahrens mit dem Broydenschen Quasi-Newton-Verfahren mit projizierten Korrekturen ist, wie es von Gay und Schnabel vorgeschlagen worden ist. Der neue Algorithmus hat die Konvergenzordnung des sequentiellen Sekantenverfahrens, und die Wahl der ersten Inkremente ist durch das Prinzip der minimalen Variation bei Quasi-Newton-Verfahren gerechtfertigt. Zwei Fassungen des Verfahrens werden an einigen bekannten Testproblemen numerisch verglichen.

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Authors and Affiliations

  1. Department of Applied Mathematics, IMECC-UNICAMP, CP 1170, 13100, Campinas, SP, Brazil

    J. M. Martínez &  T. L. Lopes

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  1. J. M. Martínez
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  2. T. L. Lopes
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Martínez, J.M., Lopes, T.L. Combination of the sequential secant method and Broyden's method with projected updates. Computing 25, 379–386 (1980). https://doi.org/10.1007/BF02285232

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  • DOI: https://doi.org/10.1007/BF02285232

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