Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations: Application to nonlinear PDEs and ODEs

Malik Zaka Ullah, S. Serra-Capizzano, Fayyaz Ahmad and Eman S. Al-Aidarous

Applied Mathematics and Computation, 2015, vol. 269, issue C, 972-987

Abstract: In the present study, we consider multi-step iterative method to solve systems of nonlinear equations. Since the Jacobian evaluation and its inversion are expensive, in order to achieve a better computational efficiency, we compute Jacobian and its inverse only once in a single cycle of the proposed multi-step iterative method. Actually the involved systems of linear equations are solved by employing the LU-decomposition, rather than inversion. The primitive iterative method (termed base method) has convergence-order (CO) five and then we describe a matrix polynomial of degree two to design a multi-step method. Each inclusion of single step in the base method will increase the convergence-order by three. The general expression for CO is 3s−1, where s is the number of steps of the multi-step iterative method. Computational efficiency is also addressed in comparison with other existing methods. The claimed convergence-rates proofs are established. The new contribution in this article relies essentially in the increment of CO by three for each added step, with a comparable computational cost in comparison with existing multi-steps iterative methods. Numerical assessments are made which justify the theoretical results: in particular, some systems of nonlinear equations associated with the numerical approximation of partial differential equations (PDEs) and ordinary differential equations (ODEs) are built up and solved.

Keywords: Multi-step; Iterative methods; Systems of nonlinear equations; Nonlinear partial differential equations; Nonlinear ordinary differential equations (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (3)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:269:y:2015:i:c:p:972-987

DOI: 10.1016/j.amc.2015.07.096

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