Search Results (42)

The Steffensen method is a very effective root-finding procedure used for solving nonlinear equations of the form $f\left(x\right)=0$ numerically. When the root of $f\left(x\right)$ is simple, this method converges with order 2. Following a critical review of its convergence analysis, in this work, we provide a modification with memory of the Steffensen method, which achieves an order substantially higher than 2, without increasing the number of function evaluations per iteration. Specifically, with an arbitrary fixed integer $k\ge 1$ and with initial approximations $x_0,x_1,\dots ,x_k$, the modified method generates a sequence of approximations to the solution of $f\left(x\right)=0$ via $x_{n+1}=x_n-f\left(x_n\right)/f[c_n,x_n]$, $n=k,k+1,\dots ,$ where $c_n=x_n-f\left(x_n\right)/p_{n,k}^{\prime }\left(x_n\right)$ and $p_{n,k}\left(x\right)$ is the polynomial of interpolation to $f\left(x\right)$ at the points $x_n,x_{n-1},\dots ,x_{n-k}$. Just as in the original method, the modified method makes use of only $f\left(x\right)$; no derivatives of $f\left(x\right)$ are needed. We prove that the order of this method is $1+\sqrt{2}\dot{=}2.414$ for $k=1$ and it can be increased towards $(3+\sqrt{5})/2\dot{=}2.618$ by increasing k. (For example, the order of the method is $2.617\cdots$ already for $k=6$.) Full article

We develop a novel Steffensen-type iterative solver to solve nonlinear scalar equations without requiring derivatives. A two-parameter one-step scheme without memory is first introduced and analyzed. Its optimal quadratic convergence is then established. To enhance the convergence rate without additional functional evaluations, we extend the scheme by incorporating memory through adaptively updated accelerator parameters. These parameters are approximated by Newton interpolation polynomials constructed from previously computed values, yielding a derivative-free method with R-rate of convergence of approximately 3.56155. A dynamical system analysis based on attraction basins demonstrates enlarged convergence regions compared to Steffensen-type methods without memory. Numerical experiments further confirm the accuracy of the proposed scheme for solving nonlinear equations. Full article

In this paper, we study majorization-type inequalities for n-convex (specifically 3-convex) functions. Numerous papers deal with such integral inequalities, in which n-convex functions are defined on compact intervals and nonnegative measures are used in the integrals. The main goal of this paper is to formulate similar results for noncompact intervals and signed measures. We follow a well-known method often used for compact intervals: approximation of n-convex functions with simple n-convex functions. After some preliminary results, we present new approximation theorems, some of which extend classical results, while others are completely unique approximations. Then we obtain some novel majorization-type inequalities, which can be applied under more general conditions than those currently known. Finally, we illustrate the applicability of our results by answering problems from different areas: discrete majorization-type inequalities, specifically one-dimensional inequality of Sherman for n-convex functions; characterization of Steffensen–Popoviciu measures for nonnegative, continuous, and increasing 3-convex functions; Hermite–Hadamard-type inequalities for 3-convex functions. Full article

A new hybrid of a Steffensen-type method and genetic algorithm is developed for the efficient simultaneous computation of roots of nonlinear equations, particularly in all cases involving non-differentiable functions and multiple roots. Traditional numerical methods often fail to handle these complexities effectively, highlighting the need for a more robust solution. The proposed algorithm combines the global search strength of the genetic algorithm (GA) with the local refinement capabilities of a derivative-free optimal fourth-order Steffensen method. This integration enhances both exploration and exploitation capabilities, leading to improved convergence and computational accuracy. By uniting the GA’s global optimization with the local refinement of iterative solvers, the algorithm forms a higher-order framework capable of locating all roots concurrently. This study validates the performance of this hybrid strategy through diverse applications in biomedical engineering problems. Full article

A local and a semi-local convergence analysis are presented for the Kurchatov-type method to solve numerically nonlinear equations in a Banach space. The method depends on a real parameter. By specializing the parameter, we obtain methods already studied in the literature under different types of conditions, such us Newton’s, and Steffensen’s, and Kurchatov’s methods, the Secant method, and other methods. This study is carried out under generalized conditions for first-order divided differences, as well as first-order derivatives. Both in the local case and in the semi-local case, the error estimates, the radii of the region of convergence, and the regions of the solution’s uniqueness are determined. A numerical majorizing sequence is constructed for studying semi-local convergence. The approach of restricted convergence regions is used to develop a convergence analysis of the considered method. The new approach allows a comparison of the convergence of different methods under a uniform set of conditions. In particular, the assumption of generalized continuity used to control the divided difference provides more precise knowledge on the location of the solution as well as tighter error estimates. Moreover, the generality of the approach makes it useful for studying other methods in an analogous way. Numerical examples demonstrate the applicability of our theoretical results. Full article

This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge to the desired root. Despite this drawback of Steffensen’s method, several researchers have developed higher-order iterative methods based on Steffensen’s scheme. Traub introduced a free parameter in Steffensen’s scheme to obtain the first parametric iteration method, which provides larger basins of attraction for specific values of the parameter. In this paper, we introduce a two-step derivative free fourth-order optimal iteration scheme based on Traub’s method by employing three free parameters and a weight function. We further extend it into a two-step eighth-order iteration scheme by means of memory with the help of suitable approximations of the involved parameters using Newton’s interpolation. The convergence analysis demonstrates that the proposed iteration scheme without memory has an order of convergence of 4, while its memory-based extension achieves an order of convergence of at least $7.993$, attaining the efficiency index $7.{993}^{1/3}\approx 2$. Two special cases of the proposed iteration scheme are also presented. Notably, the proposed methods compete with any optimal j-point method without memory. We affirm the superiority of the proposed iteration schemes in terms of efficiency index, absolute error, computational order of convergence, basins of attraction, and CPU time using comparisons with several existing iterative methods of similar kinds across diverse nonlinear equations. In general, for the comparison of iterative schemes, the basins of iteration are investigated on simple polynomials of the form $z^n-1$ in the complex plane. However, we investigate the stability and regions of convergence of the proposed iteration methods in comparison with some existing methods on a variety of nonlinear equations in terms of fractals of basins of attraction. The proposed iteration schemes generate the basins of attraction in less time with simple fractals and wider regions of convergence, confirming their stability and superiority in comparison with the existing methods. Full article

This article introduces a multi-step solver for sets of nonlinear equations. To achieve this, we consider and develop a multi-step Steffensen-type method without memory, which does not require evaluations of the Fréchet derivatives, and subsequently extend it to a method with memory. The resulting order is $\sqrt{5}+2$, utilizing the identical number of functional evaluations as the solver without memory, thereby demonstrating a higher computational index of efficiency. Finally, we illustrate the advantages of the proposed scheme with memory through various test problems. Full article

Local convergence analysis is mostly carried out using the Taylor series expansion approach, which requires the utilization of high-order derivatives, not iterative methods. There are other limitations to this approach, such as the following: the analysis is limited to finite-dimensional Euclidean spaces; no a priori computable error bounds on the distance or uniqueness of the solution results are provided. The local convergence analysis in this paper positively addresses these concerns in the more general setting of a Banach space. The convergence conditions involve only the operators in the methods. The more important semi-local convergence analysis not studied before is developed by using majorizing sequences. Both types of convergence analyses are based on the concept of generalized continuity. Although we study a certain class of methods, the same approach applies to extend the applicability of other schemes along the same lines. Full article

Numerous applications from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert, or Banach, to mention a few. Researchers worldwide are developing methodologies to handle the solutions of such equations. A plethora of these equations are not differentiable. These methodologies can also be applied to solve differentiable equations. A particular method is utilized as a sample via which the methodology is described. The same methodology can be used on other methods utilizing inverses of linear operators. The problem with existing approaches on the local convergence of iterative methods is the usage of Taylor expansion series. This way, the convergence is shown but by assuming the existence of high-order derivatives which do not appear on the iterative methods. Moreover, bounds on the error distances that can be computed are not available in advance. Furthermore, the isolation of a solution of the equation is not discussed either. These concerns reduce the applicability of iterative methods and constitute the motivation for developing this article. The novelty of this article is that it positively addresses all these concerns under weaker convergence conditions. Finally, the more important and harder to study semi-local analysis of convergence is presented using majorizing scalar sequences. Experiments are further performed to demonstrate the theory. Full article

In this manuscript, we use approximations of conformable derivatives for designing iterative methods to solve nonlinear algebraic or trascendental equations. We adapt the approximation of conformable derivatives in order to design conformable derivative-free iterative schemes to solve nonlinear equations: Steffensen and Secant-type methods. To our knowledge, these are the first conformable derivative-free schemes in the literature, where the Steffensen conformable method is also optimal; moreover, the Secant conformable scheme is also a procedure with memory. A convergence analysis is made, preserving the order of classical cases, and the numerical performance is studied in order to confirm the theoretical results. It is shown that these methods can present some numerical advantages versus their classical partners, with wide sets of converging initial estimations. Full article

Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore–Penrose inverse of a singular square matrix or an arbitrary $m\times n$ complex matrix. A Kurchatov-type scheme and Steffensen’s method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton–Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones. Full article

High-order iterative techniques without derivatives for multiple roots have wide-ranging applications in the following: optimization tasks, where the objective function lacks explicit derivatives or is computationally expensive to evaluate; engineering; design finance; data science; and computational physics. The versatility and robustness of derivative-free fourth-order methods make them a valuable tool for tackling complex real-world optimization challenges. An optimal extension of the Traub–Steffensen technique for finding multiple roots is presented in this work. In contrast to past studies, the new expanded technique effectively handles functions with multiple zeros. In addition, a theorem is presented to analyze the convergence order of the proposed technique. We also examine the convergence analysis for four real-life problems, namely, Planck’s law radiation, Van der Waals, the Manning equation for isentropic supersonic flow, the blood rheology model, and two well-known academic problems. The efficiency of the approach and its convergence behavior are studied, providing valuable insights for practical and academic applications. Full article

In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information. Full article

In this study, we suggest a new iterative family of iterative methods for approximating the roots with multiplicity in nonlinear equations. We found a lack in the approximation of multiple roots in the case that the nonlinear operator be non-differentiable. So, we present, in this paper, iterative methods that do not use the derivative of the non-linear operator in their iterative expression. With our new iterative technique, we find better numerical results of Planck’s radiation, Van Der Waals, Beam designing, and Isothermal continuous stirred tank reactor problems. Divided difference and weight function approaches are adopted for the construction of our schemes. The convergence order is studied thoroughly in the Theorems 1 and 2, for the case when multiplicity $p\ge 2$. The obtained numerical results illustrate the preferable outcomes as compared to the existing ones in terms of absolute residual errors, number of iterations, computational order of convergence (COC), and absolute error difference between two consecutive iterations. Full article

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates. Full article