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A parameterized multi-step Newton method for solving systems of nonlinear equations

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Abstract

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter θ to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of θ, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.

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Authors and Affiliations

  1. Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte d’Urgell 187, 08036, Barcelona, Spain

    Fayyaz Ahmad

  2. Department of Mathematics, Kosar University of Bojnord, P. O. Box 9415615458, Bojnord, Iran

    Emran Tohidi

  3. Departament d’Enginyeria Electrònica, Universitat Politècnica de Catalunya, Diagonal 647, planta 9, 08028, Barcelona, Spain

    Juan A. Carrasco

Authors
  1. Fayyaz Ahmad
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  2. Emran Tohidi
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  3. Juan A. Carrasco
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Correspondence to Emran Tohidi.

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Ahmad, F., Tohidi, E. & Carrasco, J.A. A parameterized multi-step Newton method for solving systems of nonlinear equations. Numer Algor 71, 631–653 (2016). https://doi.org/10.1007/s11075-015-0013-7

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