Optimization by parameters in the iterative methods for solving non-linear equations

: In this paper, we used the necessary optimality condition for parameters in a two-point iterations for solving nonlinear equations. Optimal values of these parameters fully coincide with those obtained in [6] and allow us to increase the convergence order of these iterative methods. Numerical experiments and the comparison of existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we considered a variety of real life problems from different disciplines, e.g., Kepler’s equation of motion, Planck’s radiation law problem, in order to check the applicability and effectiveness of our proposed methods.


INTRODUCTION
Many iterative methods for solving nonlinear equations often include non-zero free parameters. Their suitable choices allow to increase the convergence order of methods. There are some choices of parameters based on the error analysis, see for example [1,2,3,4,5,8,9] and references therein. Recently, researchers have also proposed several biparametric two-step simple methods with and without memory [1, 3,4,6]. On the other hand, optimization by parameters is one of the powerful techniques in science and engineering practice. The main goal of this paper is to find the optimal choices of parameters ̅ and, γ in the two points iterative methods. We obtained analytic formulae for λ and γ without symbolic computation technique.
The paper is organized as follows. In section 2, we developed necessary optimality condition for parameters in the two-point method. In the last section, we present the results of numerical experiments that confirm the theoretical conclusion about the convergence order and made a comparison with well-known methods of the same order of convergence.

PMAS
In numerical analysis and engineering applications, it is often required to solve a non- is a scalar function defined on an open interval D. Assume that f is sufficiently smooth and has a simple zero * ∈ and f' is nonsingular in D.
In this paper, we consider optimization by parameters in the two-point iterative methods [6,7].
where γ, λ are non-zero parameters. It is easy to show that the minimization problem

Necessary optimality of condition for parameters in the two-point iterations
The Taylor expansion of f(xn+1) at the point yn gives We find stationary point of f (yn(λ, γ)) solving the system From (11) and (12) clear that the parameters γ and λ are determined, so that the system of equations Thus, we find the extremum point (13) of function The parameters given by the formulae (13) are naturally said to be optimal in the sense of necessary optimality condition (8). So, We can find approximations of . , n n λ γ They can be found using the information available from the current and previous iteration steps. The methods containig such parameters are called methods with memory.
Note that (13) fully coincides with those obtained in [3] and under (13). Now we shall find iteration parameter n τ with some accuracy. To this end, we approximate As a result, we have From (4) (13) and (17).

Remarks.
Obviously, the optimal choices (13) also hold true for three-point iterative methods, the first two steps of which are the same as (1). Note that the stationary point of view is also applicable for solving system of nonlinear equations. ), ( = then given, are , ,

RESULTS AND DISCUSSION
We employed an iterative method (1) with (13), (17) and method (19) which has been selected from [5] and [8,9,10]. For 1 , 2 , we consider the initial guesses −0.6 and 1.3, respectively. In particular, 3 f is Kepler's equation which relates the eccentric anomaly E , the mean anomaly M and the eccentricity ε in an elliptic orbit.
One of the classical laws of planetary motion due to Kepler says that a planet revolves around the sun in an elliptic orbit.
Suppose one needs to find the position In [10] we have considered one of the famous classical physics problem, which is known as the Planck's radiation law problem. Fourth non-linear function 4 f arises from this problem. Planck's radiation law problem calculates the energy density within an isothermal blackboard and is given by the following formula: where λ is the wavelength of the radiation, is the absolute temperature of the blackbody, is the Boltzmann constant, is the Planck constant and is the speed of light. We are interested in determining wavelength λ which corresponds to maximum energy density � λ �. has two zeros. Obviously, one of the roots = 0 is not taken for discussion. Another root is * = 4.965114231744276303699. Moreover, the initial guesses are chosen that 0 and 6 for those problems.
The results, for methods with memory, are computed with the same initial values of γ and λ .
The calculations have been performed in the MATHEMATICA 12 using multi-precision arithmetic with 1000 digits and we used the following stopping criterion: Here, ) (ρ is calculated by the following formula [2,9]: From Tables 1-4, we see that the computed results completely support the theory of convergence discussed in previous sections. The convergence order of the method (19) with 2 = p is eight which is higher than other two-step methods with memory.

CONCLUSIONS
In this paper, a new two-point derivativefree iterative method with memory for solving nonlinear equations was introduced and studied. Using optimal values of parameters, the higher order method with memory is obtained. Exact analytical formulas for the optimal values of the parameters have been found for the first time. The convergence order increased from four to eight without any additional computations. Finally, numerical experiments have shown that the new method is eight-order and effective. We also solved some real world applications of different nature to show the effectiveness of the proposed methods.