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Iterative Methods with Memory for Solving Systems of Nonlinear Equations Using a Second Order Approximation

Author

Listed:
  • Alicia Cordero

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain
    These authors contributed equally to this work.)

  • Javier G. Maimó

    (Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic)

  • Juan R. Torregrosa

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain
    These authors contributed equally to this work.)

  • María P. Vassileva

    (Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic
    These authors contributed equally to this work.)

Abstract

Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.

Suggested Citation

  • Alicia Cordero & Javier G. Maimó & Juan R. Torregrosa & María P. Vassileva, 2019. "Iterative Methods with Memory for Solving Systems of Nonlinear Equations Using a Second Order Approximation," Mathematics, MDPI, vol. 7(11), pages 1-12, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1069-:d:284468
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    References listed on IDEAS

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    1. Soleymani, F. & Lotfi, T. & Tavakoli, E. & Khaksar Haghani, F., 2015. "Several iterative methods with memory using self-accelerators," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 452-458.
    2. Campos, Beatriz & Cordero, Alicia & Torregrosa, Juan R. & Vindel, Pura, 2015. "A multidimensional dynamical approach to iterative methods with memory," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 701-715.
    Full references (including those not matched with items on IDEAS)

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