http://scholars.ntou.edu.tw/handle/123456789/24704
標題: | Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem | 作者: | Liu, Chein-Shan Chang, Chih-Wen Kuo, Chung-Lun |
關鍵字: | optimal fourth-order one-step iterative schemes;memory-accelerating method;optimal combination function;optimal relaxation factor;Lie symmetry method | 公開日期: | 2024 | 出版社: | MDPI | 卷: | 16 | 期: | 1 | 來源出版物: | SYMMETRY-BASEL | 摘要: | In this paper, some one-step iterative schemes with memory-accelerating methods are proposed to update three critical values f '(r), f ''(r), and f '''(r) of a nonlinear equation f(x) = 0 with r being its simple root. We can achieve high values of the efficiency index (E.I.) over the bound 2(2/3) = 1.587 with three function evaluations and over the bound 2(1/2) = 1.414 with two function evaluations. The third-degree Newton interpolatory polynomial is derived to update these critical values per iteration. We introduce relaxation factors into the Dzunic method and its variant, which are updated to render fourth-order convergence by the memory-accelerating technique. We developed six types optimal one-step iterative schemes with the memory-accelerating method, rendering a fourth-order convergence or even more, whose original ones are a second-order convergence without memory and without using specific optimal values of the parameters. We evaluated the performance of these one-step iterative schemes by the computed order of convergence (COC) and the E.I. with numerical tests. A Lie symmetry method to solve a second-order nonlinear boundary-value problem with high efficiency and high accuracy was developed. |
URI: | http://scholars.ntou.edu.tw/handle/123456789/24704 | DOI: | 10.3390/sym16010120 |
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