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Keywords:
Newton's method; Banach space; rate of convergence; semilocal convergence; nondiscrete mathematical induction; estimate function
Summary:
We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.
References:
[1] Amat, S., Bermúdez, C., Busquier, S., Gretay, J.: Convergence by nondiscrete mathematical induction of a two step secant's method. Rocky Mt. J. Math. 37 (2007), 359-369. DOI 10.1216/rmjm/1181068756 | MR 2333375 | Zbl 1140.65040
[2] Amat, S., Busquier, S.: Third-order iterative methods under Kantorovich conditions. J. Math. Anal. Appl. 336 (2007), 243-261. DOI 10.1016/j.jmaa.2007.02.052 | MR 2348504 | Zbl 1128.65036
[3] Amat, S., Busquier, S., Gutiérrez, J. M., Hernández, M. A.: On the global convergence of Chebyshev's iterative method. J. Comput. Appl. Math. 220 (2008), 17-21. DOI 10.1016/j.cam.2007.07.022 | MR 2444150 | Zbl 1149.65035
[4] Argyros, I. K.: The Theory and Application of Abstract Polynomial Equations. St. Lucie/CRC/Lewis Publ. Mathematics series, Boca Raton, Florida, USA (1998).
[5] Argyros, I. K.: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298 (2004), 374-397. DOI 10.1016/j.jmaa.2004.04.008 | MR 2086964 | Zbl 1061.47052
[6] Argyros, I. K.: On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169 (2004), 315-332. DOI 10.1016/j.cam.2004.01.029 | MR 2072881 | Zbl 1055.65066
[7] Argyros, I. K.: Concerning the ``terra incognita'' between convergence regions of two Newton methods. Nonlinear Anal., Theory Methods Appl. 62 (2005), 179-194. DOI 10.1016/j.na.2005.02.113 | MR 2139363 | Zbl 1072.65079
[8] Argyros, I. K.: Approximating solutions of equations using Newton's method with a modified Newton's method iterate as a starting point. Rev. Anal. Numér. Théor. Approx. 36 (2007), 123-137. MR 2498828 | Zbl 1199.65179
[9] Argyros, I. K.: Computational Theory of Iterative Methods. Studies in Computational Mathematics 15. Elsevier, Amsterdam (2007). MR 2356038 | Zbl 1147.65313
[10] Argyros, I. K.: On a class of Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 228 (2009), 115-122. DOI 10.1016/j.cam.2008.08.042 | MR 2514268 | Zbl 1168.65349
[11] Argyros, I. K.: A semilocal convergence analysis for directional Newton methods. Math. Comput. 80 (2011), 327-343. DOI 10.1090/S0025-5718-2010-02398-1 | MR 2728982 | Zbl 1211.65057
[12] Argyros, I. K., Hilout, S.: Efficient Methods for Solving Equations and Variational Inequalities. Polimetrica Publisher, Milano, Italy (2009).
[13] Argyros, I. K., Hilout, S.: Enclosing roots of polynomial equations and their applications to iterative processes. Surv. Math. Appl. 4 (2009), 119-132. MR 2558651 | Zbl 1205.26023
[14] Argyros, I. K., Hilout, S.: Extending the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 234 (2010), 2993-3006. DOI 10.1016/j.cam.2010.04.014 | MR 2652146 | Zbl 1195.65075
[15] Argyros, I. K., Hilout, S., Tabatabai, M. A.: Mathematical Modelling with Applications in Biosciences and Engineering. Nova Publishers, New York, 2011. MR 2895345
[16] Bi, W., Wu, Q., Ren, H.: Convergence ball and error analysis of the Ostrowski-Traub method. Appl. Math., Ser. B (Engl. Ed.) 25 (2010), 374-378. MR 2679357 | Zbl 1240.65167
[17] Cătinaş, E.: The inexact, inexact perturbed, and quasi-Newton methods are equivalent models. Math. Comput. 74 (2005), 291-301. DOI 10.1090/S0025-5718-04-01646-1 | MR 2085412 | Zbl 1054.65050
[18] Chen, X., Yamamoto, T.: Convergence domains of certain iterative methods for solving nonlinear equations. Numer. Funct. Anal. Optimization 10 (1989), 37-48. DOI 10.1080/01630568908816289 | MR 0978801 | Zbl 0645.65028
[19] Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics 35. Springer, Berlin (2004). MR 2063044 | Zbl 1056.65051
[20] Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., Romero, N., Rubio, M. J.: The Newton method: from Newton to Kantorovich. Spanish Gac. R. Soc. Mat. Esp. 13 (2010), 53-76. MR 2647925 | Zbl 1195.65001
[21] Ezquerro, J. A., Hernández, M. A.: On the $R$-order of convergence of Newton's method under mild differentiability conditions. J. Comput. Appl. Math. 197 (2006), 53-61. DOI 10.1016/j.cam.2005.10.023 | MR 2256051 | Zbl 1106.65048
[22] Ezquerro, J. A., Hernández, M. A.: An improvement of the region of accessibility of Chebyshev's method from Newton's method. Math. Comput. 78 (2009), 1613-1627. DOI 10.1090/S0025-5718-09-02193-0 | MR 2501066 | Zbl 1198.65096
[23] Ezquerro, J. A., Hernández, M. A., Romero, N.: Newton-type methods of high order and domains of semilocal and global convergence. Appl. Math. Comput. 214 (2009), 142-154. DOI 10.1016/j.amc.2009.03.072 | MR 2541053 | Zbl 1173.65032
[24] Gragg, W. B., Tapia, R. A.: Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11 (1974), 10-13. DOI 10.1137/0711002 | MR 0343594 | Zbl 0284.65042
[25] Hernández, M. A.: A modification of the classical Kantorovich conditions for Newton's method. J. Comput. Appl. Math. 137 (2001), 201-205. DOI 10.1016/S0377-0427(01)00393-4 | MR 1865886 | Zbl 0992.65057
[26] Kantorovich, L. V., Akilov, G. P.: Functional Analysis. Transl. from the Russian. Pergamon Press, Oxford (1982). MR 0664597 | Zbl 0484.46003
[27] Krishnan, S., Manocha, D.: An efficient surface intersection algorithm based on lower-dimensional formulation. ACM Trans. on Graphics. 16 (1997), 74-106. DOI 10.1145/237748.237751
[28] Lukács, G.: The generalized inverse matrix and the surface-surface intersection problem. Theory and Practice of Geometric Modeling, Lect. Conf., Blaubeuren/FRG 1988 167-185 (1989). MR 1042329 | Zbl 0692.68076
[29] Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. Computer Science and Applied Mathematics. Academic Press, New York (1970). MR 0273810 | Zbl 0241.65046
[30] Ostrowski, A. M.: Sur la convergence et l'estimation des erreurs dans quelques procédés de résolution des équations numériques. French Gedenkwerk D. A. Grave, Moskau 213-234 (1940). MR 0004377 | Zbl 0023.35302
[31] Ostrowski, A. M.: La méthode de Newton dans les espaces de Banach. (The Newton method in Banach spaces). French C. R. Acad. Sci., Paris, Sér. A 272 (1971), 1251-1253. MR 0285110 | Zbl 0228.65041
[32] Ostrowski, A. M.: Solution of Equations in Euclidean and Banach Spaces. 3rd ed. of solution of equations and systems of equations. Pure and Applied Mathematics, 9. Academic Press, New York (1973). MR 0359306 | Zbl 0304.65002
[33] Păvăloiu, I.: Introduction in the Theory of Approximation of Equations Solutions. Dacia Ed. Cluj-Napoca (1976).
[34] Potra, F. A.: A characterization of the divided differences of an operator which can be represented by Riemann integrals. Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 9 (1980), 251-253. MR 0651780 | Zbl 0523.65043
[35] Potra, F. A.: An application of the induction method of V. Pták to the study of regula falsi. Apl. Mat. 26 (1981), 111-120. MR 0612668 | Zbl 0486.65038
[36] Potra, F. A.: The rate of convergence of a modified Newton's process. Apl. Mat. 26 (1981), 13-17. MR 0602398 | Zbl 0486.65039
[37] Potra, F. A.: An error analysis for the secant method. Numer. Math. 38 (1982), 427-445. DOI 10.1007/BF01396443 | MR 0654108 | Zbl 0465.65033
[38] Potra, F. A.: On the convergence of a class of Newton-like methods. Iterative solution of nonlinear systems of equations, Proc. Meeting, Oberwolfach 1982, Lect. Notes Math. 953 125-137. DOI 10.1007/BFb0069378 | MR 0678615 | Zbl 0507.65020
[39] Potra, F. A.: On the a posteriori error estimates for Newton's method. Beitr. Numer. Math. 12 (1984), 125-138. MR 0732159
[40] Potra, F. A.: On a class of iterative procedures for solving nonlinear equations in Banach spaces. Computational Mathematics, Banach Cent. Publ. 13 607-621 (1984). MR 0798124 | Zbl 0569.65042
[41] Potra, F. A.: Sharp error bounds for a class of Newton-like methods. Libertas Math. 5 (1985), 71-84. MR 0816258 | Zbl 0581.47050
[42] Potra, F. A., Pták, V.: Sharp error bounds for Newton's process. Numer. Math. 34 (1980), 63-72. DOI 10.1007/BF01463998 | MR 0560794 | Zbl 0434.65034
[43] Potra, F. A., Pták, V.: Nondiscrete induction and a double step secant method. Math. Scand. 46 (1980), 236-250. MR 0591604 | Zbl 0423.65034
[44] Potra, F. A., Pták, V.: On a class of modified Newton processes. Numer. Funct. Anal. Optimization 2 (1980), 107-120. DOI 10.1080/01630568008816049 | MR 0580387 | Zbl 0472.65049
[45] Potra, F. A., Pták, V.: A generalization of regula falsi. Numer. Math. 36 (1981), 333-346. DOI 10.1007/BF01396659 | MR 0613073 | Zbl 0478.65039
[46] Potra, F. A., Pták, V.: Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston (1984). MR 0754338 | Zbl 0549.41001
[47] Proinov, P. D.: General local convergence theory for a class of iterative processes and its applications to Newton's method. J. Complexity 25 (2009), 38-62. DOI 10.1016/j.jco.2008.05.006 | MR 2475307 | Zbl 1158.65040
[48] Proinov, P. D.: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complexity 26 (2010), 3-42. DOI 10.1016/j.jco.2009.05.001 | MR 2574570 | Zbl 1185.65095
[49] Pták, V.: Some metric aspects of the open mapping and closed graph theorems. Math. Ann. 163 (1966), 95-104. DOI 10.1007/BF02052841 | MR 0192316 | Zbl 0138.37602
[50] Pták, V.: A quantitative refinement of the closed graph theorem. Czech. Math. J. 24 (1974), 503-506. MR 0348431 | Zbl 0315.46007
[51] Pták, V.: A theorem of the closed graph type. Manuscr. Math. 13 (1974), 109-130. DOI 10.1007/BF01411490 | MR 0348430 | Zbl 0286.46008
[52] Pták, V.: Deux théoremes de factorisation. C. R. Acad. Sci., Paris, Sér. A 278 (1974), 1091-1094. MR 0341096 | Zbl 0277.46047
[53] Pták, V.: Concerning the rate of convergence of Newton's process. Commentat. Math. Univ. Carol. 16 (1975), 699-705. MR 0398092 | Zbl 0314.65023
[54] Pták, V.: A modification of Newton's method. Čas. Pěst. Mat. 101 (1976), 188-194. MR 0443326 | Zbl 0328.46013
[55] Pták, V.: Nondiscrete mathematical induction and iterative existence proofs. Linear Algebra Appl. 13 (1976), 223-238. DOI 10.1016/0024-3795(76)90098-7 | MR 0394119 | Zbl 0323.46005
[56] Pták, V.: The rate of convergence of Newton's process. Numer. Math. 25 (1976), 279-285. DOI 10.1007/BF01399416 | MR 0478587 | Zbl 0304.65037
[57] Pták, V.: Nondiscrete mathematical induction. Gen. Topol. Relat. mod. Anal. Algebra IV, Proc. 4th Prague topol. Symp. 1976, Part A, Lect. Notes Math. 609 166-178 (1977). DOI 10.1007/BFb0068681 | MR 0487618 | Zbl 0367.46007
[58] Pták, V.: What should be a rate of convergence? RAIRO, Anal. Numér. 11 (1977), 279-286. MR 0474799
[59] Pták, V.: Stability of exactness. Commentat. math., spec. Vol. II, dedic. L. Orlicz (1979), 283-288. MR 0552012 | Zbl 0445.46003
[60] Pták, V.: A rate of convergence. Numer. Funct. Anal. Optimization 1 (1979), 255-271. DOI 10.1080/01630567908816015 | MR 0537831 | Zbl 0441.46010
[61] Pták, V.: Factorization in Banach algebras. Stud. Math. 65 (1979), 279-285. MR 0567080 | Zbl 0342.46036
[62] Ren, H., Wu, Q.: Convergence ball of a modified secant method with convergence order $1.839.\. $. Appl. Math. Comput. 188 (2007), 281-285. DOI 10.1016/j.amc.2006.09.111 | MR 2327116 | Zbl 1118.65044
[63] Rheinboldt, W. C.: A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5 (1968), 42-63. DOI 10.1137/0705003 | MR 0225468 | Zbl 0155.46701
[64] Tapia, R. A.: The Kantorovich theorem for Newton's method. Am. Math. Mon. 78 (1971), 389-392. DOI 10.2307/2316909 | MR 1536290 | Zbl 0215.27404
[65] Wu, Q., Ren, H.: A note on some new iterative methods with third-order convergence. Appl. Math. Comput. 188 (2007), 1790-1793. DOI 10.1016/j.amc.2006.11.043 | MR 2335032 | Zbl 1121.65052
[66] Yamamoto, T.: A convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51 (1987), 545-557. DOI 10.1007/BF01400355 | MR 0910864 | Zbl 0633.65049
[67] Zabrejko, P. P., Nguen, D. F.: The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates. Numer. Funct. Anal. Optimization 9 (1987), 671-684. DOI 10.1080/01630568708816254 | MR 0895991 | Zbl 0627.65069
[68] Zinčenko, A. I.: Some approximate methods of solving equations with non-differentiable operators. Ukrainian Dopovidi Akad. Nauk Ukraïn. RSR (1963), 156-161. MR 0160096
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