Himani Arora

Approximating Multiple Roots of Applied Mathematical Problems Using Iterative Techniques

Axioms, Mar 6, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more

An efficient derivative free iterative method for solving systems of nonlinear equations

Applicable Analysis and Discrete Mathematics, 2013

We present a derivative free method of fourth order convergence for solving systems of nonlinear ... more We present a derivative free method of fourth order convergence for solving systems of nonlinear equations. The method consists of two steps of which first step is the well-known Traub's method. First-order divided difference operator for functions of several variables and direct computation by Taylor's expansion are used to prove the local convergence order. Computational efficiency of new method in its general form is discussed and is compared with existing methods of similar nature. It is proved that for large systems the new method is more efficient. Some numerical tests are performed to compare proposed method with existing methods and to confirm the theoretical results.

format_quoteIterative methods for nonlinear equations, like Newton's method, often require derivatives; this paper addresses the need for derivative-free approaches.format_quote

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Efficient higher order derivative-free multipoint methods with and without memory for systems of nonlinear equations

International Journal of Computer Mathematics, Mar 15, 2017

We present a family of multipoint methods without memory with sixth order convergence for solving... more We present a family of multipoint methods without memory with sixth order convergence for solving systems of nonlinear equations. The methods use first order divided difference operators and are derivative free. Extending the work further, we explore a family of methods with memory with increasing order of convergence from the basic family without memory. The increase in convergence order is attained by varying a free parameter step to step using information available from the previous step. It is proved that the convergence order of the family with memory is increased from 6 to 3 + √ 10 ≈ 6.162, and in some special cases to 3 + 2 √ 3 ≈ 6.464 and 3 + √ 13 ≈ 6.606. Computational efficiency of the methods is discussed and compared with existing methods. Numerical examples, including those arise from integral equations and boundary value problems, are considered to verify the theoretical results. A comparison with the existing methods shows that the new methods are more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.

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CMMSE: A novel scheme having seventh-order convergence for nonlinear systems

Journal of Computational and Applied Mathematics

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions

Mathematics

The construction of derivative-free iterative methods for approximating multiple roots of a nonli... more 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.

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A Family of Fifth-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

Numerical Analysis and Applications, 2021

Abstract We present a family of fifth-order iterative methods for finding multiple roots of nonli... more Abstract We present a family of fifth-order iterative methods for finding multiple roots of nonlinear equations. Numerical examples are considered to check the validity of the theoretical results. It is shown that the new methods are competitive with other methods for finding multiple roots. Basins of attraction for the new methods and some existing methods to observe the dynamics in the complex plane are drawn.

Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

Mathematical and Computational Applications, 2018

In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear... more In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , … . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.

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Efficient higher order derivative-free multipoint methods with and without memory for systems of nonlinear equations

International Journal of Computer Mathematics, 2017

We present a family of multipoint methods without memory with sixth order convergence for solving... more We present a family of multipoint methods without memory with sixth order convergence for solving systems of nonlinear equations. The methods use first order divided difference operators and are derivative free. Extending the work further, we explore a family of methods with memory with increasing order of convergence from the basic family without memory. The increase in convergence order is attained by varying a free parameter step to step using information available from the previous step. It is proved that the convergence order of the family with memory is increased from 6 to 3 + √ 10 ≈ 6.162, and in some special cases to 3 + 2 √ 3 ≈ 6.464 and 3 + √ 13 ≈ 6.606. Computational efficiency of the methods is discussed and compared with existing methods. Numerical examples, including those arise from integral equations and boundary value problems, are considered to verify the theoretical results. A comparison with the existing methods shows that the new methods are more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.

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Some novel optimal eighth order derivative-free root solvers and their basins of attraction

Applied Mathematics and Computation, 2016

We present two families of derivative-free methods with eighth order convergence for solving nonl... more We present two families of derivative-free methods with eighth order convergence for solving nonlinear equations. Each method of the families requires four function evaluations per full iteration, that means, the families are optimal in the sense of the hypothesis of Kung-Traub (1974). Computational results and comparison (including CPU time) with existing methods confirm the efficient and robust character of new methods. Moreover, the presented basins of attraction also confirm equal or better performance of the methods as compared to other established methods in literature.

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An efficient derivative free iterative method for solving systems of nonlinear equations

Applicable Analysis and Discrete Mathematics, 2013

We present a derivative free method of fourth order convergence for solving systems of nonlinear ... more We present a derivative free method of fourth order convergence for solving systems of nonlinear equations. The method consists of two steps of which first step is the well-known Traub's method. First-order divided difference operator for functions of several variables and direct computation by Taylor's expansion are used to prove the local convergence order. Computational efficiency of new method in its general form is discussed and is compared with existing methods of similar nature. It is proved that for large systems the new method is more efficient. Some numerical tests are performed to compare proposed method with existing methods and to confirm the theoretical results.

format_quoteThe derivative-free method proposed is more efficient than existing methods, particularly for large systems, due to minimized function evaluations.format_quote

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Efficient Jarratt-like methods for solving systems of nonlinear equations

Calcolo, 2013

ABSTRACT We present the iterative methods of fourth and sixth order convergence for solving syste... more ABSTRACT We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed fourth order scheme and requires one extra function evaluation in addition to the evaluations of fourth order method. Computational efficiency in its general form is discussed. A comparison between the efficiencies of proposed techniques with existing methods of similar nature is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in the examples. It is shown that the present methods are more efficient than their existing counterparts, particularly when applied to the large systems of equations.

A simple yet efficient derivative free family of seventh order methods for systems of nonlinear equations

SeMA Journal, 2015

Based on Traub-Steffensen scheme, we present a family of seventh order derivative free methods fo... more Based on Traub-Steffensen scheme, we present a family of seventh order derivative free methods for solving systems of nonlinear equations. Computational efficiency of the methods of the family is considered and compared with existing methods of similar nature. Furthermore, numerical experiments are performed to confirm the theoretical results concerning order of convergence and computational efficiency. It is shown that, in general, the proposed methods are more efficient than their existing counterparts.

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