Some Efficient Algorithms to Raise Order of Convergence of  Iterative Methods for Nonlinear Equations with Applications: Some Efficient Algorithms to Raise Order of Convergence

Hameer Akhtar  Abro; Muhammad Mujtaba  Shaikh

Authors

Hameer Akhtar Abro Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro
Muhammad Mujtaba Shaikh Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro , Supply Chain and Operations Management Research Group, Mehran University of Engineering and Technology, Jamshoro

Keywords:

Nonlinear Equations, Efficient Algorithms, Iterative Methods, CPU Time, Order of Convergence, Computational Efficiency

Abstract

Two algorithms to raise the order of convergence of nonlinear solvers for scalar nonlinear equations are proposed. The suggested algorithms comprise of three steps: first two being any existing iterative method for nonlinear equations with nth order convergence, and the proposed third step being free from any new derivative. The third step in the proposed algorithms uses two different divided difference approximations to replace new derivatives. The order of convergence is raised to n + 3 and n + 4, respectively, when the third steps are combined with any two-step order convergent method. The extension in orders of convergence of methods is proved theoretically. As an application of proposed algorithms, proposed third steps are combined with some well-known existing third, fourth and fifth order two-step methods from literature. The consequent improvement in order of convergence is justified for five new methods derived using proposed algorithms. The computational performance of the proposed methods and some other similar order methods from literature are examined on several nonlinear equations of different nature including engineering problems and real mechanical system. All the proposed methods exhibit encouraging performance for test examples, and also for some applied nonlinear equations, like NASA’s launched satellite, real mechanical system, catenary cable and thermodynamics application.

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Some Efficient Algorithms to Raise Order of Convergence of Iterative Methods for Nonlinear Equations with Applications