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.

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Figures/Tables 5

References 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

A Class of Efficient Modified Algorithms Based onHalley-Newton Methods

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