Semi-local convergence of the Newton-HSS method under the center Lipschitz condition

Hongxiu Zhong; Guoliang Chen; Xueping Guo

doi:10.3934/naco.2019007

Article Contents

2019, Volume 9, Issue 1: 85-99. Doi: 10.3934/naco.2019007

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Semi-local convergence of the Newton-HSS method under the center Lipschitz condition

1.
School of Science, Jiangnan University, Wuxi, 214122, P. R. China
2.
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China

* Corresponding author: Hongxiu Zhong
* Corresponding author: Hongxiu Zhong

The first author is supported by the National Natural Science Foundation of China (No. 11701225), the Fundamental Research Funds for the Central Universities (No. JUSRP11719) and the Natural Science Foundation of Jiangsu Province (No. BK20170173). The second author is supported by the National Natural Science Foundation of China (No. 11471122) and the Science and Technology Commission of Shanghai Municipality (STCSM) (No. 13dz2260400)

Received: February 2018

Revised: June 2018

Published: March 2019

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Abstract

Newton-type methods have gained much attention in the past decades, especially for the semilocal convergence based on no information around the solution $x_*$ of the target nonlinear equation. For large sparse non-Hermitian positive definite systems of nonlinear equation, assuming that the nonlinear operator satisfies the center Lipschitz condition, which is wider than usual Lipschtiz condition and H$ö$lder continuous condition, we establish a new Newton-Kantorovich convergence theorem for the Newton-HSS method. Once the convergence criteria is satisfied, the iteration sequence $\{x_k\}_{k = 0}^∞$ generated by the Newton-HSS method is well defined, and converges to the solution $x_*$. Numerical results illustrate the effect.

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Mathematics Subject Classification: Primary: 65F10, 65H10; Secondary: 65F50.

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Table 1. Parameters' values for different initial point $x_0$ in the case of m = 3, N = 30, q = 600

$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
1	20.2504	1.1624	75.9105	0.9956	0.004	4925	5.0839e+04
2	20.2504	1.1542	151.8310	0.9936	0.006	5577	5.1564e+04
10	20.2504	0.9331	761.3715	0.9339	0.070	3322	7.8891e+04
20	20.2504	0.5542	1.5515e+03	0.7685	0.301	9798	2.2368e+05

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Table 2. Parameters' values for different initial point $x_0$ in the case of m = 3, N = 40, q = 600

$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
1	15.5883	1.9189	66.8682	0.9966	0.003	5242	5.7083e+04
2	15.5883	1.9062	133.7451	0.9951	0.004	2542	5.7845e+04
10	15.5883	1.5606	670.5120	0.9496	0.053	24923	8.6297e+04
20	15.5883	0.9485	1.3620e+03	0.8198	0.219	3142	2.3364e+05

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Table 3. Parameters' values for different initial point $x_0$ in the case of m = 3, N = 50, q = 600

$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
1	12.7405	2.8827	60.7730	0.9972	0.002	2886	6.0552e+04
2	12.7405	2.8643	121.5538	0.9960	0.003	2357	6.1334e+04
10	12.7405	2.3595	609.2850	0.9592	0.042	4785	9.0384e+04
20	12.7405	1.4504	1.2351e+03	0.8525	0.173	76858	2.3920e+05

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Table 4. Parameters' values for different initial point $x_0$ in the case of m = 3, N = 30, q = 800

$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
1	26.6689	0.9065	100.6620	0.9967	0.0030	6591	8.3595e+04
2	26.6689	0.9015	201.3317	0.9952	0.0040	2863	8.4514e+04
10	26.6689	0.7627	1.0083e+03	0.9500	0.0520	4083	1.1808e+05
20	26.6689	0.4943	2.0385e+03	0.8212	0.2170	3563	2.8113e+05

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Table 5. Parameters' values for different initial point $x_0$ in the case of m = 3, N = 40, q = 800

$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
1	20.4520	1.4830	88.3329	0.9974	0.0020	3983	9.5568e+04
2	20.4520	1.4754	176.6724	0.9963	0.0030	3396	9.6556e+04
10	20.4520	1.2610	884.7154	0.9619	0.0390	4261	1.3219e+05
20	20.4520	0.8368	1.7854e+03	0.8617	0.1600	5757	3.0013e+05

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Table 6. Parameters' values for different initial point $x_0$ in the case of m = 3, N = 50, q = 800

$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
1	16.6546	2.2085	79.9039	0.9979	0.0020	23235	1.0317e+05
2	16.6546	2.1976	159.8137	0.9970	0.0020	2347	1.0419e+05
10	16.6546	1.8906	800.2221	0.9693	0.0310	3352	1.4081e+05
20	16.6546	1.2714	1.6131e+03	0.8873	0.1270	79278	3.1131e+05

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Table 7. Numerical results for different $x_0$ and $N$(m = 3, q = 600)

$x_0$	Error estimates			CPU			Outer IT
$x_0$	$N=30$	$N=40$	$N=50$	$N=30$	$N=40$	$N=50$	$N=30$	$N=40$	$N=50$
1	6.15e-07	4.90e-07	4.56e-07	1.60	5.57	6.20	4	4	4
2	2.42e-07	1.63e-07	1.39e-07	1.849	2.98	5.73	5	5	5
10	3.64e-07	2.70e-07	1.88e-07	1.12	28.15	10.85	6	6	6
20	2.60e-07	2.31e-07	1.37e-07	3.14	3.83	162.82	7	7	7

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Table 8. Numerical results for different $x_0$ and $N$(m = 3, q = 800)

$x_0$	Error estimates			CPU			Outer IT
$x_0$	$N=30$	$N=40$	$N=50$	$N=30$	$N=40$	$N=50$	$N=30$	$N=40$	$N=50$
1	4.02e-07	3.80e-07	3.38e-07	2.02	3.38	74.69	4	4	4
2	1.41e-07	1.48e-07	1.15e-07	0.93	2.95	8.30	5	5	5
10	3.47e-07	2.16e-07	2.54e-07	1.31	3.73	12.50	6	6	6
20	1.53e-07	1.33e-07	1.06e-07	1.15	4.94	260.67	7	7	7

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