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 On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

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 Modifications of the continuation method for the solution of systems of nonlinear equations

1979 ◽  
Vol 2 (2) ◽  
pp. 299-308
Author(s):  
G. R. Lindfield ◽  
D. C. Simpson
Keyword(s):  
Nonlinear Equations ◽  
Continuation Method ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Modified Method ◽  
Wide Range

Modifications are proposed to the Davidenko-Broyden algorithm for the solution of a system of nonlinear equations. The aim of the modifications is to reduce the overall number of function evaluations by limiting the number of function evaluations for any one subproblem. To do this alterations are made to the strategy used in determining the subproblems to be solved. The modifications are compared with other methods for a wide range of test problems, and are shown to significantly reduce the number of function evaluations for the difficult problems. For the easier problems the modified method is equivalent to the Davidenko-Broyden algorithm.

scholarly journals En enhanced matrix-free method via double step length approach for solving systems of nonlinear equations

2017 ◽  
Vol 6 (4) ◽  
pp. 147 ◽  
Author(s):  
Abubakar Sani Halilu ◽  
H. Abdullahi ◽  
Mohammed Yusuf Waziri
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Step Length ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Numerical Comparison ◽  
Inexact Line Search ◽  
Derivative Free ◽  
Variant Method

A variant method for solving system of nonlinear equations is presented. This method use the special form of iteration with two step length parameters, we suggest a derivative-free method without computing the Jacobian via acceleration parameter as well as inexact line search procedure. The proposed method is proven to be globally convergent under mild condition. The preliminary numerical comparison reported in this paper using a large scale benchmark test problems show that the proposed method is practically quite effective.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

2021 ◽  
pp. 2150059
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Nonlinear Equations ◽  
Inverse Operator ◽  
General Setting ◽  
Basins Of Attraction ◽  
Function Evaluation ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Additional Function ◽  
Numerical Experimentation ◽  
The Cost

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

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.

Accelerated iterative methods for finding solutions of a system of nonlinear equations

2007 ◽  
Vol 190 (2) ◽  
pp. 1815-1823 ◽  
Author(s):  
Miquel Grau-Sánchez ◽  
Josep M. Peris ◽  
José M. Gutiérrez
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
System Of Nonlinear Equations

scholarly journals Finding the Roots of System of Nonlinear Equations by a Novel Filled Function Method

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Xiang Wang ◽  
You-Lin Shang ◽  
Wei-Gang Sun ◽  
Ying Zhang
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Function Method ◽  
Optimization Problem ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Global Optimization Problem ◽  
Filled Function ◽  
Filled Function Method ◽  
Constrained System

We present a novel filled function approach to solve box-constrained system of nonlinear equations. The system is first transformed into an equivalent nonsmooth global minimization problem, and then a new filled function method is proposed to solve this global optimization problem. Numerical experiments on several test problems are conducted and the computational results are also reported.

scholarly journals An improved differential evolution algorithm with a restart technique to solve systems of nonlinear equations

2020 ◽  
Vol 10 (1) ◽  
pp. 118-136
Author(s):  
Jeerayut Wetweerapong ◽  
Pikul Puphasuk
Keyword(s):  
Differential Evolution ◽  
Nonlinear Equations ◽  
Nonlinear Models ◽  
Differential Evolution Algorithm ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
De Algorithm ◽  
New Strategy ◽  
Evolution Algorithm ◽  
Improved Differential Evolution Algorithm

In this research, an improved differential evolution algorithm with a restart technique (DE-R) is designed for solutions of systems of nonlinear equations which often occurs in solving complex computational problems involving variables of nonlinear models. DE-R adds a new strategy for mutation operation and a restart technique to prevent premature convergence and stagnation during the evolutionary search to the basic DE algorithm. The proposed method is evaluated on various real world and synthetic problems and compared with the recently developed methods in the literature. Experiment results show that DE-R can successfully solve all the test problems with fast convergence speed and give high quality solutions. It also outperforms the compared methods.

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