Semilocal Convergence Theorem Research Articles

A MODIFIED INEXACT NEWTON METHOD

In this paper, we consider a modified inexact Newton method for solving a nonlinear system F(x) = 0 where F(x) : R n → R n . The basic idea is to accelerate convergence. A semi-local convergence theorem for the modified inexact Newton method is established and an affineinvariant version is also given. Moreover, we test three numerical examples which show that the modified inexact scheme is more efficient than theclassical inexact Newton strategy.

Semilocal convergence theorem for the inverse-free Jarratt method under new Hölder conditions.

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.

On the convergence of Halley’s method for simultaneous computation of polynomial zeros

AbstractIn this paper we study the convergence of Halley’s method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley’s method for simultaneous computation of polynomial zeros.

Convergence of the two-point Weierstrass root-finding method

In this paper we present new local and semilocal convergence theorems for the two-point Weierstrass method for the simultaneous computation of polynomial zeros. Our local convergence result improves and complements the previous one obtained by Kanno et al. (Jpn J Indus Appl Math 13, 267–288, 1996).

A new semilocal convergence theorem for the Weierstrass method for finding zeros of a polynomial simultaneously

In this paper we study the convergence of the famous Weierstrass method for simultaneous approximation of polynomial zeros over a complete normed field. We present a new semilocal convergence theorem for the Weierstrass method under a new type of initial conditions. Our result is obtained by combining ideas of Weierstrass (1891) and Proinov (2010). A priori and a posteriori error estimates are also provided under the new initial conditions.

MODIFICATION OF THE KANTOROVICH-TYPE CONDITIONS FOR NEWTON'S METHOD INVOLVING TWICE FRECHET DIFFERENTIABLE OPERATORS

We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217]). This way, our convergence criteria can be weaker; the error estimates tighter and the information on the location of the solution more precise. Numerical examples are presented to show that our results apply in cases not covered before such as [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217].

Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.

Chebyshev-Secant-type Methods for Non-differentiable Operators

In this paper we give a semilocal convergence theorem for a family of iterative methods for solving nonlinear equations defined between two Banach spaces. This family is obtained as a combination of the well known Secant method and Chebyshev method. We give a very general convergence result that allow the application of these methods to non-differentiable problems.

On the semi-local convergence of Halley’s method under a center-Lipschitz condition on the second Fréchet derivative

We expand the applicability of Halley’s method for solving nonlinear equations in a Banach space setting. We assume the existence of the center-Lipschitz condition on the second Fréchet-derivative of the operator involved instead of Lipschitz condition used extensively in the literature [1,2,4,5]. The center-Lipschitz condition is satisfied in many interesting cases, where the Lipschitz condition is not satisfied [3,4,6,7,13]. We show that the semi-local convergence theorem established in [X.B. Xu, Y.H. Ling, Semilocal convergence for Halley’s method under weak Lipschitz condition, Appl. Math. Comput. 215 (2009) 3057–3067] is not true. A new semi-local convergence theorem is established for Halley’s method under the same condition. Our results are illustrated using a nonlinear Hammerstein integral equation of the second kind where our convergence criteria are satisfied but convergence criteria in earlier studies such as [1,2] are not satisfied.

A variant of the Newton–Kantorovich theorem for nonlinear integral equations of mixed Hammerstein type

We study nonlinear integral equations of mixed Hammerstein type using Newton’s method as follows. We investigate the theoretical significance of Newton’s method to draw conclusions about the existence and uniqueness of solutions of these equations. After that, we approximate the solutions of a particular nonlinear integral equation by Newton’s method. For this, we use the majorant principle, which is based on the concept of majorizing sequence given by Kantorovich, and milder convergence conditions than those of Kantorovich. Actually, we prove a semilocal convergence theorem which is applicable to situations where Kantorovich’s theorem is not.

Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

Under the hypothesis that the first derivative satisfies some kind of weak Lipschitz conditions, a new semilocal convergence theorem for inexact Newton method is presented. Unified convergence criteria ensuring the convergence of inexact Newton method are also established. Applications to some special cases such as the Kantorovich type conditions and γ‐Conditions are provided and some well‐known convergence theorems for Newton′s method are obtained as corollaries.

A unifying theorem for Newton’s method on spaces with a convergence structure

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) [1–3,5], Meyer (1984, 1987, 1992) [13–15]. Numerical examples are provided for solving equations in cases not covered before.

On a third‐order Newton‐type method free of bilinear operators

AbstractThis paper is devoted to the study of a third‐order Newton‐type method. The method is free of bilinear operators, which constitutes the main limitation of the classical third‐order iterative schemes. First, a global convergence theorem in the real case is presented. Second, a semilocal convergence theorem and some examples are analyzed, including quadratic equations and integral equations. Finally, an approximation using divided differences is proposed and used for the approximation of boundary‐value problems. Copyright © 2009 John Wiley & Sons, Ltd.

A semi-local convergence theorem for a robust revised Newton’s method

It is well known that Newton’s iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular. In this paper, we study a robust revised Newton’s method for solving nonlinear equations, which can be carried out with a starting point with a degenerate derivative at an iterative step. It is proved that the method is convergent under the conditions of the Newton–Kantorovich theorem, which implies a larger convergence domain of the method. We also show that our method inherits the fast convergence of Newton’s method. Numerical experiments are performed to show the robustness of the proposed method in comparison with the standard Newton’s method.

A new semilocal convergence theorem for a fast iterative method with nondifferentiable operators

A new semilocal convergence theorem for a fast iterative method in Banach spaces is provided for approximating a solution of a nondifferentiable operator equation. A condition for divided differences of order one is considered in this paper, which generalizes the usual ones, i.e., Lipschitz continuous or Holder continuous conditions. Note that no conditions of divided differences of order two are used. Therefore our results are of theoretical and practical interest. Finally, a numerical example is provided to show that the new iterative method compares favorably with earlier ones.

On Newton’s method for solving equations containing Fréchet-differentiable operators of order at least two

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].

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.

High order iterative schemes for quadratic equations

High order iterative methods for quadratic equations are studied. A bi-parametric family that includes some well known iterative schemes is introduced. A unified semilocal convergence theorem is presented. The implementation and some applications to partial differential equations are finally discussed.

A new semi-local convergence theorem for the inexact Newton methods

We present a new semi-local convergence theorem for the inexact Newton methods in the assumption that the derivative satisfies some kind of weak Lipschitz conditions. As special cases of our main result we re-obtain some well-known convergence theorems for Newton methods.

A new semilocal convergence theorem of Müller’s method

A semilocal convergence theorem of Müller’s method is established under the γ-condition. An error estimation is given which matches the convergence order of the method. Numerical examples are provided to show that our results apply, where earlier ones fail.