Derivative-free Iterative Methods Research Articles

Derivative free processes of high order for nondifferentiable equations in Banach spaces

In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm.

Two-step inertial derivative-free projection method for solving nonlinear equations with application

Large-scale systems of nonlinear equations play a fundamental role in various applications, spanning from differential equations to economics, engineering, management science, probability theory, and various other applied sciences. This paper proposes a derivative-free iterative method to find ϵ-approximate solutions of large-scale nonlinear equations. The proposed scheme combines the hyperplane projection technique with a two-step inertial extrapolation, presenting a new approach. The proposed method does not require Jacobian information and thus can be applied to solve nonsmooth equations. Under standard assumptions, we prove the proposed method’s global convergence and nonasymptotic O(1/k) convergence rate. Furthermore, we include numerical results and comparisons to demonstrate the efficiency of the proposed method. Finally, we demonstrate the applicability of the proposed method in solving regularized decentralized logistic regression, a popular problem in machine learning applications.

Ball convergence of derivative free iterative methods with or without memory using weight operator technique

A method without memory as well as a method with memory are developed free of derivatives for solving Banach space valued equations. Their ball convergence analysis is provided using only the derivative and the divided difference of order one in contrast to earlier works on the real line using the fifth as well as the seventh derivative. This way the applicability is expanded for these methods.

Optimal Fourth-Order Methods for Multiple Zeros: Design, Convergence Analysis and Applications

Nonlinear equations are frequently encountered in many areas of applied science and engineering, and they require efficient numerical methods to solve. To ensure quick and precise root approximation, this study presents derivative-free iterative methods for finding multiple zeros with an ideal fourth-order convergence rate. Furthermore, the study explores applications of the methods in both real-life and academic contexts. In particular, we examine the convergence of the methods by applying them to the problems, namely Van der Waals equation of state, Planck’s law of radiation, the Manning equation for isentropic supersonic flow and some academic problems. Numerical results reveal that the proposed derivative-free methods are more efficient and consistent than existing methods.

Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of methods successfully works even in such scenarios. We extended this idea to memory-based iterative methods by utilizing self-accelerating parameters derived from the current and previous approximations. As a result, we increased the convergence order from five to ten without requiring additional function evaluations. Analytical proofs of the proposed family of derivative-free methods, both with and without memory, are provided. Furthermore, numerical experimentation on diverse problems reveals the effectiveness and good performance of the proposed methods when compared with well-known existing methods.

A New Second Derivative Free Iterative Method of Fifth Order of Convergence and Its Applications

The primary objective of this article is to derive a new free from second derivative iterative method for solving nonlinear equations. The proposed method is proven to have fifth order of convergence. Comparisons with other iterative methods represent the advantage of the modified method. Observation on applications of the method in problem of chemical equilibrium , binary azeotropic problem, volume from van der Waals equations and eccentric anomaly in Kepler's law exhibits that our method is applicable and preferable.

An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

Recent researches have shown a great interest in developing novel numerical methods for solving nonlinear equations of the form arising from real world phenomena. However, very little attention has been given to the study of condition numbers which is an important aspect in measuring the sensitivity of the problem in response to slight perturbations in the input data. In this article, we present an efficient free derivative iterative scheme constructed by refining Newton-Raphson method standard form in which the derivative term is approximated by using finite difference scheme; hence making it derivative free. We also conducted an in-depth analysis of the condition numbers to explore sensitivity and efficiency comparisons between the proposed algorithm and existing methods for the given problems. Our investigation focused on iteration numbers, residuals, and convergence under mild error tolerances. Based on five numerical case studies, results revealed that the proposed Algorithm 2 outperforms the existing Algorithm 1 in terms of accuracy of the approximate solution. The results for the condition numbers indicated that all problems considered were ill-conditioned, highlighting the significance of studying condition numbers in the context of solving nonlinear equations.

Convergence of Derivative-Free Iterative Methods with or without Memory in Banach Space

A method without memory as well as a method with memory are developed free of derivatives for solving equations in Banach spaces. The convergence order of these methods is established in the scalar case using Taylor expansions and hypotheses on higher-order derivatives which do not appear in these methods. But this way, their applicability is limited. That is why, in this paper, their local and semi-local convergence analyses (which have not been given previously) are provided using only the divided differences of order one, which actually appears in these methods. Moreover, we provide computable error distances and uniqueness of the solution results, which have not been given before. Since our technique is very general, it can be used to extend the applicability of other methods using linear operators with inverses along the same lines. Numerical experiments are also provided in this article to illustrate the theoretical results.

Seventh Order Derivative-Free Methods for Non-differentiable Operator Equations

In nonlinear problems where function’s derivatives are difficult or expensive to compute, derivative-free iterative methods are good options to find the numerical solution. One of the important parts in the development of such methods is to study their convergence properties. In this paper, we review the concepts of local and semi-local convergence for a derivative-free method for nonlinear equations. In the earlier study of the considered method, the convergence analysis was carried out assuming the existence of higher order derivatives while no derivative is used in the method. Such assumptions certainly restrict its applicability. The present study further provides the estimate of convergence radius and bounds on the error for the given method. Thus, the applicability of the method clearly seems to be extended over the wider class of problems. We also review some of the recent developments in this area. The results presented in this paper can be useful for practitioners and researchers in developing and analyzing derivative-free numerical algorithms.

Three-Step Derivative-Free Method of Order Six

Derivative-free iterative methods are useful to approximate the numerical solutions when the given function lacks explicit derivative information or when the derivatives are too expensive to compute. Exploring the convergence properties of such methods is crucial in their development. The convergence behavior of such approaches and determining their practical applicability require conducting local as well as semi-local convergence analysis. In this study, we explore the convergence properties of a sixth-order derivative-free method. Previous local convergence studies assumed the existence of derivatives of high order even when the method itself was not utilizing any derivatives. These assumptions imposed limitations on its applicability. In this paper, we extend the local analysis by providing estimates for the error bounds of the method. Consequently, its applicability expands across a broader range of problems. Moreover, the more important and challenging semi-local convergence not investigated in earlier studies is also developed. Additionally, we survey recent advancements in this field. The outcomes presented in this paper can be proved valuable to practitioners and researchers engaged in the development and analysis of derivative-free numerical algorithms. Numerical tests illuminate and validate further the theoretical results.

Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1.9847. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.

Unified Convergence Criteria of Derivative-Free Iterative Methods for Solving Nonlinear Equations

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.

Higher order Traub–Steffensen type methods and their convergence analysis in Banach spaces

Abstract In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized q-step scheme with increasing convergence order 2q + 2 is proposed and analyzed. Novelty of the q-step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation.

An optimal fourth-order second derivative free iterative method for nonlinear scientific equations

In the present paper, we develop an efficient second derivative free two-step optimal fourth-order iterative method for nonlinear equations. We explore the convergence criteria of the proposed method and also exhibit its validity and efficiency by considering some test problems. We present both numerical as well as graphical comparisons. Further, the dynamical behavior of the proposed method is explored.

Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions

The construction of derivative-free iterative methods for approximating multiple roots of a nonlinear equation is a relatively new line of research. This paper presents a novel family of one-parameter second-order techniques. Our schemes are free from derivatives and have been designed to find multiple roots (m≥2). The new techniques involve the weight function approach. The convergence analysis for the new family is presented in the main theorem. In addition, some special cases of the new class are discussed. We also illustrate the applicability of our methods on van der Waals, Planck’s radiation, root clustering, and eigenvalue problems. We also contrast them with the known methods. Finally, the dynamical study of iterative schemes also provides a good overview of their stability.

Derivative-Free Kurchatov-Type Accelerating Iterative Method for Solving Nonlinear Systems: Dynamics and Applications

Two novel Kurchatov-type first-order divided difference operators were designed, which were used for constructing the variable parameter of three derivative-free iterative methods. The convergence orders of the new derivative-free methods are 3, (5+17)/2≈4.56 and 5. The new derivative-free iterative methods with memory were applied to solve nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) in numerical experiments. The dynamical behavior of our new methods with memory was studied by using dynamical plane. The dynamical planes showed that our methods had good stability.

A Novel Root-Finding Algorithm With Engineering Applications and its Dynamics via Computer Technology

Root-finding of non-linear equations is one of the most appearing problems in engineering sciences. Most of the complicated engineering problems can be modeled easily by means of non-linear functions. The role of iterative algorithms via computers for solving such functions is much important and cannot be denied in the modern age. In an iterative algorithm, the convergence order and the computational cost per iteration are the main characteristics that depict its efficiency and performance i.e., a method with higher-order and lower computational cost will be more efficient and vice versa. Keeping these facts into consideration, the main goal of this paper is to introduce a new derivative-free iterative method that performs better. We develop this algorithm by utilizing the forward- and finite-difference schemes on well-known Househölder’s method, resulting in an efficient and derivative-free algorithm with a low per iteration computing cost. We also look at the developed algorithm’s convergence criterion and show that it is quartic-order convergent. We investigate nine test-examples and solve them to demonstrate its correctness, validity, and efficiency numerically. Some real-world engineering problems in civil and chemical engineering are also included in these examples. The numerical results of the test-examples reveal that the newly constructed method outperforms the existing similar algorithms found in the literature. We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other comparable existing approaches. The graphical findings show that the developed method has a faster convergence speed than the other comparable algorithms.

Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.

A multi-root solver for discontinuous and non-differentiable equations by integrating genetic algorithm and derivative-free iterative methods

Genetic Algorithm (GA) has a strong global searching ability but limited convergence efficiency at later stage, while derivative-free iterative methods have high local convergence efficiency but strict requirements on the initial approximation. Combining GA and derivative-free iterative methods, a multi-root solver is proposed for a class of complex nonlinear functions with discontinuity, non-differentiability and multi-root in this work. Firstly, an improved GA (IGA) is presented by integrating an adaptive crossover operator and three kinds of manual intervention measures into the standard GA to further improve the global searching ability. Then, a global search in the domain is implemented by using the proposed IGA, and the qualified individuals are selected as initial approximations. Finally, based on this, local accurate convergence is achieved by combining the derivative-free iterative method. To demonstrate the effectiveness of the proposed method, a series of numerical experiments are conducted, and the results show that the proposed multi-root solver has better performance in efficiency, stability as well as applicability.

Sparse signal reconstruction in compressive sensing via derivative-free iterative method

Finding sparse solutions to under-determined or ill-conditioned equations is a problem that usually arise in compressive sensing. In this article, a derivative-free iterative method is presented for recovering sparse signal in compressive sensing by approximating the solution to a convex constrained nonlinear equation. The proposed method is derived from the modified Polak-Ribiere-Polyak conjugate gradient method for unconstrained optimization. The global convergence is established under mild assumptions. Preliminary numerical results in recovering sparse signal are given to show that the proposed method is efficient.