# American Institute of Mathematical Sciences

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

## 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

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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math., 25 (2007), 231-242. [13] X. Guo, On the convergence of Newton's method in Banach space, Journal of Zhejiang University(Sciences Edition), 27 (2000), 484-492. [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. [15] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Oxford, Pergamon, 1964. [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. [17] J. M. Ortega and W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [18] W. C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations, 2$^{nd}$ edition, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970012. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math., 25 (2007), 231-242. [13] X. Guo, On the convergence of Newton's method in Banach space, Journal of Zhejiang University(Sciences Edition), 27 (2000), 484-492. [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. [15] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Oxford, Pergamon, 1964. [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. [17] J. M. Ortega and W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [18] W. C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations, 2$^{nd}$ edition, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970012. [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. [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. [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. [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. [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. [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.
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
 $x_0$ $\beta$ $\gamma$ $\delta$ $\theta$ $\tau$ $l_*$ $r_0$ 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
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
 $x_0$ $\beta$ $\gamma$ $\delta$ $\theta$ $\tau$ $l_*$ $r_0$ 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
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
 $x_0$ $\beta$ $\gamma$ $\delta$ $\theta$ $\tau$ $l_*$ $r_0$ 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
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
 $x_0$ $\beta$ $\gamma$ $\delta$ $\theta$ $\tau$ $l_*$ $r_0$ 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
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
 $x_0$ $\beta$ $\gamma$ $\delta$ $\theta$ $\tau$ $l_*$ $r_0$ 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
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
 $x_0$ $\beta$ $\gamma$ $\delta$ $\theta$ $\tau$ $l_*$ $r_0$ 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
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
 $x_0$ 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
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
 $x_0$ 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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