Steffensen's Method Research Articles

A One-step Steffensen-type Method with Super-cubic Convergence for Solving Nonlinear Equations

In this paper, a one-step Steffensen-type method of order 3.383 is designed and proved for solving nonlinear equations. This super-cubic convergence is obtained by self-accelerating second-order Steffensen's method twice with memory, but without any new function evaluations. The proposed method is very efficient and convenient, since it is still a derivative-free two-point method. Numerical examples confirm the theoretical results and high computational efficiency.

Loudness Calculation and its Application Based on Moore Model

Loudness describes the sound intensity sensations of the human hearing. A study about loudness calculation which is based on the Moore model shows us that starting from the mechanism of human`s ear, the basic principle and calculation skills of algorithm have been researched. By using Matlab to optimize the algorithm, the specific loudness curve and loudness level of any audio file can be calculated; equal-loudness curve by using Steffensen method is calculated compared with that from the acoustic standard, In order to validating the program. The loudness of the sewing machine under different conditions is calculated.

How to Improve the Domain of Starting Points for Steffensen's Method

We analyze the semilocal convergence of Steffensen's method, using a novel technique, which is based on recurrence relations, for solving systems of nonlinear equations. This technique allows analyzing the convergence of Steffensen's method to solutions of equations, where the function involved can be both differentiable and nondifferentiable. Moreover, this technique also allows enlarging the domain of starting points for Steffensen's method from certain predictions with the simplified Steffensen method.

An hybrid method that improves the accessibility of Steffensen’s method

Steffensen's method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen's method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen's method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.

Accelerating convergence in minisum location problem with [formula omitted] norms

This paper presents a procedure for accelerating convergence of the Weiszfeld algorithm in the classical single facility location median problem in which the distances are measured by ℓp-norms. To this end, we combined Steffensen's method, a generic acceleration scheme applied to iterative processes for solving fixed point equations, with the acceleration methods based on the transformation of the Weiszfeld algorithm by a factor which is a function of the parameter p. The convergence of the proposed methodology and the conditions under which it is guaranteed are analyzed. The computational results show that the total number of iterations to meet a given stopping criterion will be reduced with respect to the results obtained in other algorithms proposed in the literature. The running times are either reduced or quite similar with respect to the existing algorithms for which no results of convergence are provided.

Fast FE–BEM algorithms for orthotropic frictional contact

SUMMARYThis paper presents a new and efficient methodology for solving 3D frictional contact problems considering an orthotropic friction law. The contact methodology is based on a proposed augmented Lagrangian formulation for orthotropic frictional contact problems, and a new discrete contact operator, which allows to reduce the number of unknowns in a Newton‐like algorithm that accelerates the attainment of the solution. A fast Uzawa scheme is also proposed on the basis of the Steffensen's method. Both algorithms prove to be very robust and efficient to solve orthotropic frictional contact problems. The proposed formalism has the advantage of being very compact and valid for both the FEM and the BEM. Numerical results are given to demonstrate the validity of the formulation and algorithms proposed. Copyright © 2013 John Wiley & Sons, Ltd.

On a Newton–Steffensen type method

In this paper we study the convergence of a Newton–Steffensen type method for solving nonlinear equations in R , introduced by Sharma [J.R. Sharma, A composite third order Newton–Steffensen method for solving nonlinear equations, Appl. Math. Comput. 169 (2005), 242–246]. Under simplified assumptions regarding the smoothness of the nonlinear function, we show that the q -convergence order of the iterations is 3. The efficiency index of the method is 3 3 , and is larger than I 2 = 2 , which corresponds to the Newton method or the Steffensen method. Moreover, we show that if the nonlinear function maintains the same monotony and convexity on an interval containing the solution, and the initial approximation satisfies the Fourier condition, then the iterations converge monotonically to the solution. We also obtain a posteriori formulas for controlling the errors. The numerical examples confirm the theoretical results.

A fast algorithm for finding the first intersection with a non-convex yield surface

A major task in the numerical modelling of soils using complex elastoplastic material models is stress updating. This paper proposes a fast and robust numerical algorithm for locating the first intersection between a non-convex yield surface and an elastic trial stress path. The intersection problem is cast into a problem of finding the smallest positive root of a nonlinear function. Such a function may have multiple roots within the interval of interest. The method is based on the modified Steffensen method, with important modifications to address the issues arising from the non-convexity. Numerical examples demonstrate that the proposed M 2 Steffensen method is indeed computationally efficient and robust.

Convergence of inexact difference methods under the generalized Lipschitz conditions

We study the convergence of the inexact chord method and Steffensen method for the solution of systems of nonlinear equations under the generalized Lipschitz conditions for first-order divided differences. We consider methods with a check of the relative discrepancy. The results obtained easily provide an estimate of the convergence sphere for inexact methods. For special cases, these results coincide with the known ones.

Convergence and Error Estimate of the Steffensen Method

We present the Steffensen method in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math> space. The convergence theorem is established by using the technique of majorizing function. Meanwhile, an error estimate is given. It avoids the calculus of derivative but has the same convergence order 2 with Newton&#39;s method. Finally, illustrative examples are included to demonstrate the validity and applicability of the technique.

An improved local convergence analysis for Newton–Steffensen-type method

We provide a local convergence analysis for Newton–Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces. Under new center–conditions and the Aubin continuity property, we obtain the linear local convergence of Newton–Steffensen method. Our results compare favorably with related obtained in (Argyros and Hilout, 2007 submitted; Hilout in J. Math. Anal. Appl. 339:753–761, 2008).

On the Steffensen method under the generalized Lipschitz conditions for the divided difference operator

AbstractConvergence of the Steffensen method for solving nonlinear operator equations in the Banach spaces under the generalized Lipschitz condition for the first–order divided differences is investigated. The conditions and quadratic speed of convergence of this method are found. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

JULIA SETS OF GENERALIZED NEWTON'S METHOD

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

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.

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.

Correction of sonic anemometer angle of attack errors

An improved method of correcting for the angle of attack error resulting from the imperfect (co)sine response of ultrasonic anemometers is proposed. The angle of attack, which was calculated as the arctangent of observed wind vectors, contains the angle of attack errors in the vectors themselves, and hence this angle was ‘false’. The ‘true’ angle of attack should be calculated from the corrected or ‘true’ wind vectors. In the improved method, the ‘true’ angle of attack is derived by solving a nonlinear equation which connects the ‘false’ angle of attack to the ‘true’ one. In applying this method to the case of R2- and R3-type Solent ultrasonic anemometers, the fit of the function for the sine responses to wind tunnel data was improved, and the cosine response function was also improved to consider the effect of the difference of the vertical positions of the transducers. The nonlinear equation was solved using the Steffensen method; robustly and adequately fast for practical use in calculating eddy fluxes. The accuracy of the correction method is improved over a previous one, especially at large angles of attack. Applying our correction method to field data from two forests and one peat bog, the eddy fluxes of sensible heat, latent heat and CO 2were increased and the energy balance closure rates were improved. These results indicate that a large portion of energy imbalance can be accounted for by the ultrasonic anemometer angle of attack dependent errors.

A composite third order Newton–Steffensen method for solving nonlinear equations

In this paper, we suggest a third-order method formed by the composition of Newton and Steffensen methods for finding simple and real roots of a nonlinear equation in single variable. Per iteration the formula requires two evaluations of the function and single evaluation of the derivative. Experiments show that the method is suitable in the cases where Newton and Steffensen methods fail.

Local convergence of general Steffensen type methods

We study the local convergence of a generalized Steffensen method. We show that this method substantially improves the convergence order of the classical Steffensen method. The convergence order of our method is greater or equal to the number of the controlled nodes used in the Lagrange-type inverse interpolation, which, in its turn, is substantially higher than the convergence orders of the Lagrange type inverse interpolation with uncontrolled nodes (since their convergence order is at most \(2\)).

Aitken-Steffensen type methods for nonsmooth functions (III)

We provide sufficient conditions for the convergence of the Steffensen method for solving the scalar equation \(f(x)=0\), without assuming differentiability of \(f\) at other points than the solution \(x^\ast\). We analyze the cases when the Steffensen method generates two sequences which approximate bilaterally the solution.

Analysis of Henrici's Transformation for Singular Problems

Henrici's transformation is the underlying scheme that generates, by cycling, Steffensen's method for the approximation of the solution of a nonlinear equation in several variables. The aim of this paper is to analyze the asymptotic behavior of the obtained sequence (s * ) by applying Henrici's transformation when the initial sequence (s n ) behaves sublinearly. We extend the work done in the regular case by Sadok [17] to vector sequences in the singular case. Under suitable conditions, we show that the slowest convergence rate of (s * ) is to be expected in a certain subspace N of R p . More precisely, if we write s * =s * ,N+s * ,N⊥, the orthogonal decomposition into N and N ⊥, then the convergence is linear for (s * ,N) but ( * ,N⊥) converges to the same limit faster than the initial one. In certain cases, we can have N=R p and the convergence is linear everywhere.