# zbMATH — the first resource for mathematics

New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. (English) Zbl 1185.65095
The author proposes a lot of new general convergence theorems for the Picard iteration, applied to a mapping $$T$$ in a complete metric space. To elaborate this new theory, he uses the concepts of quasi-homogeneous functions, gauge functions of high order, a function of initial conditions of the mapping $$T$$, a convergence function of the mapping $$T$$ and the initial points of a mapping. The function of the initial conditions of a mapping represents a generalization of the concept of contraction.
Four new convergence theorems for the Picard iteration are proved (Theorems 5.4, 5.5, 5.6, 5.7); each of these theorems gives the radius of the convergence ball, error estimates (a priori and a posteriori) and the existence of a fixed point for the mapping $$T$$. These results are then applied to obtain fixed point theorems for the iterated contraction mapping (with respect to a function of initial conditions). Also, these results are applied to study the convergence of the Newton-Kantorovich method for operator equations in Banach spaces. Three Newton-Kantorovich type theorems which generalize, extend, or complete some results from the literature are proved.
In the last section, the theory is applied to Newton’s iteration for the zeros of an analytic function and also, many published results are extended (especially the results of S. Smale, Newton’s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics, Proc. Symp. Honor G. S. Young, Laramie/Wyo. 1985, 185–196 (1986; Zbl 0613.65058)).

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 47J05 Equations involving nonlinear operators (general) 47H10 Fixed-point theorems 47J25 Iterative procedures involving nonlinear operators 65H05 Numerical computation of solutions to single equations 65E05 General theory of numerical methods in complex analysis (potential theory, etc.) 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
Full Text:
##### References:
 [1] Proinov, P.D., A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear anal., 67, 2361-2369, (2007) · Zbl 1130.54021 [2] Proinov, P.D., General local convergence theory for a class of iterative processes and its applications to newton’s method, J. complexity, 25, 38-62, (2009) · Zbl 1158.65040 [3] Kantorovich, L.V., On newton’s method for functional equations, Dokl. acad. nauk SSSR, 59, 1237-1240, (1948), (in Russian) [4] Smale, S., Newton’s method estimates from data at one point, (), 185-196 [5] Ortega, J.M.; Rheinboldt, W.C., Iterative solutions of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046 [6] Hicks, T.; Rhoades, B.E., A Banach type fixed point theorem, Math. japonica, 24, 327-330, (1979) · Zbl 0432.47036 [7] Park, S., A unified approach to fixed points of contractive maps, J. Korean math. soc., 16, 95-105, (1980) · Zbl 0431.54028 [8] Ćirić, Lj., Generalized contractions and fixed-point theorems, Publ. inst. math., 12, 19-26, (1971) · Zbl 0234.54029 [9] Berinde, V., On the approximation of fixed points of weak contractive mappings, Carpathian J. math., 19, 1, 7-12, (2003) · Zbl 1114.47045 [10] Berinde, V., Approximating fixed points of weak contractions using Picard iteration, Nonlinear anal. forum, 9, 1, 43-53, (2004) · Zbl 1078.47042 [11] Ezquerro, J.A.; Hernández, M.A., Generalized differentiability conditions for newton’s method, IMA J. numer. anal., 22, 187-205, (2002) · Zbl 1006.65051 [12] 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, 53-61, (2006) · Zbl 1106.65048 [13] Gragg, W.B.; Tapia, R.A., Optimal error bounds for the newton – kantorovich theorem, SIAM J. numer. anal., 11, 10-13, (1974) · Zbl 0284.65042 [14] Ostrowski, A.M., La méthode de Newton dans LES espaces de Banach, C. R. acad. sci. Paris seŕ. A, 272, 1251-1253, (1971) · Zbl 0228.65041 [15] Miel, G.J., An update version of the Kantorovich theorem for newton’s method, Computing, 27, 237-244, (1981) · Zbl 0458.65046 [16] Yamamoto, T., A method for finding sharp error bounds for newton’s method under the Kantorovich assumptions, Numer. math., 49, 203-220, (1986) · Zbl 0607.65033 [17] Appell, J.; De Pascale, E.; Lysenko, Ju.V.; Zabrejko, P.P., New results on newton – kantorovich approximations with applications to nonlinear integral equations, Numer. funct. anal. optim., 18, 1-17, (1997) · Zbl 0881.65049 [18] Argyros, I.K; Gutiérrez, J.M., On the semilocal convergence of newton’s method under unifying conditions, (), 1-4 [19] Ferreira, O.P.; Svaiter, B.F., Kantorovich’s majorants principle for newton’s method, Comput. optim. appl., 42, 213-229, (2009) · Zbl 1191.90095 [20] Galantai, A., The theory of newton’s method, J. comput. appl. math., 124, 25-44, (2000) · Zbl 0965.65080 [21] Galperin, A., On convergence domain of newton’s and modified Newton methods, Numer. funct. anal. optim., 26, 385-405, (2005) · Zbl 1077.65057 [22] Gutiérrez, J.M., A new semilocal convergence theorem for newton’s method, J. comput. appl. math., 79, 131-145, (1997) · Zbl 0872.65045 [23] Huang, Z.D., A note on the Kantorovich theorem for newton’s iteration, J. comput. appl. math., 79, 211-217, (1993) · Zbl 0782.65071 [24] Yamamoto, T., Historical development in convergence analysis for newton’s and Newton-like methods, J. comput. appl. math., 124, 1-23, (2000) · Zbl 0965.65079 [25] Rheinboldt, W.C., On a theorem of S. Smale about newton’s method for analytic mappings, Appl. math. lett., 1, 69-72, (1988) · Zbl 0631.65064 [26] Wang, X.H.; Han, D.F., On dominating sequences method in the point estimate and Smale theorem, Sci. China (ser. A), 33, 135-144, (1990) [27] Wang, D.; Zhao, F., The theory of smale’s point estimation and its applications, J. comput. appl. math., 60, 253-269, (1995) · Zbl 0871.65046 [28] Wang, X.H., Convergence of newton’s method and inverse function theorem in Banach space, Math. comp., 68, 169-186, (1999) · Zbl 0923.65028 [29] Dedieu, J.-P.; Priouret, P.; Malajovich, G., Newton’s method on Riemannian manifolds, IMA J. numer. anal., 23, 395-419, (2003) · Zbl 1047.65037 [30] Alvarez, F.; Bolte, J.; Munier, J., A unified local convergence result for newton’s method in Riemannian manifolds, Found. comput. math., 8, 197-226, (2008) · Zbl 1147.58008 [31] Li, C.; Wang, J.-H., Newton’s method for sections on Riemannian manifolds: generalized covariant $$\alpha$$-theory, J. complexity, 24, 423-451, (2008) · Zbl 1153.65059 [32] Li, C.; Wang, J.-H.; Dedieu, J.-P., Smale’s point estimate theory for newton’s method on Lie groups, J. complexity, 25, 128-151, (2009) · Zbl 1170.65040 [33] Bianchini, R.M.; Grandolfi, M., Transformazioni di tipo contracttivo generalizzato in uno spazio metrico, Atti accad. naz. lincei rend. cl. sci. fiz. mat. natur., 45, 212-216, (1968) · Zbl 0205.27202 [34] Matkowski, J., Integrable solutions of functional equations, () · Zbl 0383.39002 [35] Zitarosa, A., Una generalizzazione del teorema di Banach sulle contrazioni, Matematiche, 23, 417-424, (1968) · Zbl 0199.25902 [36] Proinov, P.D., Semilocal convergence of two iterative methods for simultaneous computation of polynomial zeros, C. R. acad. bulg. sci., 59, 705-712, (2006) · Zbl 1119.65043 [37] Cheney, W.; Goldstein, A.A., Proximity maps for convex sets, Proc. amer. math. soc., 10, 448-450, (1959) · Zbl 0092.11403 [38] Rheinboldt, W.C., A unified convergence theory for a class of iterative processes, SIAM J. numer. anal., 5, 42-63, (1968) · Zbl 0155.46701 [39] Gel’man, A., A certain fixed-point principle, Dokl. akad. nauk SSSR, 198, 506-508, (1971), (in Russian). [Engl. transl. in: Soviet Math. Dokl. 12 (1971) 813-816] [40] I.A. Rus, Teoria punctului fix, II, Univ. Babes, Cluj, 1973 [41] Hicks, T.L., Another view of fixed point theory, Math. japonica, 35, 231-234, (1990) · Zbl 0701.47032 [42] Kornstaedt, H.-J., Funktionalungleichungen und iterationsverfahren, Aequ. math., 13, 21-45, (1975) [43] Proinov, P.D., A new semilocal convergence theorem for the Weierstrass method from data at one point, C. R. acad. bulg. sci., 59, 131-136, (2006) · Zbl 1101.65049 [44] Vertgeim, B.A., On the conditions for the applicability of newton’s method, Dokl. akad. nauk SSSR, 110, 719-722, (1956), (in Russian) · Zbl 0072.13601 [45] Vertgeim, B.A., On some methods of the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi mat. nauk, 12, 166-169, (1957), (in Russian) [Engl. Transl.: Amer. Math. Soc. Transl. 16 (1960) 378-382] · Zbl 0079.33305 [46] Keller, H.B., Newton’s method under mild differentiability conditions, J. comput. system sci., 4, 15-28, (1970) · Zbl 0191.16001 [47] Lysenko, Yu.V., The convergence conditions of the newton – kantorovich method for nonlinear equations with Hölder linearizations, Dokl. akad. nauk BSSR, 38, 20-24, (1994), (in Russian) · Zbl 0819.65095 [48] DePascale, E.; Zabrejko, P.P., The convergence of the newton – kantorovich method under vertgeim conditions: A new improvement, Z. anal. anwend., 17, 271-280, (1998) · Zbl 0907.65054 [49] Cianciaruso, F.; De Pascale, E., Newton – kantorovich approximations when the derivative is Hölderian: old and new results, Numer. funct. anal. optim., 24, 713-723, (2003) · Zbl 1037.65059 [50] Cianciaruso, F.; De Pascale, E., Estimates of majorizing sequences in the newton – kantorovich method, Numer. funct. anal. optim., 27, 529-538, (2006) · Zbl 1101.65053 [51] Cianciaruso, F.; De Pascale, E., Estimates of majorizing sequences in the newton – kantorovich method: A further improvement, J. math. anal. appl., 322, 329-335, (2006) · Zbl 1123.65049 [52] Jankó, B., Rezolvarea equatiilor operationale neliniare in spatii Banach, (1969), Editura Academiei republicii socialiste Romania Bucuresti [53] Petcu, D., On the Kantorovich hypothesis for newton’s method, Informatika (Vilnius), 4, 1-2, 188-198, (1993) · Zbl 0905.65060 [54] Hernández, M.A., The Newton method for operators with Hölder continuous first derivative, J. optim. theory appl., 109, 631-648, (2001) · Zbl 1012.65052 [55] Yamamoto, T., A unified derivation of several error bounds for newton’s process, J. comput. appl. math., 12-13, 179-191, (1985) · Zbl 0582.65047 [56] Miel, G.J., Majorizing sequences and error bounds for iterative methods, Math. comp., 34, 185-202, (1980) · Zbl 0425.65033 [57] Potra, F.A., On the a posteriori error estimates for newton’s method, Beitraege numer. math., 12, 125-138, (1984) · Zbl 0556.65048 [58] Blum, L.; Cucker, F.; Shub, M.; Smale, S., Complexity and real computation, (1998), Springer New York [59] Dedieu, J.-P., () [60] Potra, F.A.; Pták, V., Sharp error bounds for newton’s process, Numer. math., 34, 63-72, (1980) · Zbl 0434.65034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.