共查询到20条相似文献,搜索用时 15 毫秒
1.
O.P. Ferreira 《Journal of Computational and Applied Mathematics》2011,235(5):1515-1522
A local convergence analysis of Newton’s method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications. 相似文献
2.
Ioannis K. Argyros 《Journal of Complexity》2011,27(1):39-54
We present a semilocal convergence theorem for Newton’s method (NM) on spaces with a convergence structure. Using our new idea of recurrent functions, we provide a tighter analysis, with weaker hypotheses than before and with the same computational cost as for Argyros (1996, 1997, 1997, 2007) [1], [2], [3] and [5], Meyer (1984, 1987, 1992) [13], [14] and [15]. Numerical examples are provided for solving equations in cases not covered before. 相似文献
3.
Hongmin Ren 《Applied mathematics and computation》2009,213(2):440-448
In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem. 相似文献
4.
We use Newton’s method to approximate a locally unique solution of an equation in a Banach space setting. We introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton’s method than before [J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1–17; I.K. Argyros, The theory and application of abstract polynomial equations, in: Mathematics Series, St. Lucie/CRC/Lewis Publ., Boca Raton, Florida, USA, 1998; I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005) 179–194; I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New York, 2008; S. Chandrasekhar, Radiative Transfer, Dover Publ., New York, 1960; F. Cianciaruso, E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003) 713–723; N.T. Demidovich, P.P. Zabrejko, Ju.V. Lysenko, Some remarks on the Newton–Kantorovich method for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993) 22–26. (in Russian); E. De Pascale, P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998) 271–280; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994) 20–24. (in Russian); B.A. Vertgeim, On conditions for the applicability of Newton’s method, (Russian), Dokl. Akad. Nauk., SSSR 110 (1956) 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957) 166–169. (in Russian); English transl.:; Amer. Math. Soc. Transl. 16 (1960) 378–382] provided that the Fréchet-derivative of the operator involved is p-Hölder continuous (p∈(0,1]). 相似文献
5.
From Kantorovich’s theory we establish a general semilocal convergence result for Newton’s method based fundamentally on a generalization required to the second derivative of the operator involved. As a consequence, we obtain a modification of the domain of starting points for Newton’s method and improve the a priori error estimates. Finally, we illustrate our study with an application to a special case of conservative problems. 相似文献
6.
Sergio Plaza Natalia Romero 《Journal of Computational and Applied Mathematics》2011,235(10):3238-3244
We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=zm−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one. 相似文献
7.
Ioannis K. Argyros 《Applied mathematics and computation》2009,215(4):1553-1560
We provide sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation containing operators that are Fréchet-differentiable of order at least two, in a Banach space setting. Numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot [J.M. Gutiérrez, A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79(1997) 131-145; Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica 5 (1985) 71-84]. 相似文献
8.
Changbum Chun 《Applied mathematics and computation》2009,211(2):474-479
In this paper, we present a new third-order modification of Newton’s method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots. 相似文献
9.
José L. Hueso Eulalia Martínez Juan R. Torregrosa 《Journal of Computational and Applied Mathematics》2009
It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method. 相似文献
10.
In this paper, the convergence of a general algorithm with θ-type stabilization form for the variational multiscale (VMS) method is presented. Meanwhile, explicit-type and implicit-type algorithms with linear convergence and quadratic convergence are derived from the θ-type algorithm, respectively. The combination of explicit-type and implicit-type algorithms are applied to adaptive VMS, which shows good efficiency. Finally, some numerical tests are shown to support the convergence analysis. 相似文献
11.
We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the nonlinear operator under consideration. Using this result we show that the Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance. 相似文献
12.
Weifeng Yang 《Applied mathematics and computation》2010,217(6):2490-2494
The symmetries of Julia sets of Newton’s method is investigated in this paper. It is shown that the group of symmetries of Julia set of polynomial is a subgroup of that of the corresponding standard, multiple and relax Newton’s method when a nonlinear polynomial is in normal form and the Julia set has finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relax Newton’s method to be horizontal line is obtained. 相似文献
13.
Newton’s method is a basic tool in numerical analysis and numerous applications, including operations research and data mining. We survey the history of the method, its main ideas, convergence results, modifications, its global behavior. We focus on applications of the method for various classes of optimization problems, such as unconstrained minimization, equality constrained problems, convex programming and interior point methods. Some extensions (non-smooth problems, continuous analog, Smale’s results, etc.) are discussed briefly, while some others (e.g., versions of the method to achieve global convergence) are addressed in more details. 相似文献
14.
A generalization of the variants of Newton’s method based on interpolation rules of quadrature is obtained, in order to solve systems of nonlinear equations. Under certain conditions, convergence order is proved to be 2d+1, where d is the order of the partial derivatives needed to be zero in the solution. Moreover, different numerical tests confirm the theoretical results and allow us to compare these variants with Newton’s classical method, whose convergence order is d+1 under the same conditions. 相似文献
15.
In this paper, a variant of Steffensen’s method of fourth-order convergence for solving nonlinear equations is suggested. Its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth-order convergence. Its applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs are showed as well in the numerical examples. 相似文献
16.
Ioannis K. Argyros 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):331-341
We approximate a locally unique solution of an equation on a Banach space setting using Newton’smethod.Motivated by the work
by Ferreira and Svaiter [5] but using more precise majorization sequences, and under the same computational cost we provide:
a larger convergence region; finer error bounds on the distances involved, and an at least as precise information on the location
of the solution than in [5]. The results can also compare favorably to the corresponding ones given byWang in [10]. Finally
we complete the study with two concrete applications.
相似文献
17.
In this paper, a modified Newton’s method for the best rank-one approximation problem to tensor is proposed. We combine the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm with the iterative matrix of GRQ-Newton method, and present a modified version of GRQ-Newton algorithm. A line search along the projective direction is employed to obtain the global convergence. Preliminary numerical experiments and numerical comparison show that our algorithm is efficient. 相似文献
18.
The convergence region of Traub’s method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results. 相似文献
19.
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.
相似文献
20.
Ioannis K. Argyros 《Central European Journal of Mathematics》2007,5(2):205-214
We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator
equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s
method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example
is also provided.
相似文献