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.

scholarly journals Eighteenth Order Convergent Method for Solving Non-Linear Equations

2017 ◽  
Vol 10 (1) ◽  
pp. 144-150 ◽  
Author(s):  
V.B Vatti ◽  
Ramadevi Sri ◽  
M.S Mylapalli
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Subject Classification ◽  
Order Convergence ◽  
Numerical Examples ◽  
Convergent Method ◽  
Non Linear Equations

In this paper, we suggest and discuss an iterative method for solving nonlinear equations of the type f(x)=0 having eighteenth order convergence. This new technique based on Newton’s method and extrapolated Newton’s method. This method is compared with the existing ones through some numerical examples to exhibit its superiority. AMS Subject Classification: 41A25, 65K05, 65H05.

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.

scholarly journals On an new Open type variant of Newton´s method

2014 ◽  
Vol 10 (2) ◽  
pp. 21-31
Author(s):  
Manoj Kumar
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Open Type ◽  
Numerical Tests ◽  
Comparison Of The Results

Abstract The aim of the present paper is to introduce and investigate a new Open type variant of Newton's method for solving nonlinear equations. The order of convergence of the proposed method is three. In addition to numerical tests verifying the theory, a comparison of the results for the proposed method and some of the existing ones have also been given.

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.

scholarly journals A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 83
Author(s):  
José M. Gutiérrez ◽  
Miguel Á. Hernández-Verón
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Integral Equations ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Iterative Scheme ◽  
Fredholm Integral Equations

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.

Applied-Information Technology with Two Modified Newton-Type Methods in Order of Convergence Six for Solving Nonlinear Equations

2014 ◽  
Vol 540 ◽  
pp. 435-438
Author(s):  
Liang Fang
Keyword(s):  
Information Technology ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Rapid Development ◽  
Nonlinear Problems ◽  
Efficiency Index ◽  
Important Direction ◽  
Better Than

With the rapid development of information technology and wide application of science and technology, nonlinear problems become an important direction of research in the field of numerical calculation. In this paper, we mainly study the iterative algorithm of nonlinear equations. We present and analyze two modified Newton-type methods with order of convergence six for solving nonlinear equations. The methods are free from second derivatives. Both of them require three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented methods is 1.431 which is better than that of the classical Newton’s method 1.414. Some numerical results illustrate that the proposed methods are more efficient and perform better than the classical Newton's method.

scholarly journals Why using symbolic computation for proving the convergence of iterative methods?

2013 ◽  
Vol 22 (2) ◽  
pp. 127-134
Author(s):  
GHEORGHE ARDELEAN ◽  
◽  
LASZLO BALOG ◽  
Keyword(s):  
Nonlinear Equations ◽  
Symbolic Computation ◽  
Newton’S Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Higher Order ◽  
Order Convergence ◽  
Convergence Error ◽  
Higher Order Convergence ◽  
Appl Math

In [YoonMe Ham et al., Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477–486], some higher-order modifications of Newton’s method for solving nonlinear equations are presented. In [Liang Fang et al., Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math., 228 (2009) 296–303], the authors point out some flaws in the results of YoonMe Ham et al. and present some modified variants of the method. In this paper we point out that the paper of Liang Fang et al. itself contains some flaw results and we correct them by using symbolic computation in Mathematica. Moreover, we show that the main result in Theorem 3 of Liang Fang et al. is wrong. The order of convergence of the method is’nt 3m+2, but is 2m+4. We give the general expression of convergence error too.

A New Iterative Method with Sixth-Order Convergence for Solving Nonlinear Equations

2012 ◽  
Vol 542-543 ◽  
pp. 1019-1022
Author(s):  
Han Li
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Efficiency Index ◽  
Order Convergence ◽  
Second Derivatives ◽  
New Iterative Method ◽  
Better Than

In this paper, we present and analyze a new iterative method for solving nonlinear equations. It is proved that the method is six-order convergent. The algorithm is free from second derivatives, and it requires three evaluations of the functions and two evaluations of derivatives in each iteration. The efficiency index of the presented method is 1.431 which is better than that of classical Newton’s method 1.414. Some numerical experiments illustrate that the proposed method is more efficient and performs better than classical Newton's method and some other methods.

scholarly journals Geometrically Constructed Families of Newton's Method for Unconstrained Optimization and Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Sanjeev Kumar ◽  
Vinay Kanwar ◽  
Sushil Kumar Tomar ◽  
Sukhjit Singh
Keyword(s):  
Iterative Method ◽  
Unconstrained Optimization ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Optimization Problems ◽  
Newton’S Iterative Method ◽  
Parabolic Curve ◽  
Unconstrained Optimization Problems ◽  
Straight Line

One-parameter families of Newton's iterative method for the solution of nonlinear equations and its extension to unconstrained optimization problems are presented in the paper. These methods are derived by implementing approximations through a straight line and through a parabolic curve in the vicinity of the root. The presented variants are found to yield better performance than Newton's method, in addition that they overcome its limitations.

A Variant of Newton Method with Eighth-Order Convergence for Solving Nonlinear Equations

2012 ◽  
Vol 490-495 ◽  
pp. 51-55
Author(s):  
Liang Fang
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Eighth Order ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a variant of Newton method with order of convergence eight for solving nonlinear equations. The method is free from second derivatives. It requires three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented method is 1.5157 which is better than that of classical Newton’s method 1.4142. Some numerical experiments illustrate that the proposed method is more efficient and performs better than classical Newton's method.

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