The MULTIPLISITAS NEWTON DAN TITIK TETAP ATRAKTIF DALAM MENENTUKAN KEKONVERGENAN

Newton's method is one of the numerical methods used in finding polynomial roots. This method will be very effective to use, if the initial estimate of the roots for the Newton iteration function satisfies sufficient Newtonian convergence, [11]. In this article we will analyze the efficacy of this method by looking at the relationship between the fixed point method and Newton's iteration function. When the iteration of the function converges to the root, the velocity of convergence can also be determined. In terms of the speed of convergence, it turns out to be very dependent on the multiplicity of Newton's method itself.

An Efficient Derivative Free One-Point Method with Memory for Solving Nonlinear Equations

We propose a derivative free one-point method with memory of order 1.84 for solving nonlinear equations. The formula requires only one function evaluation and, therefore, the efficiency index is also 1.84. The methodology is carried out by approximating the derivative in Newton’s iteration using a rational linear function. Unlike the existing methods of a similar nature, the scheme of the new method is easy to remember and can also be implemented for systems of nonlinear equations. The applicability of the method is demonstrated on some practical as well as academic problems of a scalar and multi-dimensional nature. In addition, to check the efficacy of the new technique, a comparison of its performance with the existing techniques of the same order is also provided.

Kinematical Research of Free-Floating Space-Robot System at Position Level Based on Screw Theory

Kinematics of a free-floating space-robot system is a fundamental and complex subject. Problems at the position level, however, are not considered sufficiently because of the nonholonomic property of the system. Current methods cannot handle these problems simply and efficiently. A novel and systematical modeling approach is provided; forward and inverse kinematics at the position level are deduced based on the product of exponentials (POE) formula and conservation of linear momentum. The whole deduction process is concise and clear. More importantly, inertial tensor parameters are not introduced. Then, three situations with different known variables are mainly studied. Due to the complexity of inverse kinematical equations, a numerical method is proposed based on Newton’s iteration method. Two calculation examples are given, a dual-arm planar model and a single-arm spatial model; both forward and inverse kinematical solutions are given, while inverse kinematical results are compared with simulation results of Adams. The results indicate that the proposed methods are quite accurate and efficient.

An introduction to an ancient Chinese algorithm and its modification

Purpose Every student knows Newton’s iteration method from a textbook, which is widely used in numerical simulation, what few may know is that its ancient Chinese partner, Ying Buzu Shu, in about second century BC has much advantages over Newton’s method. The purpose of this paper is to introduce the ancient Chinese algorithm and its modifications for numerical simulation. Design/methodology/approach An example is given to show that the ancient Chinese algorithm is insensitive to initial guess, while a fast convergence rate is predicted. Findings Two new algorithms, which are suitable for numerical simulation, are introduced by absorbing the advantages of the Newton iteration method and the ancient Chinese algorithm. Research limitations/implications This paper focuses on a single algebraic equation; however, it is easy to extend the theory to algebraic systems. Practical implications The Newton iteration method can be updated in numerical simulation. Originality/value The ancient Chinese algorithm is elucidated to have modern applications in various numerical methods.

A comparison of some fixed point iteration procedures by using the basins of attraction

Several iterative processes have been defined by researchers to approximate the fixed points of various classes operators. In this paper we present, by using the basins of attraction for the roots of some complex polynomials, an empirical comparison of some iteration procedures for fixed points approximation of Newton’s iteration operator. Some numerical results are presented. The Matlab m-files for generating the basins of attraction are presented, too.