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

Hongxiu Zhong; Guoliang Chen; Xueping Guo

doi:10.3934/naco.2019007

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March 2019, 9(1): 85-99. doi: 10.3934/naco.2019007

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

Hongxiu Zhong ^1,, , Guoliang Chen ^2, and Xueping Guo ^2,

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

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 October 2018

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.

Keywords: Large sparse systems, nonlinear equations, Newton-HSS method, Newton-Kantorovich theorem.

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

Citation: Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007

References:

[1]	H. An and Z. Bai, A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57 (2007), 235-252. doi: 10.1016/j.apnum.2006.02.007. Google Scholar
[2]	H. An, Z. Mo and X. Liu, A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math., 20 (2007), 47-60. doi: 10.1016/j.cam.2005.12.030. Google Scholar
[3]	O. Axelsson and G. F. Carey, On the numerical solution of two-point singularly perturbed boundary value problems, Comput. Method Appl. Mech. Eng., 50 (1985), 217-229. doi: 10.1016/0045-7825(85)90094-5. Google Scholar
[4]	O. Axelsson and M. Nikolova, Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D, Computing, 61 (1998), 331-357. doi: 10.1007/BF02684384. Google Scholar
[5]	Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626. doi: 10.1137/S0895479801395458. Google Scholar
[6]	Z. Bai and X. Guo, On Newton-HSS method for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math., 28 (2010), 235-260. doi: 10.4208/jcm.2009.10-m2836. Google Scholar
[7]	S. Bellavia and B. Morini, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM J. Sci. Comput., 23 (2001), 940-960. doi: 10.1137/S1064827599363976. Google Scholar
[8]	M. Chen, R. Lin and Q. Wu, Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition, J. Comput. Appl. Math., 264 (2014), 115-130. doi: 10.1016/j.cam.2013.12.047. Google Scholar
[9]	M. Chen, Q. Wu and R. Lin, Semilocal convergence analysis for the Modified Newton-HSS method under the H$ö$lder condition, Numer. Algor., 72 (2016), 667-685. doi: 10.1007/s11075-015-0061-z. Google Scholar
[10]	R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408. doi: 10.1137/0719025. Google Scholar
[11]	O. P. Ferreira and B. F. Svaiter, Kantorovich's majorants principle for Newton's method, Comput. Optim. Appl., 42 (2009), 213-229. doi: 10.1007/s10589-007-9082-4. Google Scholar
[12]	X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math., 25 (2007), 231-242. Google Scholar
[13]	X. Guo, On the convergence of Newton's method in Banach space, Journal of Zhejiang University(Sciences Edition), 27 (2000), 484-492. Google Scholar
[14]	X. Guo and I. S. Duff, Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl., 18 (2011), 299-315. doi: 10.1002/nla.713. Google Scholar
[15]	L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Oxford, Pergamon, 1964. Google Scholar
[16]	C. Li and K. F. NG, Majorizing functions and convergence of the Gauss-Newton method for convex composite optimation, SIAM J. Optim., 18 (2007), 613-642. doi: 10.1137/06065622X. Google Scholar
[17]	J. M. Ortega and W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. Google Scholar
[18]	W. C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations, 2$^{nd}$ edition, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970012. Google Scholar
[19]	Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. Sci. Stat. Comput., 7 (1986), 856-869. doi: 10.1137/0907058. Google Scholar
[20]	W. Shen and C. Li, Kantorovich-type convergence criterion for inexact Newton methods, Appl. Numer. Math., 59 (2009), 1599-1611. doi: 10.1016/j.apnum.2008.11.002. Google Scholar
[21]	X. Wang, Convergence of Newton's method and inverse function theorem in Banach space, Math. Comput., 68 (1999), 169-185. doi: 10.1090/S0025-5718-99-00999-0. Google Scholar
[22]	A. Yang and Y. Wu, Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algebra, Control. Optim., 2 (2012), 839-853. doi: 10.3934/naco.2012.2.839. Google Scholar
[23]	H. Zhong, G. Chen and X. Guo, On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algor., 69 (2015), 553-567. doi: 10.1007/s11075-014-9912-2. Google Scholar
[24]	H. Zhong, G. Wu and G. Chen, A flexible and adaptive simpler block GMRES with deflated restarting for linear systems with multiple right-hand sides, J. Comput. Appl. Math., 282 (2015), 139-156. doi: 10.1016/j.cam.2014.12.040. Google Scholar

show all references

References:

[1]	H. An and Z. Bai, A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57 (2007), 235-252. doi: 10.1016/j.apnum.2006.02.007. Google Scholar
[2]	H. An, Z. Mo and X. Liu, A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math., 20 (2007), 47-60. doi: 10.1016/j.cam.2005.12.030. Google Scholar
[3]	O. Axelsson and G. F. Carey, On the numerical solution of two-point singularly perturbed boundary value problems, Comput. Method Appl. Mech. Eng., 50 (1985), 217-229. doi: 10.1016/0045-7825(85)90094-5. Google Scholar
[4]	O. Axelsson and M. Nikolova, Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D, Computing, 61 (1998), 331-357. doi: 10.1007/BF02684384. Google Scholar
[5]	Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626. doi: 10.1137/S0895479801395458. Google Scholar
[6]	Z. Bai and X. Guo, On Newton-HSS method for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math., 28 (2010), 235-260. doi: 10.4208/jcm.2009.10-m2836. Google Scholar
[7]	S. Bellavia and B. Morini, A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM J. Sci. Comput., 23 (2001), 940-960. doi: 10.1137/S1064827599363976. Google Scholar
[8]	M. Chen, R. Lin and Q. Wu, Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition, J. Comput. Appl. Math., 264 (2014), 115-130. doi: 10.1016/j.cam.2013.12.047. Google Scholar
[9]	M. Chen, Q. Wu and R. Lin, Semilocal convergence analysis for the Modified Newton-HSS method under the H$ö$lder condition, Numer. Algor., 72 (2016), 667-685. doi: 10.1007/s11075-015-0061-z. Google Scholar
[10]	R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408. doi: 10.1137/0719025. Google Scholar
[11]	O. P. Ferreira and B. F. Svaiter, Kantorovich's majorants principle for Newton's method, Comput. Optim. Appl., 42 (2009), 213-229. doi: 10.1007/s10589-007-9082-4. Google Scholar
[12]	X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math., 25 (2007), 231-242. Google Scholar
[13]	X. Guo, On the convergence of Newton's method in Banach space, Journal of Zhejiang University(Sciences Edition), 27 (2000), 484-492. Google Scholar
[14]	X. Guo and I. S. Duff, Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl., 18 (2011), 299-315. doi: 10.1002/nla.713. Google Scholar
[15]	L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Oxford, Pergamon, 1964. Google Scholar
[16]	C. Li and K. F. NG, Majorizing functions and convergence of the Gauss-Newton method for convex composite optimation, SIAM J. Optim., 18 (2007), 613-642. doi: 10.1137/06065622X. Google Scholar
[17]	J. M. Ortega and W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. Google Scholar
[18]	W. C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations, 2$^{nd}$ edition, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970012. Google Scholar
[19]	Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. Sci. Stat. Comput., 7 (1986), 856-869. doi: 10.1137/0907058. Google Scholar
[20]	W. Shen and C. Li, Kantorovich-type convergence criterion for inexact Newton methods, Appl. Numer. Math., 59 (2009), 1599-1611. doi: 10.1016/j.apnum.2008.11.002. Google Scholar
[21]	X. Wang, Convergence of Newton's method and inverse function theorem in Banach space, Math. Comput., 68 (1999), 169-185. doi: 10.1090/S0025-5718-99-00999-0. Google Scholar
[22]	A. Yang and Y. Wu, Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algebra, Control. Optim., 2 (2012), 839-853. doi: 10.3934/naco.2012.2.839. Google Scholar
[23]	H. Zhong, G. Chen and X. Guo, On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algor., 69 (2015), 553-567. doi: 10.1007/s11075-014-9912-2. Google Scholar
[24]	H. Zhong, G. Wu and G. Chen, A flexible and adaptive simpler block GMRES with deflated restarting for linear systems with multiple right-hand sides, J. Comput. Appl. Math., 282 (2015), 139-156. doi: 10.1016/j.cam.2014.12.040. Google Scholar

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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$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
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

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$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
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

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$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
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

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$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
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

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$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
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

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$x_0$	$\beta$	$\gamma$	$\delta$	$\theta$	$\tau$	$l_*$	$r_0$
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
$x_0$	$N=30$	$N=40$	$N=50$	$N=30$	$N=40$	$N=50$	$N=30$	$N=40$	$N=50$
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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$x_0$	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

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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$x_0$	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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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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