On the R-order of convergence of Newton's method under mild differentiability conditions

A new technique is used instead of the classical majorant principle to analyze the R-order of convergence of the Newton process when more general conditions than the Kantorovich ones are considered.     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

The Ulm method under mild differentiability conditions

The Ulm method is considered to approximate a solution of a nonlinear operator equation F(x) = 0. We study the convergence of this method when F′ is ω-conditioned and prove that the R-order of convergence is at least 1 + p if ω is quasi-homogeneous of type ω(tz)≤ t ^p ω(z), for z > 0, tϵ[0,1] and pϵ[0,1]. Preparation of this paper was partly supported by the Ministry of Education and Science (MTM 2005-03091).     

Newton's method under mild differentiability conditions with error analysis

Summary The concept of majorizing sequences introduced by Rheinboldt (SIAM J.N.A. 1968) is used to prove convergence for Newton's method for operator equations of the formT f= when the operator satisfied the condition that the Fréchet derivative is Hölder continuous.A detailed analysis of computational errors is given for Newton's method applied to operators with Hölder continuous derivatives. This analysis is shown to reduce the analysis of Lancaster (Num. Math. 1968) when the operator has a continuous second derivative.The above analysis is applied to an example of a second order differential equation.The author is grateful to National Research Council of Canada and National Defency Research Board of Canada for financial support.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

The Newton—Kantorovich method under mild differentiability conditions and the Ptâk error estimates

We study the Newton-Kantorovich method under mild differentiability conditions. Using Zabrejko-Nguen assumptions we extend the results obtained byZabrejko andNguen in [11]. We also derive Ptâk error estimates which compare favorably with the ones obtained previously byKeller [4],Rokne [7], andArgyros in [1].     

Semilocal convergence for a family of Chebyshev-Halley like iterations under a mild differentiability condition

The semilocal convergence of a family of Chebyshev-Halley like iterations for nonlinear operator equations is studied under the hypothesis that the first derivative satisfies a mild differentiability condition. This condition includes the usual Lipschitz condition and the H?lder condition as special cases. The method employed in the present paper is based on a family of recurrence relations. The R-order of convergence of the methods is also analyzed. As well, an application to a nonlinear Hammerstein integral equation of the second kind is provided. Furthermore, two numerical examples are presented to demonstrate the applicability and efficiency of the convergence results.     

Optimal convergence properties of kernel density estimators without differentiability conditions

Let X ₁, X ₂, ..., X _n be independent observations from an (unknown) absolutely continuous univariate distribution with density f and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmOzayaajaGaaiikaiaadIhacaGGPaGaeyypa0Jaaiikaiaad6ga% caWGObGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqadaba% Gaam4saiaacUfadaWcgaqaaiaacIcacaWG4bGaeyOeI0Iaamiwamaa% BaaaleaacaWGPbaabeaakiaacMcaaeaacaWGObGaaiyxaaaaaSqaai% aadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa!5356!\[\hat f(x) = (nh)^{ - 1} \sum\nolimits_{i = 1}^n {K[{{(x - X_i )} \mathord{\left/ {\vphantom {{(x - X_i )} {h]}}} \right. \kern-\nulldelimiterspace} {h]}}} \] be a kernel estimator of f(x) at the point x, \s-<x<, with h=h _n (h _n O and nh _n , as n) the bandwidth and K a kernel function of order r. Optimal rates of convergence to zero for the bias and mean square error of such estimators have been studied and established by several authors under varying conditions on K and f. These conditions, however, have invariably included the assumption of existence of the r-th order derivative for f at the point x. It is shown in this paper that these rates of convergence remain valid without any differentiability assumptions on f at x. Instead some simple regularity conditions are imposed on the density f at the point of interest. Our methods are based on certain results in the theory of semi-groups of linear operators and the notions and relations of calculus of finite differences.This research was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and the University of Alberta Central Research Fund.     

Generalized differentiability conditions for Newton's method

The use of majorizing sequences is the usual way to prove theconvergence of Newton's method. An alternative technique tomajorizing sequences is provided in this paper, in which threescalar sequences are used, so that the analysis of convergenceis simplified when the traditional convergence condition isrelaxed. An application to a nonlinear integral equation isalso given, which is also solved and the solution approximatedby a discretization process.     

Convergence of Directional Methods under mild differentiability and applications

A semilocal convergence analysis for Directional Methods under mild differentiability conditions is provided in this study. Using our idea of recurrent functions, we provide sufficient convergence conditions as well as the corresponding errors bounds. The results are extended to hold in a Hilbert space setting and a favorable comparison is provided with earlier works [6], [7], [8], [9], [10], [11] and [20]. Numerical examples are also provided in this study.     

A two-step Steffensen's method under modified convergence conditions

A simplification of a third order iterative method is proposed. The main advantage of this method is that it does not need to evaluate neither any Fréchet derivative nor any bilinear operator. A semilocal convergence theorem in Banach spaces, under modified Kantorovich conditions, is analyzed. A local convergence analysis is also performed. Finally, some numerical results are presented.     

Moderate deviations for estimators under exponentially stochastic differentiability conditions

We introduce two exponentially stochastic differentiability conditions to study moderate deviations for M-estimators. Under a generalized exponentially stochastic differentiability condition, a moderate deviation principle is established. Some sufficient conditions of the exponentially stochastic differentiability and examples are also given.     

Weaker convergence conditions for the secant method

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.     

Practical convergence conditions for the Davidon-Fletcher-Powell method

The convergence properties of the Davidon-Fletcher-Powell method when applied to the minimization of convex functions are considered for the case where the one-dimensional minimization required at each iteration is not solved exactly. Conditions on the error incurred at each iteration are given which are sufficient for the original method to have a linear or superlinear rate of convergence, and for the restarted version to have ann-step quadratic rate of convergence.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

New sufficient convergence conditions for the secant method

We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.