Affiliations: [a] School of Mathematics and Information Technology, Yuncheng University, Yuncheng, Shanxi, China | [b] College of Mathematics and Statistics, Taishan University, Tai’an, Shandong, China
Correspondence:
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Corresponding author: Liang Fang, College of Mathematics and Statistics, Taishan University, Tai’an, Shandong, China. E-mail: [email protected].
Abstract: The root-fnding problem of a univariate nonlinear equation is a fundamental and long-studied problem, and has wide applications in mathematics and engineering computation. In this paper, two modified Newton-type methods and two variants of Super-Halley’s method to solve nonlinear equations are brought forward. Analytical discussions are reported and the theoretical efficiency of the methods is studied. Both of Algorithms 1 and 2 require three evaluations of the functions and two evaluations of derivatives at each iteration. Therefore the efficiency indices of Algorithms 1 and 2 are 1.431 and 1.476, respectively. The presented variants of Super-Halley’s methods require two evaluations of the functions and two evaluations of derivatives at each iteration. Therefore the efficiency indices of Algorithms 3 and 4 are 1.565. Hence, all efficiency indices of the proposed algorithms are better than that of classical Newton’s method 1.414. All of the proposed algorithms in this paper are free from second derivatives. Some numerical results are finally provided to support the theoretical discussions of the proposed methods.
Keywords: Iterative method, nonlinear equations, order of convergence, efficiency index, Newton’s method
