• 论文 • 上一篇    下一篇

两类五阶解非线性方程组的迭代算法

裕静静, 江平, 刘植   

  1. 合肥工业大学 数学学院, 合肥 230009
  • 收稿日期:2016-01-21 出版日期:2017-05-15 发布日期:2017-07-18
  • 通讯作者: 刘植,E-mail:liuzhi314@126.com
  • 基金资助:

    国家自然科学基金(11471093).

裕静静, 江平, 刘植. 两类五阶解非线性方程组的迭代算法[J]. 计算数学, 2017, 39(2): 151-166.

Yu Jingjing, Jiang Ping, Liu Zhi. THE TWO KIND OF ITERATIVE METHODS WITH FIFTH-ORDER FOR SOLVING THE SYSTEM OF NONLINEAR EQUATIONS[J]. Mathematica Numerica Sinica, 2017, 39(2): 151-166.

THE TWO KIND OF ITERATIVE METHODS WITH FIFTH-ORDER FOR SOLVING THE SYSTEM OF NONLINEAR EQUATIONS

Yu Jingjing, Jiang Ping, Liu Zhi   

  1. School of Mathematics, Hefei University of Technology, Hefei 230009, China
  • Received:2016-01-21 Online:2017-05-15 Published:2017-07-18
本文首先根据Runge-Kutta方法的思想,结合Newton迭代法,提出了一类带参数的解非线性方程组F(x)=0的迭代算法,然后基于解非线性方程fx)=0的King算法,给出第二类解非线性方程组的迭代算法,收敛性分析表明这两类算法都是五阶收敛的.其次给出了本文两类算法的效率指数,以及一些已知算法的效率指数,并且将本文算法的效率指数与其它方法进行详细的比较,通过效率比率Ri,j可知本文算法具有较高的计算效率.最后给出了四个数值实例,将本文两类算法与现有的几种算法进行比较,实验结果说明本文算法收敛速度快,迭代次数少,有明显的优势.
In this paper,for solving system of nonlinear equations,we use the Runge-Kutta method to achieve a family of iterative methods with parameters,which are based on Newton s method.And the other iterative method is built up which is based on the King s method for solving the nonlinear equation.And then,we prove that these two methods are convergent with fifth order.These two iterative method s efficiency indexes and some other recently published method s efficiency indexes are given.Compared with other methods,the two methods have higher computational efficiency.Finally,four numerical examples are given to show that our methods are effective.

MR(2010)主题分类: 

()
[1] 李庆扬, 莫孜中, 祁力群等. 非线性方程组的数值解法[M]. 北京:科学出版社, 1978.

[2] Darvishi M T, Barati A. A third-order Newton-type method to solve systems of nonlinear equations[J]. Applied Mathematics and Computation, 2007, 187(2):630-635.

[3] Darvishi M T, Barati A. Super cubic iterative methods to solve systems of nonlinear equations[J]. Applied Mathematics and Computation, 2007, 188(2):1678-1685.

[4] Noor M A, Waseem M. Some iterative methods for solving a system of nonlinear equations[J]. Computers and Mathematics with Applications, 2009, 57(1):101-106.

[5] Hafiz M A, Bahgat M S M. An efficient two-step iterative method for solving system of nonlinear equations[J]. Journal of Mathematics Research, 2012, 4(4):28-34.

[6] Coddero A, Torregrosa J R. Variants of Newton's method using fifth-order quadrature formulas[J]. Applied Mathematics and Computation, 2007, 190(1):686-689.

[7] Khirallah M Q, Hafiz M A. Novel three order methods for solving a system of nonlinear equations[J]. Bulletin of Society for Mathematical Services and Standards, 2012, 1(2):1-14.

[8] 代璐璐,檀结庆.两种解非线性方程组的四阶迭代算法[J]. 数值计算与计算机应用, 2012, 33(2):121-128.

[9] 张旭,檀结庆. 三步五阶迭代方法解非线性方程组[J]. 计算数学,2013, 35(3):297-304.

[10] Kou J, Li Y, Wang X. Some modifications of Newton's method with fifth-order convergence[J]. Journal of Computational and Applied Mathematics, 2007, 209(2):146-152.

[11] Homeier H H H. A modified Newton method with cubic convergence:the multivarite case[J]. Journal of Computational and Applied Mathematics, 2004, 169(1):161-169.

[12] Sharma J R, Gupta P. An efficient fifth order method for solving systems of nonlinear equations[J]. Computer and Mathematics with Applications, 2014, 67(3):591-601.

[13] Grau-S'anchez M, Grau A, Noguera M. On the computational efficiency index and some iterative methods for solving systems of nonlinear equations[J]. Journal of Computational and Applied Mathematics, 2011, 236:1259-1266.

[14] King R F. A family of fourth order methods for nonlinear equations[J]. Numer. Anal, 1973, 10:876-879.144.
[1] 郭俊, 吴开腾, 张莉, 夏林林. 一种新的求非线性方程组的数值延拓法[J]. 计算数学, 2017, 39(1): 33-41.
[2] 张旭, 檀结庆, 艾列富. 一种求解非线性方程组的3p阶迭代方法[J]. 计算数学, 2017, 39(1): 14-22.
[3] 肖飞雁, 李旭旭, 陈飞盛. 非线性延迟积分微分方程连续Runge-Kutta方法的稳定性分析[J]. 计算数学, 2017, 39(1): 1-13.
[4] 刘晴, 檀结庆, 张旭. 一种基于Chebyshev迭代解非线性方程组的方法[J]. 计算数学, 2015, 37(1): 14-20.
[5] 王洋, 伍渝江, 付军. 一类弱非线性方程组的Picard-MHSS迭代方法[J]. 计算数学, 2014, 36(3): 291-302.
[6] 张旭, 檀结庆. 三步五阶迭代方法解非线性方程组[J]. 计算数学, 2013, 35(3): 297-304.
[7] 杨爱利, 伍渝江, 李旭, 孟玲玲. 一类非线性方程组的Newton-PSS迭代法[J]. 计算数学, 2012, 34(4): 329-340.
[8] 陈传淼, 胡宏伶, 雷蕾, 曾星星. 非线性方程组的Newton流线法[J]. 计算数学, 2012, 34(3): 235-258.
[9] 杨柳, 陈艳萍. 求解非线性方程组的一种新的全局收敛的Levenberg-Marquardt算法[J]. 计算数学, 2008, 30(4): 388-396.
[10] 陈全发,肖爱国,. Runge-Kutta-Nystrm方法的若干新性质[J]. 计算数学, 2008, 30(2): 201-212.
[11] 袁功林,鲁习文,韦增欣,. 具有全局收敛性的求解对称非线性方程组的一个修改的信赖域方法[J]. 计算数学, 2007, 29(3): 225-234.
[12] 孙建强,秦孟兆,. 解Burgers方程的一种显式稳定性方法[J]. 计算数学, 2007, 29(1): 67-72.
[13] 冷欣,刘德贵,宋晓秋,陈丽容,. 一类求解奇异延迟微分方程的两步连续Runge-Kutta方法的收敛性[J]. 计算数学, 2006, 28(1): 1-12.
[14] 安恒斌,白中治. NGLM:一类全局收敛的Newton-GMRES方法[J]. 计算数学, 2005, 27(2): 151-174.
[15] 杨柳,陈艳萍. 一种新的Levenberg-Marquardt算法的收敛性[J]. 计算数学, 2005, 27(1): 55-62.
阅读次数
全文


摘要