Increasing the applicability of Steffensen's method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensen's method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensen's method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensen's method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors.     

A Steffensen's type method in Banach spaces with applications on boundary-value problems

In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed.     

Monotone enclosing of solutions of the Steffensen‐type

In this paper we consider two iterative methods of the Steffensen‐type. The iteration sequences which approximate the solution of f(x) = 0 from opposite sides are generated by Steffensen's method and a secant method, respectively. We show two enclosing theorems and establish orders of convergence. The iteration sequences converge with order two and three, respectively. Numerical examples complete the paper.     

Numerical study of modified Adomian's method applied to Burgers equation

The application of Adomian's decomposition method to partial differential equations, when the exact solution is not reached, demands the use of truncated series. But the solution's series may have small convergence radius and the truncated series may be inaccurate in many regions. In order to enlarge the convergence domain of the truncated series, Padé approximants (PAs) to the Adomian's series solution have been tested and applied to partial and ordinary differential equations, with good results. In this paper, PAs, both in x$x$ and t$t$ directions, applied to the truncated series solution given by Adomian's decomposition technique for Burgers equation, are tested. Numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution. It is also shown in this paper, that the application of Adomian's method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences, the so-called method of lines, may reduce the convergence domain of the solution's series.     

A Short Note on the Quasilinearization Method for Fractional Differential Equations

Recent literature shows that for certain classes of fractional differential equations the monotone iterative technique fails to guarantee the quadratic convergence of the quasilinearization method. The present work proves the quadratic convergence of the quasilinearization method and the existence and uniqueness of the solution of such a class of fractional differential equations. Our analysis depends upon the classical Kantorovich theorem on Newton's method. Various examples are discussed in order to illustrate our approach.     

CONVERGENCE OF NEWTON＇S METHOD FOR A MINIMIZATION PROBLEM IN IMPULSE NOISE REMOVAL

Recently, two-phase schemes for removing salt-and-pepper and random-valued impulse noise are proposed in [6, 7]. The first phase uses decision-based median filters to locate those pixels which are likely to be corrupted by noise (noise candidates). In the second phase, these noise candidates are restored using a detail-preserving regularization method which allows edges and noise-free pixels to be preserved. As shown in [18], this phase is equivalent to solving a one-dimensional nonlinear equation for each noise candidate.One can solve these equations by using Newton‘s method. However, because of the edgepreserving term, the domain of convergence of Newton‘s method will be very narrow. In this paper, we determine the initial guesses for these equations such that Newton‘s method will always converge.     

Kantorovich's Majorization and Functional Equations

Abstract

We consider systems of nonlinear difference equations arising when convergence analysis of an iterative method for solving operator equations in Banach spaces is carried out via Kantorovich's technique of majorization. The main challenge in this context is to determine the convergence domain of the corresponding majorant generator. As it turns out, dealing with this task leads to solution of functional equations of a certain kind. After considering several examples, we formulate two generic models and develop an approach to their solution.     

A general class of inequalities containing Steffensen's inequality

Mathematical Notes - A general inequality for sourcewise representable functions is proved which contains a class of inequalities for monotonic functions and, in particular, Steffensen's...     

Local convergence in Fermat's problem

The general Fermat problem is to find the minimum of the weighted sum of distances fromm destination points in Euclideann-space. Kuhn recently proved that a classical iterative algorithm converges to the unique minimizing point , for any choice of the initial point except for a denumerable set. In this note, it is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination. If  is not a destination, then locally convergence is always linear with upper and lower asymptotic convergence boundsλ andλ′ (λ ≥ 1/2, whenn=2). If  is a destination, then convergence can be either linear, quadratic or sublinear. Three numerical examples which illustrate the different possibilities are given and comparisons are made with the use of Steffensen's scheme to accelerate convergence.     

Solving nonlinear partial differential equations using the modified variational iteration Padé technique

In this paper, the modified variational iteration method (MVIM) is reintroduced with the enhancement of Padé approximants to lengthen the interval of convergence of VIM or MVIM when used alone in solving nonlinear problems. KdV, mKdV, Burger's and Lax's equations are used as examples to illustrate the effectiveness and convenience of the proposed technique.     

Some variants of Cauchy's method with accelerated fourth-order convergence

In this paper, we present some variants of Cauchy's method for solving non-linear equations. Analysis of convergence shows that the methods have fourth-order convergence. Per iteration the new methods cost almost the same as Cauchy's method. Numerical results show that the methods can compete with Cauchy's method.     

Use of Halley's Method in the Nonlinear Finite Element Analysis

Halley's method is a higher order iteration method for the solution of nonlinear systems of equations. Unlike Newton's method, which converges quadratically in the vicinity of the solution, Halley's method can exhibit a cubic order of convergence. The equations of Halley's method for multiple dimensions are derived using Padé approximants and inverse one-point interpolation, as proposed by Cuyt. The investigation of the performance of Halley's method concentrates on eight-node volume elements for nonlinear deformations using Staint Venant-Kirchhoff's constitutive law, as well as a geometric linear theory of von Mises plasticity. The comparison with Newton's method reveals the sensibility of Halley's method, in view of the radius of attraction but also demonstrates the advantages of Halley's method considering simulation costs and the order of convergence. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kepler equation and accelerated Newton method

We study the efficiency of the accelerated Newton method (Garlach, SIAM Rev. 36 (1994) 272–276) for several orders of convergence versus Danby's method for the resolution of Kepler's equation; we find that the cited method of order three is competitive with Danby's method and the classical Newton's method. We also generalize the accelerated Newton method for the resolution of system of algebraic equations, obtaining a formula of order three and a proof of its convergence; its application to several examples shows that its efficiency is greater than Newton's method.     

On the second-order asymptotical regularization of linear ill-posed inverse problems

ABSTRACT

In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the corresponding convergence rates. We prove that, under the appropriate source conditions and by using Morozov's conventional discrepancy principle, SOAR exhibits the same power-type convergence rate as the classical version of asymptotical regularization (Showalter's method). Moreover, we propose a new total energy discrepancy principle for choosing the terminating time of the dynamical solution from SOAR, which corresponds to the unique root of a monotonically non-increasing function and allows us to also show an order optimal convergence rate for SOAR. A damped symplectic iterative regularizing algorithm is developed for the realization of SOAR. Several numerical examples are given to show the accuracy and the acceleration effect of the proposed method. A comparison with other state-of-the-art methods are provided as well.     

On a Newton-Kurchatov-type Iterative Process

Newton's method and Kurchatov's method are iterative processes known for their fast speed of convergence. We construct from both methods an iterative method to approximate solutions of nonlinear equations given by a nondifferentiable operator, and we study its semilocal convergence in Banach spaces. Finally, we consider several applications of this new iterative process.     

One-Step Hybrid Block Method Containing Third Derivatives and Improving Strategies for Solving Bratu's and Troesch's Problems

In this paper, we develop a one-step hybrid block method for solving boundary value problems, which is applied to the classical one-dimensional Bratu's and Troesch's problems. The convergence analysis of the new technique is discussed, and some improving strategies are considered to get better performance of the method. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. The solution of this system is obtained through a homotopy-type strategy used to provide the starting points needed by Newton's method. Some numerical experiments are presented to show the performance and effectiveness of the proposed approach in comparison with other methods that appeared in the literature.     

Third-order modification of Newton's method

In this paper, we present a new modification of Newton's method for solving non-linear equations. Analysis of convergence shows that the new method is cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.     

On Newton's method for solving generalized equations

In this paper, we study the convergence properties of a Newton-type method for solving generalized equations under a majorant condition. To this end, we use a contraction mapping principle. More precisely, we present semi-local convergence analysis of the method for generalized equations involving a set-valued map, the inverse of which satisfying the Aubin property. Our analysis enables us to obtain convergence results under Lipschitz, Smale and Nesterov-Nemirovski's self-concordant conditions.