一类基于Halley-Newton型的有效修正算法

Abstract

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

CLC Number:

O224.2

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

Trendmd

Figures/Tables 5

References 21

1	Andrei N . An unconstrained optimization test functions collection. Adv Model Optim, 2008, 10, 147- 161
2	An H B , Bai Z Z . A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. Appl Numer Math, 2007, 57, 235- 252 doi: 10.1016/j.apnum.2006.02.007
3	Bai Z Z . Block and asynchronous two-stage methods for mildly nonlinear systems. Numer Math, 1999, 82, 1- 20 doi: 10.1007/s002110050409
4	Burden R L , Faires J D . Numerical Analysis (Seventh Edition). Boston: Thomson Learning Inc, 2001, 600- 635
5	Schnabel R B . Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Philadelphia: SIAM, 1996
6	Kelley C T . Solving Nonlinear Equations with Newton's Method. Philadelphia: SIAM, 2003
7	Kelley C T . Iterative Methods for Linear and Nonlinear Equations. North Carolina: North Carolina State University, 1995
8	Levin Y , Ben-Israelb A . Directional Halley and quasi-Halley methods in n variables. Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, 2001, 8, 345- 367
9	Narang M , Bhatia S , Kanwar V . New two-parameter Chebyshev-Halley-like family of fourth and sixthorder methods for systems of nonlinear equations. Appl Math Comput, 2016, 275, 394- 403
10	Nedzhibov G H . A family of multi-pointiterative methods for solving systems of nonlinear equations. J Comput Appl Math, 2008, 222, 244- 250 doi: 10.1016/j.cam.2007.10.054
11	Noor M A . New classes of iterative methods for nonlinear equations. Appl Math Comput, 2007, 191, 128- 131
12	Noor M A , Shah F A . A family of iterative schemes for finding zeros of nonlinear equations having unknowm multiplicity. Appl Math Inf Sci, 2014, 8, 2367- 2373 doi: 10.12785/amis/080532
13	More J J , Grabow B S , Hillstrom K E . Testing unconstrained optimization software. ACM Trans Math Software, 1981, 7, 17- 41 doi: 10.1145/355934.355936
14	Shah F A , Noor M A . Higher order iterative schemes for nonlinear equations using decomposition technique. Appl Math Comput, 2015, 266, 414- 423
15	Sun Z , Zhang K . Solving nonlinear systems of equations based on social cognitive optimization. Comput Eng Appl, 2008, 44, 42- 46
16	Traub J F . Iterative Methods for the Solution of Equations. Englewood: Prentice Hall, 1964
17	Woodbury M A. Inverting modified matrices[R]. Princeton NJ: Statistical Research Group, 1950
18	Xiao X , Yin H . A new class of methods with higher order of convergence for solving systems of nonlinear equations. Appl Math Comput, 2015, 264, 300- 309
19	Yang X J , Gao F . New technology for solving diffusion and heat equations. Thermal Science, 2017, 21, 133- 140 doi: 10.2298/TSCI160411246Y
20	Yang X J . A new integral transform with an application in heat-transfer problem. Thermal Science, 2016, 20, 677- 681 doi: 10.2298/TSCI16S3677Y
21	Gao F , Yang X J , Zhang Y F . Exact traveling wave solutions for a new nonlinear heat-transfer equation. Thermal Science, 2016, 21, 1833- 1838

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

A Class of Efficient Modified Algorithms Based onHalley-Newton Methods

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 5

References 21

Related Articles 5

Metrics

Comments

Recommended 10

[1]	Yu Dongmei, Liu Dayi. The Levenberg-Marquardt Algorithm for Solving the Generalized Complementarity Problems [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1311-1326.
[2]	Ma Changfeng, Xie Yajun, Bu Fan. The Tensor Scheme BCGSTAB Algorithm for Solving Stein Tensor Equations [J]. Acta mathematica scientia,Series A, 2024, 44(6): 1652-1664.
[3]	Yu Yikang,Niu Jing. A Kind of Numerical Algorithm for Elliptic Interface Problem [J]. Acta mathematica scientia,Series A, 2023, 43(3): 883-895.
[4]	Zhihao Ge,Ruihua Li. Numerical Methods for the Critical Temperature and Gap Solution of Bogoliubov-Tolmachev-Shirkov Model [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1699-1711.
[5]	Lixin Feng,Xiaoxu Yang. Spectral Regularization Method for Volterra Integral Equation of the First Kind with Noise Data [J]. Acta mathematica scientia,Series A, 2020, 40(3): 650-661.