scholarly journals On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
H. Montazeri ◽  
F. Soleymani ◽  
S. Shateyi ◽  
S. S. Motsa
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
The Other ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Numerical Tests ◽  
Programming Package ◽  
New Iterative Method

We consider a system of nonlinear equationsF(x)=0. A new iterative method for solving this problem numerically is suggested. The analytical discussions of the method are provided to reveal its sixth order of convergence. A discussion on the efficiency index of the contribution with comparison to the other iterative methods is also given. Finally, numerical tests illustrate the theoretical aspects using the programming package Mathematica.

scholarly journals Multi-Step Preconditioned Newton Methods for Solving Systems of Nonlinear Equations

2017 ◽  
Author(s):  
Fayyaz Ahmad ◽  
Malik Zaka Ullah ◽  
Ali Saleh Alshomrani ◽  
Shamshad Ahmad ◽  
Aisha M. Alqahtani ◽  
...  
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Numerical Stability ◽  
Order Of Convergence ◽  
Convergence Order ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Newton Methods ◽  
Rapid Convergence ◽  
Base Method

The study of different forms of preconditioners for solving a system of nonlinear equations, by using Newton’s method, is presented. The preconditioners provide numerical stability and rapid convergence with reasonable computation cost, whenever chosen accurately. Different families of iterative methods can be constructed by using a different kind of preconditioners. The multi-step iterative method consists of a base method and multi-step part. The convergence order of base method is quadratic and each multi-step add an additive factor of one in the previously achieved convergence order. Hence the convergence of order of an m-step iterative method is m + 1. Numerical simulations confirm the claimed convergence order by calculating the computational order of convergence. Finally, the numerical results clearly show the benefit of preconditioning for solving system of nonlinear equations.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

scholarly journals New Iterative Method for Solving Nonlinear Equations

2014 ◽  
Vol 11 (4) ◽  
pp. 1649-1654 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
New Method ◽  
New Iterative Method

The aim of this paper is to propose an efficient three steps iterative method for finding the zeros of the nonlinear equation f(x)=0 . Starting with a suitably chosen , the method generates a sequence of iterates converging to the root. The convergence analysis is proved to establish its five order of convergence. Several examples are given to illustrate the efficiency of the proposed new method and its comparison with other methods.

scholarly journals New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations

Sensors ◽  
2020 ◽  
Vol 20 (21) ◽  
pp. 5976
Author(s):  
Kalyanasundaram Madhu ◽  
Arul Elango ◽  
René Jr Landry ◽  
Mo’tassem Al-arydah
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Computation Time ◽  
Efficiency Index ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Additional Function ◽  
Computation Efficiency ◽  
Global Navigation Satellite

A two-step fifth and a multi-step 5+3r order iterative method are derived, r≥1 for finding the solution of system of nonlinear equations. The new two-step fifth order method requires two functions, two first order derivatives, and the multi-step methods needs a additional function per step. The performance of this method has been tested with finding solutions to several test problems then applied to solving pseudorange nonlinear equations on Global Navigation Satellite Signal (GNSS). To solve the problem, at least four satellite’s measurements are needed to locate the user position and receiver time offset. In this work, a number of satellites from 4 to 8 are considered such that the number of equations is more than the number of unknown variables to calculate the user position. Moreover, the Geometrical Dilution of Precision (GDOP) values are computed based on the satellite selection algorithm (fuzzy logic method) which could be able to bring the best suitable combination of satellites. We have restricted the number of satellites to 4 to 6 for solving the pseudorange equations to get better GDOP value even after increasing the number of satellites beyond six also yields a 0.4075 GDOP value. Actually, the conventional methods utilized in the position calculation module of the GNSS receiver typically converge with six iterations for finding the user position whereas the proposed method takes only three iterations which really decreases the computation time which provide quicker position calculation. A practical study was done to evaluate the computation efficiency index (CE) and efficiency index (IE) of the new model. From the simulation outcomes, it has been noted that the new method is more efficient and converges 33% faster than the conventional iterative methods with good accuracy of 92%.

scholarly journals A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Moin-ud-Din Junjua ◽  
Saima Akram ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Basins Of Attraction ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Numerical Tests ◽  
New Methods ◽  
New Family ◽  
Engineering Problems ◽  
Fourth Order Method

Solving systems of nonlinear equations plays a major role in engineering problems. We present a new family of optimal fourth-order Jarratt-type methods for solving nonlinear equations and extend these methods to solve system of nonlinear equations. Convergence analysis is given for both cases to show that the order of the new methods is four. Cost of computations, numerical tests, and basins of attraction are presented which illustrate the new methods as better alternates to previous methods. We also give an application of the proposed methods to well-known Burger's equation.

scholarly journals A New Sixth Order Method for Nonlinear Equations inR

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Sukhjit Singh ◽  
D. K. Gupta
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Performance Measure ◽  
Sixth Order ◽  
Order Method ◽  
Numerical Examples ◽  
Real Roots ◽  
New Iterative Method ◽  
Number Of Iterations

A new iterative method is described for finding the real roots of nonlinear equations inR. Starting with a suitably chosenx0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newton’s method and other sixth order methods considered.

An Efficient Iterative Method with Order of Convergence Seven for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2574-2577
Author(s):  
Yun Hong Hu ◽  
Fang Liang ◽  
Li Fang Guo ◽  
Zhong Yong Hu
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Results ◽  
Type Method ◽  
Second Derivatives ◽  
Seventh Order ◽  
And Performance ◽  
Better Than

In this paper, we present a modified seventh-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 at each step. Therefore the efficiency index of the presented method is 1.47577 which is better than that of classical Newton’s method 1.41421. Some numerical results demonstrate the efficiency and performance of the presented method.

scholarly journals A Class of Steffensen-Type Iterative Methods for Nonlinear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
F. Soleymani ◽  
M. Sharifi ◽  
S. Shateyi ◽  
F. Khaksar Haghani
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
Numerical Tests ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

A class of iterative methods without restriction on the computation of Fréchet derivatives including multisteps for solving systems of nonlinear equations is presented. By considering a frozen Jacobian, we provide a class ofm-step methods with order of convergencem+1. A new method named as Steffensen-Schulz scheme is also contributed. Numerical tests and comparisons with the existing methods are included.

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.

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