A reliable algorithm to check the accuracy of iterative schemes for solving nonlinear equations: an application of the C
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A reliable algorithm to check the accuracy of iterative schemes for solving nonlinear equations: an application of the CESTAC method Mohammad Ali Fariborzi Araghi1 Received: 26 December 2019 / Accepted: 12 March 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract The aim of this study is to apply the discrete stochastic arithmetic (DSA) to validate the class of muli-step iterative methods and find the optimal numerical solution of nonlinear equations. To this end, the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are applied. By using this approach, the optimal number of iteration and the optimal solution with its accuracy are found. In this case, the usual stopping termination in the iterative procedure is replaced by a new criterion which is independent of the given tolerance () such that the optimal results are evaluated computationally. A main theorem is proved which shows the accuracy of the iterative schemes by means of the concept of common significant digits. The numerical results are presented to illustrate the efficiency and importance of using the DSA in place of the floating-point arithmetic (FPA). Keywords Discrete stochastic arithmetic (DSA) · CESTAC method · CADNA library · Multi-step methods · Nonlinear equations · Floating-point arithmetic (FPA) Mathematics Subject Classification 65H05 · 65J15 · 62L15 · 34A34 · 47J25
1 Introduction Iterative methods play an important role to solve nonlinear equations in the point of numerical view. A considerable topic, in this case, is to choose a method with high accuracy and order of convergence to find the optimal number of iterations and the best numerical solution according to the machine precision in companion with a proper termination criterion. Let the following nonlinear equation: f (x) = 0, (1) where f : D ⊂ R → R is a function on an open interval D and α ∈ D is the simple zero of f.
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Mohammad Ali Fariborzi Araghi [email protected]; [email protected] Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran 19558-47881, Iran
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M. A. F. Araghi
In recent years, some new iterative methods have been introduced to solve Eq. (1). These methods can be classified as k-step, k = 1, 2, 3, ..., methods. An improvement of Newton’s method was discussed in [15]. A higher order 2-point iterative methods for solving quadratic equations was developed in [5]. King in [23] introduced a family of 4-th order method. A family of 3-step method with cubic convergence for solving nonlinear equations was suggested in [27]. The other class of 3-step iterative methods with 16-th order of convergence, was developed by Li et al. [25]. Also, in [9], a family of 8-th order iterative schemes was proposed. Behl et al. [7] presented a highly efficient family of iterative methods with 6-th order of convergence and in [6], an optimal family of 16-th order methods to solve Eq. (1) was discussed. Cordero et al. [16],
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