Article Contents
Article Contents

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

• * 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)

• 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.

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

 Citation:

• 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

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

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

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

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

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

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

 $x_0$ Error estimates CPU Outer IT $N=30$ $N=40$ $N=50$ $N=30$ $N=40$ $N=50$ $N=30$ $N=40$ $N=50$ 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

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

 $x_0$ Error estimates CPU Outer IT $N=30$ $N=40$ $N=50$ $N=30$ $N=40$ $N=50$ $N=30$ $N=40$ $N=50$ 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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