scholarly journals Proving the convergence of the iterative methods by using symbolic computation in Maple

2012 ◽  
Vol 28 (1) ◽  
pp. 1-8
Author(s):  
GHEORGHE ARDELEAN ◽  
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Symbolic Computation ◽  
High Order ◽  
High Order Methods ◽  
Convergence Error

The proofs of the convergence for some high-order methods for solving nonlinear equations, by using symbolic computation in Maple, is presented. Also, the convergence error for some Newton-type methods is evaluated by symbolic computation.

scholarly journals Multistep High-Order Methods for Nonlinear Equations Using Padé-Like Approximants

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
High Order ◽  
High Order Methods ◽  
Optimal Methods ◽  
Numerical Tests ◽  
Theoretical Results

We present new high-order optimal iterative methods for solving a nonlinear equation, f(x)=0, by using Padé-like approximants. We compose optimal methods of order 4 with Newton’s step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.

scholarly journals Effective High-Order Iterative Methods via the Asymptotic Form of the Taylor-Lagrange Remainder

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Isaac Fried
Keyword(s):  
Iterative Methods ◽  
Asymptotic Form ◽  
Nonlinear Function ◽  
High Order ◽  
High Order Methods ◽  
High Order Iterative Methods

The asymptotic form of the Taylor-Lagrange remainder is used to derive some new, efficient, high-order methods to iteratively locate the root, simple or multiple, of a nonlinear function. Also derived are superquadratic methods that converge contrarily and superlinear and supercubic methods that converge alternatingly, enabling us not only to approach, but also to bracket the root.

scholarly journals A class of efficient high‐order iterative methods with memory for nonlinear equations and their dynamics

2018 ◽  
Vol 41 (17) ◽  
pp. 7263-7282 ◽  
Author(s):  
Cory L. Howk ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Carles Teruel
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
High Order ◽  
With Memory ◽  
High Order Iterative Methods

scholarly journals On a Family of High-Order Iterative Methods under Kantorovich Conditions and Some Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
S. Amat ◽  
C. Bermúdez ◽  
S. Busquier ◽  
M. J. Legaz ◽  
S. Plaza
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
High Order ◽  
High Order Iterative Methods

This paper is devoted to the study of a class of high-order iterative methods for nonlinear equations on Banach spaces. An analysis of the convergence under Kantorovich-type conditions is proposed. Some numerical experiments, where the analyzed methods present better behavior than some classical schemes, are presented. These applications include the approximation of some quadratic and integral equations.

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.

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.

scholarly journals Using symbolic computation in Mathematica for verifying the convergence of the iterative methods

2013 ◽  
Vol 22 (1) ◽  
pp. 9-13
Author(s):  
GHEORGHE ARDELEAN ◽  
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Symbolic Computation ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Correct Result ◽  
Modified Newton’S Method ◽  
Second Derivatives ◽  
Appl Math

In [Jisheng Kou, The improvements of modified Newton’s method, Appl. Math. Comput., 189 (2007) 602–609], the improvements of some thirdorder modifications of Newton’s method for solving nonlinear equations are presented. In this paper we point out some flaws in the results of Jisheng Kou and we correct them by using symbolic computation in Mathematica. In [M. A. Noor et al., A new modified Halley method without second derivatives for nonlinear equations, Appl. Math. Comput., 189 (2007) 1268–1273] , the error equation obtained for the new method presented is wrong. We present the correct result by using symbolic computation, too. Finally, we present two examples of very simply proofs for the convergence of iterative methods by using symbolic computation. We consider that the Mathematica programs in this paper are good examples for other authors to prove the convergence of the iterative methods or to verify their results.

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