一类基于Halley-Newton型的有效修正算法

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 1066-1078.

一类基于Halley-Newton型的有效修正算法

谢亚君()

^{福州外语外贸学院理工学院 & 福建省高校工程研究中心福州 350202}

收稿日期:2019-12-30 出版日期:2021-08-26 发布日期:2021-08-09
作者简介:谢亚君, E-mail: xyj@fzfu.edu.cn
基金资助:
福建省自然科学基金(2019J01879);国科大重点实验室研发项目H2020003(20A01246ZY);福建省重大教改项目(FBJG20200310);福州外语外贸学院高层次人才项目(FWKQJ2020002)

A Class of Efficient Modified Algorithms Based onHalley-Newton Methods

Yajun Xie()

^{School of Technology, Fuzhou University of International Studies and Trade & Engineering Research Center of Universities of Fujian Province, Fuzhou 350202}

Received:2019-12-30 Online:2021-08-26 Published:2021-08-09
Supported by:
the NSF of Fujian Province(2019J01879);the Key Research and Development Projects of University of Chinese Academy of Science H2020003(20A01246ZY);the Major Educational Reform Projects of Fujian Province(FBJG20200310);the High-Level Talents Project of Fuzhou University of International Studies and Trade(FWKQJ2020002)

摘要/Abstract

摘要：

基于Halley方法及经典的牛顿法，通过引入适当参数和线搜索技术，该文提出了求解非线性方程组的一类新的牛顿型算法，并给出两种具体修正迭代格式.在适当假设下，证明了新算法的全局收敛性.数值实验结果表明该方法是可行有效的.

关键词: 非线性方程组, Halley法, Newton法, 收敛性, 数值实验

Abstract:

In this paper, based on the Halley method and the classical Newton method, a class of new modified Halley-Newton method is presented for solving the systems of nonlinear equations with two concrete modified iteration schemes. The convergence performances of the two new variants of Newton iteration method are analyzed in details under appropriate assumptions. Some numerical experiments are given to illustrate the efficiency of the proposed methods.

Key words: Systems of nonlinear equations, Halley method, Newton iteration method, Convergence analysis, Numerical tests

中图分类号:

O224.2

谢亚君. 一类基于Halley-Newton型的有效修正算法[J]. 数学物理学报, 2021, 41(4): 1066-1078.

Yajun Xie. A Class of Efficient Modified Algorithms Based onHalley-Newton Methods[J]. Acta mathematica scientia,Series A, 2021, 41(4): 1066-1078.

图/表 5

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表 5

参考文献 21

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方法		MHN1	MHN2	NM	Halley
参数		α= 2, β = 2, γ = 2	β = 2, γ = 2
初值	It	6	6	7	10
(2, -0.5)^T	CPU	0.013431	0.011241	0.013870	0.013895
	Val	2.516e-010	2.516e-010	1.0195e-009	1.7879e-007
	It	34	20	32	26
(500, 50)^T	CPU	0.017173	0.012883	0.015482	0.014889
	Val	3.4362e-011	5.0377e-011	3.3054e-010	5.1109e-007
	It	26	26	-	24
(100, 100)^T	CPU	0.01655	0.014466	-	0.013880
	Val	1.9425e-010	1.9425e-010	-	5.3603e-007

方法		MHN1	MHN2	NM	Halley
参数		α= 2, β = 2, γ = 1.8	β = 2, γ = 1.8
初值	It	15	11	-	14
(2, 1, 1)^T	CPU	0.019879	0.020301	-	0.023726
	Val	1.2505e-007	2.9591e-007	-	3.4397e-007
	It	15	14	-	22
(1, 0, 1)^T	CPU	0.019571	0.019100	-	0.019471
	Val	9.4170e-007	5.3605e-007	-	6.9823e-007
	It	11	11	-	18
(10, 10, 10)^T	CPU	0.020074	0.018517	-	0.021271
	Val	9.4538e-007	5.5233e-007	-	1.6925e-007

方法		MHN1	MHN2	NM	Halley
参数		α= 3, β = 3, γ = 3	β = 3, γ = 3
初值	It	4	4	-	12
(0.1, 0.1, 0.1)^T	CPU	0.020039	0.019090	-	0.020479
	Val	1.3221e-009	1.2899e-009	-	6.2214e-007
	It	2	2	10	8
(0.01, 0.01, 0.01)^T	CPU	0.018182	0.027469	0.020194	0.020317
	Val	9.3543e-010	9.3543e-010	2.3049e-007	2.9993e-007

		MHN1	MHN2	NM	Halley
	It	23	21	-	59
E4	CPU	0.012871	0.024354	-	0.026726
	Val	3.9611e-007	2.6462e-007	-	5.1715e-007
	It	10	14	26	22
E5	CPU	0.011445	0.010914	0.011730	0.011648
	Val	1.1887e-007	6.0749e-007	2.0876	6.7433e-007
	It	33	11	100	100
E6	CPU	0.014556	0.023012	0.018521	0.018835
	Val	2.7697e-007	2.7240e-007	2.7136-007	1.522e-007
	It	46	45	60	58
E7	CPU	0.01534	0.02774	0.01554	0.015613
	Val	3.9978e-007	2.6487e-007	1.8625-007	5.2191e-007
	It	12	9	-	23
E8	CPU	0.01500	0.01189	-	0.015119
	Val	1.9616e-007	7.2264e-007	-	4.2569e-007
	It	10	9	-	-
E9	CPU	0.021609	0.027686	-	-
	Val	7.7727e-007	6.6277e-007	-	-

方法		MHN2	NM	Halley
参数		α= 3.0, β = 2.9, γ = 2.1
初值	It	4	-	4
(1, 1, ...)^T	CPU	14.10	-	18.16
	Val	1.272e-011	-	1.588e-010
	It	3	-	4
-0.85*(1, 1, ...)^T	CPU	13.08	-	18.16
	Val	7.302e-011	-	1.588e-010
	It	4	-	4
-5*(1, 1, ...)^T	CPU	15.07	-	25.16
	Val	5.302e-011	-	2.818e-010