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The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

Author

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  • Bilel Kchouk

    (University of Sherbrooke)

  • Jean-Pierre Dussault

    (University of Sherbrooke)

Abstract

We present a method, based on the Chebyshev third-order algorithm and accelerated by a Shamanskii-like process, for solving nonlinear systems of equations. We show that this new method has a quintic convergence order. We will also focus on efficiency of high-order methods and more precisely on our new Chebyshev–Shamanskii method. We also identify the optimal use of the same Jacobian in the Shamanskii process applied to the Chebyshev method. Some numerical illustrations will confirm our theoretical analysis.

Suggested Citation

  • Bilel Kchouk & Jean-Pierre Dussault, 2013. "The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 148-167, April.
    • Handle: RePEc:spr:joptap:v:157:y:2013:i:1:d:10.1007_s10957-012-0159-6
    DOI: 10.1007/s10957-012-0159-6
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    References listed on IDEAS

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    1. J. A. Ezquerro & M. Grau-Sánchez & A. Grau & M. A. Hernández & M. Noguera & N. Romero, 2011. "On Iterative Methods with Accelerated Convergence for Solving Systems of Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 163-174, October.
    2. F. Lampariello & M. Sciandrone, 2001. "Global Convergence Technique for the Newton Method with Periodic Hessian Evaluation," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 341-358, November.
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