scholarly journals On the Distance to a Root of Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Somjate Chaiya
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Newton’S Method ◽  
Upper Bound ◽  
Newton's Method ◽  
Explicit Estimate ◽  
Roots Of Polynomials ◽  
Number Of Iterations

In 2002, Dierk Schleicher gave an explicit estimate of an upper bound for the number of iterations of Newton's method it takes to find all roots of polynomials with prescribed precision. In this paper, we provide a method to improve the upper bound given by D. Schleicher. We give here an iterative method for finding an upper bound for the distance between a fixed pointzin an immediate basin of a rootαtoα, which leads to a better upper bound for the number of iterations of Newton's method.

Efficiency of Solution Methods for Kepler’s Equation

2016 ◽  
Vol 851 ◽  
pp. 587-592
Author(s):  
João Francisco Nunes de Oliveira ◽  
Roberta Veloso Garcia ◽  
Hélio Koiti Kuga ◽  
Estaner Claro Romão
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Kepler's Equation ◽  
Eccentric Orbits ◽  
Kepler’S Equation ◽  
Solution Methods ◽  
Number Of Iterations

This article discusses, in the case of eccentric orbits, some solution methods for Kepler's equation, for instance: Newton's method, Halley method and the solution by Fourire-Bessel expansion. The efficiency of solution methods is evaluated according to the number of iterations that each method needs to lead to a solution within the specified tolerance. The solution using Fourier-Bessel series is not an iterative method, however, it was analyzed the number of terms required to achieve the accuracy of the prescribed solution.

An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2585-2588
Author(s):  
Zhong Yong Hu ◽  
Fang Liang ◽  
Lian Zhong Li ◽  
Rui Chen
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Sixth Order ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

scholarly journals A fourth order method for finding a simple root of univariate function

2015 ◽  
Vol 34 (2) ◽  
pp. 197-211
Author(s):  
D. Sbibih ◽  
Abdelhafid Serghini ◽  
A. Tijini ◽  
A. Zidna
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Simple Root ◽  
Newton's Method ◽  
Order Method ◽  
Numerical Examples ◽  
Quadratic Interpolation ◽  
Univariate Function ◽  
Fourth Order Method

In this paper, we describe an iterative method for approximating asimple zero $z$ of a real defined function. This method is aessentially based on the idea to extend Newton's method to be theinverse quadratic interpolation. We prove that for a sufficientlysmooth function $f$ in a neighborhood of $z$ the order of theconvergence is quartic. Using Mathematica with its high precisioncompatibility, we present some numerical examples to confirm thetheoretical results and to compare our method with the others givenin the literature.

scholarly journals Optimal Sixteenth Order Convergent Method Based on Quasi-Hermite Interpolation for Computing Roots

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Fiza Zafar ◽  
Nawab Hussain ◽  
Zirwah Fatimah ◽  
Athar Kharal
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Hermite Interpolation ◽  
Convergent Method ◽  
Basins Of Attractions

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton’s method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the method.

A Robust Iterative Method for Flash Calculations Using the Soave-Redlich-Kwong or the Peng-Robinson Equation of State

1983 ◽  
Vol 23 (03) ◽  
pp. 521-530 ◽  
Author(s):  
L.X. Nghiem ◽  
K. Aziz ◽  
Y.K. Li
Keyword(s):  
Equation Of State ◽  
Iterative Method ◽  
Critical Point ◽  
Newton’S Method ◽  
Newton's Method ◽  
Single Phase ◽  
Saturation Pressure ◽  
Phase Region ◽  
Single Phase Region ◽  
Poor Convergence

Abstract A robust algorithm for flash calculations that uses an equation of state(EOS) is presented. It first uses a special version of the successive substitution(SS) method and switches to Powell's method if poor convergence is observed. Criteria are established for an efficient switch from one method to the other. Experience shows that this method converges near the critical point and also detects the single-phase region without computing the saturation pressure. The Soave-Redlich-Kwong (SRK) and the Peng-Robinson (PR) EOS's are used in this work, but the method is general and applies to any EOS. Introduction The calculation of vapor/liquid equilibrium using an EOS in multicomponent systems yields a system of nonlinear equations that must be solved iteratively. The SS method is commonly used, but it exhibits poor rate of convergence near the critical point. To overcome convergence problems, Newton's method has been used by Fussell and Yanosik to solve the equations. The drawback of Newton's method is the necessity of computing a complicated Jacobian matrix and its inverse at every iteration. Hence, for systems removed from their critical point it involves more work to arrive at the solution than the SS method. Furthermore, the radius of convergence of Newton's method is relatively small when compared to that of the SS method; hence, a good initial guess is required before convergence can be achieved. The single-phase region usually is determined by computing the saturation pressure and comparing it with the pressure of the system. This requires additional work, pressure of the system. This requires additional work, and it is sometimes difficult to decide whether a dewpoint or bubblepoint pressure, which involve different equations, should be computed. This paper presents a robust iterative method for flash calculations using either the SRK or the PR EOS, both of which have received much interest in recent years. The proposed method combines SS with Powell's iteration, proposed method combines SS with Powell's iteration, which is a hybrid algorithm consisting of a quasi-Newton method and a steepest-descent method. The SS method is used initially and is replaced by Powell's method if it demonstrates poor convergence, thus taking advantage of the simplicity of the former method and the robustness of the latter. The SS method has been modified so that the single-phase region can be detected without having to compute the saturation pressure. The nonlinear equations to be solved by an iteration scheme could behave differently, depending on their form and the variables for which they are solved. In this paper three different approaches are considered with paper three different approaches are considered with Powell's method. One of the three approaches is based Powell's method. One of the three approaches is based on the minimization of the Gibbs free energy. The convergence properties of the proposed schemes are demonstrated by three example problems. SPEJ P. 521

scholarly journals The Newtonian Operator and Global Convergence Balls for Newton’s Method

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1074
Author(s):  
José A. Ezquerro ◽  
Miguel A. Hernández-Verón
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Global Convergence ◽  
Point Theorem ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Integral Equation ◽  
The Fixed Point Theorem ◽  
Linear Integral

We obtain results of restricted global convergence for Newton’s method from ideas based on the Fixed-Point theorem and using the Newtonian operator and auxiliary points. The results are illustrated with a non-linear integral equation of Davis-type and improve the results previously given by the authors.

A New SPH Iterative Method for Solving Nonlinear Equations

2019 ◽  
Vol 17 (01) ◽  
pp. 1843005 ◽  
Author(s):  
Rahmatjan Imin ◽  
Ahmatjan Iminjan
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Basic Principle ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Taylor Series Expansion ◽  
Numerical Examples ◽  
First Order ◽  
Kernel Approximation

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

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