New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

DEVELOPMENT OF NEW ITERATIVE SCHEME FOR SOLVING NON-LINEAR EQUATIONS

Within our study a special type of 〖iterative method〗^ω is developed by upgrading Newton-Raphson method. We have modified Newton’s method by using our newly developed quadrature rule which is obtained by blending Trapezoidal rule and open type Newton-cotes two point rule. Our newly developed method gives better result than the Newton’s method. Order of convergence of our newly discovered quadrature rule and iterative method is 3.

Calculation of spatial target coordinates in range-difference passive radars by the Levenberg – Marquardt method

This article studies and analyzes the results of applying numerical iterative methods for solving nonlinear equation systems (Newton, modified Newton's method, gradient descent, sequential iterations, Levenberg – Marquardt), compiled and used to calculate the rectangular spatial coordinates of radio emission sources in range-difference passive radars of various configurations (incorporating from 3 to 4 receiving points). The aim of the research was to determine the optimal number of receiving points and to select the most effective algorithm for coordinate transformations of the vector of observed parameters (a set of range difference estimates from radio emission sources to the corresponding pairs of receiving points) into the vector of measured parameters (rectangular spatial coordinates). The following parameters were used as comparison criteria: passive radar working area (a part of space where the deviation of target coordinate estimates from their true values does not exceed the maximum tolerable values); average error in calculating spatial coordinates in the working area; iterations number of coordinate calculation in the analyzed part of space. Upon completing a comparative analysis of obtained characteristics and dependencies, we concluded that it is optimal to include four receiving points in a range-difference passive radar and use the Levenberg – Marquardt method to calculate the spatial coordinates of radio emission sources.

HIGHER-ORDER FAMILIES OF MULTIPLE ROOT FINDING METHODS SUITABLE FOR NON-CONVERGENT CASES AND THEIR DYNAMICS

In this paper, we present many new one-parameter families of classical Rall’s method (modified Newton’s method), Schröder’s method, Halley’s method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations.

Iterative Learning Method for Drilling Depth Optimization in Peck Deep-Hole Drilling

Due to the enclosed chip evacuation space in deep hole drilling process, chips are accumulated in drill flutes as drilling depth increases, resulting in the increase of drilling torque and lead to drill breakage. Peck drilling is a widely used method to periodically alleviate the drilling torque caused by chip evacuation; the drilling depth in each step directly determines both drill life and machining efficiency. The existing drilling depth optimization methods face problems including low accuracy of the prediction model, the hysteresis of signal diagnosis, and onerous experiments. To overcome these problems, a novel drilling depth optimization method for peck drilling based on the iterative learning optimization is proposed. First, the chip evacuation torque coefficients (CETCs) are introduced into the chip evacuation torque model to simplify the model for learning. Then, the effect of chip removal process in peck drilling on drilling depth is analyzed. The extended depth coefficient by chip removal (EDCbCR) is introduced to develop the relationship between the extended depth in each drilling step and drilling depth. On the foundation of the modeling above, an iterative learning method for drilling depth optimization in peck drilling is developed, in which a modified Newton's method is proposed to maximize machining efficiency and avoid drill breakage. In experiments with different cutting parameters, the effectiveness of the proposed method is validated by comparing the optimized and measured results. The results show that the presented learning method is able to obtain the maximum drilling depth accurately with the error less than 10%.