A stable class of modified Newton-like methods for multiple roots and their dynamics

Munish Kansal 1 , Alicia Cordero 2 , Juan R. Torregrosa 2  and Sonia Bhalla 3
  • 1 School of Mathematics, Thapar Institute of Engineering and Technology, 147004, Patiala, India
  • 2 Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022, Valencia, Spain
  • 3 Department of Mathematics, Chandigarh University, Gharuan, Mohali, India
Munish Kansal
  • Corresponding author
  • School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab, 147004, India
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, Alicia Cordero
  • Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, 46022, Spain
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, Juan R. Torregrosa
  • Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, 46022, Spain
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and Sonia Bhalla

Abstract

There have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

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The International Journal of Nonlinear Sciences and Numerical Simulation publishes original papers on all subjects relevant to nonlinear sciences and numerical simulation. The journal is directed at researchers in nonlinear sciences, engineers, and computational scientists, economists, and others, who either study the nature of nonlinear problems or conduct numerical simulations of nonlinear problems.

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