scholarly journals Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Alicia Cordero ◽  
Esther Gómez ◽  
Juan R. Torregrosa
Keyword(s):  
Heat Conduction ◽  
Nonlinear Systems ◽  
Iterative Methods ◽  
Finite Differences ◽  
Diffusion Problem ◽  
Eighth Order ◽  
New Methods ◽  
Theoretical Results ◽  
High Order Iterative Methods ◽  
Methods Numerical

For solving nonlinear systems of big size, such as those obtained by applying finite differences for approximating the solution of diffusion problem and heat conduction equations, three-step iterative methods with eighth-order local convergence are presented. The computational efficiency of the new methods is compared with those of some known ones, obtaining good conclusions, due to the particular structure of the iterative expression of the proposed methods. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and a nonlinear one-dimensional heat conduction equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the presented schemes.

scholarly journals Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 943
Author(s):  
Xiaofeng Wang ◽  
Yingfanghua Jin ◽  
Yali Zhao
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Nonlinear Ordinary Differential Equations ◽  
New Methods ◽  
Derivative Free ◽  
Initial Scheme ◽  
With Memory ◽  
Theoretical Results

Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

SOME NEWTON-TYPE ITERATIVE METHODS WITH AND WITHOUT MEMORY FOR SOLVING NONLINEAR EQUATIONS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350078 ◽  
Author(s):  
XIAOFENG WANG ◽  
TIE ZHANG
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Type Method ◽  
New Methods ◽  
Hermite Interpolating Polynomial ◽  
With Memory ◽  
Theoretical Results ◽  
Very High ◽  
High Computational Efficiency

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, [Formula: see text] and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

scholarly journals An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

scholarly journals Optimal High-Order Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Artidiello ◽  
A. Cordero ◽  
Juan R. Torregrosa ◽  
M. P. Vassileva
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Order ◽  
Function Technique ◽  
Numerical Tests ◽  
New Methods ◽  
Theoretical Results ◽  
Made In

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

New SOR-like methods for solving the Sylvester equation

2014 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakub Kierzkowski
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Relaxation Method ◽  
Sylvester Equation ◽  
Numerical Experimentation ◽  
Successive Over Relaxation ◽  
Theoretical Results ◽  
Methods Numerical

AbstractWe present new iterative methods for solving the Sylvester equation belonging to the class of SOR-like methods, based on the SOR (Successive Over-Relaxation) method for solving linear systems. We discuss convergence characteristics of the methods. Numerical experimentation results are included, illustrating the theoretical results and some other noteworthy properties of the Methods.

scholarly journals New Predictor-Corrector Methods with High Efficiency for Solving Nonlinear Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Numerical Test ◽  
Efficiency Index ◽  
High Order ◽  
The Other ◽  
Predictor Corrector ◽  
Theoretical Results

A new set of predictor-corrector iterative methods with increasing order of convergence is proposed in order to estimate the solution of nonlinear systems. Our aim is to achieve high order of convergence with few Jacobian and/or functional evaluations. Moreover, we pay special attention to the number of linear systems to be solved in the process, with different matrices of coefficients. On the other hand, by applying the pseudocomposition technique on each proposed scheme we get to increase their order of convergence, obtaining new efficient high-order methods. We use the classical efficiency index to compare the obtained procedures and make some numerical test, that allow us to confirm the theoretical results.

scholarly journals A Family of Iterative Methods with Accelerated Eighth-Order Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Alicia Cordero ◽  
Mojtaba Fardi ◽  
Mehdi Ghasemi ◽  
Juan R. Torregrosa
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Function Method ◽  
Order Convergence ◽  
Eighth Order ◽  
Weight Function Method ◽  
Theoretical Results

We propose a family of eighth-order iterative methods without memory for solving nonlinear equations. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Their efficiency indices are all found to be 1.682. Several examples allow us to compare our algorithms with known ones and confirm the theoretical results.

scholarly journals Two new effective iteration methods for nonlinear systems with complex symmetric Jacobian matrices

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Lv Zhang ◽  
Qing-Biao Wu ◽  
Min-Hong Chen ◽  
Rong-Fei Lin
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Newton Method ◽  
Convergence Properties ◽  
Jacobian Matrices ◽  
New Methods ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Complex Symmetric ◽  
Inner Iteration

AbstractIn this paper, we mainly discuss the iterative methods for solving nonlinear systems with complex symmetric Jacobian matrices. By applying an FPAE iteration (a fixed-point iteration adding asymptotical error) as the inner iteration of the Newton method and modified Newton method, we get the so–called Newton-FPAE method and modified Newton-FPAE method. The local and semi-local convergence properties under Lipschitz condition are analyzed. Finally, some numerical examples are given to expound the feasibility and validity of the two new methods by comparing them with some other iterative methods.

scholarly journals Optimization by parameters in the iterative methods for solving non-linear equations

2021 ◽  
pp. 16-22
Author(s):  
Tugal Zhanlav ◽  
Khuder Otgondorj
Keyword(s):  
Iterative Methods ◽  
Linear Equations ◽  
Real Life ◽  
Robust Methods ◽  
Life Problems ◽  
Optimal Values ◽  
Non Linear Equations ◽  
Theoretical Results ◽  
High Computational Efficiency ◽  
Methods Numerical

In this paper, we used the necessary optimality condition for parameters in a two-point iterations for solving nonlinear equations. Optimal values of these parameters fully coincide with those obtained in [6] and allow us to increase the convergence order of these iterative methods. Numerical experiments and the comparison of existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we considered a variety of real life problems from different disciplines, e.g., Kepler’s equation of motion, Planck’s radiation law problem, in order to check the applicability and effectiveness of our proposed methods.

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