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.

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

SERRA CAPIZZANO, STEFANO;
2015-01-01

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.
2015
Iterative methods; Multi-step; Nonlinear ordinary differential equations; Nonlinear partial differential equations; Systems of nonlinear equations
Ullah, M. Z.; SERRA CAPIZZANO, Stefano; Ahmad, F.; Al Aidarous, E. S.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2022751
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 4
social impact