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 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 Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

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 Second Derivative Free Eighteenth Order Convergent Method for Solving Non-Linear Equations

2017 ◽  
Vol 10 (04) ◽  
pp. 829-835
Author(s):  
V.B. Kumar Vatti ◽  
Ramadevi Sri ◽  
M.S.Kumar Mylapalli
Keyword(s):  
Linear Equations ◽  
Efficiency Index ◽  
Subject Classification ◽  
New Method ◽  
Second Derivative ◽  
Numerical Examples ◽  
Derivative Free ◽  
Convergent Method ◽  
And Performance ◽  
Non Linear Equations

In this paper, the Eighteenth Order Convergent Method (EOCM) developed by Vatti et.al is considered and this method is further studied without the presence of second derivative. It is shown that this method has same efficiency index as that of EOCM. Several numerical examples are given to illustrate the efficiency and performance of the new method. AMS Subject Classification: 41A25, 65K05, 65H05.

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 Newton Type Iterative Method with Fourth-order Convergence

2017 ◽  
Vol 12 (1) ◽  
pp. 87-95
Author(s):  
Jivandhar Jnawali
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton's Method ◽  
Second Order ◽  
Secant Method ◽  
Order Derivative ◽  
Single Variable ◽  
Order Convergence ◽  
Numerical Examples

The aim of this paper is to propose a fourth-order Newton type iterative method for solving nonlinear equations in a single variable. We obtained this method by combining the iterations of contra harmonic Newton’s method with secant method. The proposed method is free from second order derivative. Some numerical examples are given to illustrate the performance and to show this method’s advantage over other compared methods.Journal of the Institute of Engineering, 2016, 12 (1): 87-95

scholarly journals DEVELOPMENT OF NEW ITERATIVE SCHEME FOR SOLVING NON-LINEAR EQUATIONS

2021 ◽  
Author(s):  
Tusar singh ◽  
Dwiti Behera
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Iterative Scheme ◽  
Linear Equations ◽  
Quadrature Rule ◽  
Modified Newton’S Method ◽  
Newton Raphson ◽  
Raphson Method ◽  
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.

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.

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.

Quantum Newton’s Method for Solving the System of Nonlinear Equations

SPIN ◽  
2021 ◽  
pp. 2140004
Author(s):  
Cheng Xue ◽  
Yuchun Wu ◽  
Guoping Guo
Keyword(s):  
Quantum Computing ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Data Conversion ◽  
System Of Nonlinear Equations ◽  
Dimensional System ◽  
System Of Linear Equations ◽  
Whole Process

While quantum computing provides an exponential advantage in solving the system of linear equations, there is little work to solve the system of nonlinear equations with quantum computing. We propose quantum Newton’s method (QNM) for solving [Formula: see text]-dimensional system of nonlinear equations based on Newton’s method. In QNM, we solve the system of linear equations in each iteration of Newton’s method with quantum linear system solver. We use a specific quantum data structure and [Formula: see text] tomography with sample error [Formula: see text] to implement the classical-quantum data conversion process between the two iterations of QNM, thereby constructing the whole process of QNM. The complexity of QNM in each iteration is [Formula: see text]. Through numerical simulation, we find that when [Formula: see text], QNM is still effective, so the complexity of QNM is sublinear with [Formula: see text], which provides quantum advantage compared with the optimal classical algorithm.

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