A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

Abstract

There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models.

Development of a New Multi‐step Iteration Scheme for Solving Non‐Linear Models with Complex Polynomiography

“…Hence, the twelfth-order convergence of the proposed multi-step (four-step) hybrid method P12 mentioned by (9) for the nonlinear functions in multi-variable (f(x) � 0) case is proved. □ □…”

“…In this subsection, we theoretically prove the local order of convergence for the proposed method given in (9). Theorem 1.…”

“…Theorem 1. Suppose that α ∈ Q be the required simple root for a sufficiently differentiable function f: Q⊆R ⟶ R within an open real interval Q. en, the proposed four-step numerical method (9) possesses twelfth-order convergence with the error equation given by:…”

“…Suppose α be a simple root of f(x i ) � 0, where x i be the i th approximation for the root by the proposed method (9), and ζ i � x i − α be the error term in variable x at the i th iteration step. Employing the multi variable Taylor's series given in the theorem [9] for f(x i ) around α, we obtain…”

“…Another higher-order three-step iterative method as denoted by P10 is in [9]. e method is shown to be three-step that requires only 6 function evaluations per iteration, as depicted in the following computational scheme:…”