An Iterative Hybrid Algorithm for Roots of Non-Linear Equations

Chaman Lal Sabharwal

doi:10.3390/eng2010007

Abstract

Finding the roots of non-linear and transcendental equations is an important problem in engineering sciences. In general, such problems do not have an analytic solution; the researchers resort to numerical techniques for exploring. We design and implement a three-way hybrid algorithm that is a blend of the Newton–Raphson algorithm and a two-way blended algorithm (blend of two methods, Bisection and False Position). The hybrid algorithm is a new single pass iterative approach. The method takes advantage of the best in three algorithms in each iteration to estimate an approximate value closer to the root. We show that the new algorithm outperforms the Bisection, Regula Falsi, Newton–Raphson, quadrature based, undetermined coefficients based, and decomposition-based algorithms. The new hybrid root finding algorithm is guaranteed to converge. The experimental results and empirical evidence show that the complexity of the hybrid algorithm is far less than that of other algorithms. Several functions cited in the literature are used as benchmarks to compare and confirm the simplicity, efficiency, and performance of the proposed method.

Highlights

The variations of continuity-based Bisection and False Position methods are due to Dekker [1,2], Brent [3], Press [4], and several variants of False Position including the reverse quadratic interpolation (RQI) method
The new hybrid algorithm has been tested to ensure that it performs better than other existing methods by optimizing the number of iterations required for approximations, the computation order of convergence, and the efficiency index for the test cases
All tables show that the hybrid algorithm performs better with respect to number of iterations, number of function evaluations, efficiency index, and computational order of convergence

Summary

Introduction

New efficient methods for solving nonlinear equations are evolving frequently, and are ubiquitously explored and exploited in applications. Their purpose is to improve the existing methods, such as classical Bisection, False Position, Newton–Raphson, and their variant methods for efficiency, simplicity, and approximation reliability in the solution. There are various ways to approach a problem; some methods are based on only continuous functions while others take advantage of the differentiability of functions The algorithms such as Bisection using midpoint, False Position using secant line intersection, and Newton–Raphson using tangent line intersection are ubiquitous. The variations of derivative based quadratic order Newton–Raphson method are 3rd, 4th, 5th, 6th order methods From these algorithms, the researcher has to find a suitable algorithm that works best for every function [5,6]. If the same algorithm is used on a different, smaller interval [0, 3], iterations will go on forever to get to 2; it will need some tolerance on error or on the number of iterations to terminate the algorithm

Methods

Results

Conclusion