Modified Newton's method with third-order convergence and multiple roots
M. Frontini,E. Sormani +1 more
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In this paper, the authors studied the order of convergence of the mNm when the multiplicity p is known and showed that the two most efficient methods in the family of the nNm converge faster than the classical Newton's method when p is unknown.About:
This article is published in Journal of Computational and Applied Mathematics.The article was published on 2003-07-15 and is currently open access. It has received 85 citations till now. The article focuses on the topics: Newton's method & Rate of convergence.read more
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On Newton-type methods with cubic convergence
TL;DR: Weerakoon and Fernando as mentioned in this paper showed that one can modify the Werrakoon-Fernando approach by using Newton's theorem for the inverse function and derive a new class of cubically convergent Newton-type methods.
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Third-order methods from quadrature formulae for solving systems of nonlinear equations
M. Frontini,E. Sormani +1 more
TL;DR: A modification of the Newton method, based on quadrature formulas of order at least one, is extended, which produces iterative methods with order of convergence three that may be more efficient then other third-order methods as they do not require the use of the second-order Frechet derivative.
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A modification of Newton method with third-order convergence
TL;DR: Analysis of convergence shows that the new modification of Newton method for solving non-linear equations is cubically convergent.
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An analysis of the properties of the variants of Newton's method with third order convergence
TL;DR: This paper investigates the relationship between the variants of the Newton's method with cubic convergence and proves that they are different forms of the Halley method and are all contractive iterative methods in a common neighbourhood.
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The improvements of modified Newton’s method
Jisheng Kou,Jisheng Kou +1 more
TL;DR: Some third-order modifications of Newton’s method are improved and many new methods for solving non-linear equations are obtained that can compete with Newton's method.
References
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Book
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Book
Numerical methods for unconstrained optimization and nonlinear equations
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
Journal ArticleDOI
A variant of Newton's method with accelerated third-order convergence
TL;DR: It is shown that the order of convergence of the new method is three, and computed results support this theory.
Book
Numerical analysis: an introduction
TL;DR: In this article, the authors present a text in numerical analysis which is taken to mean the branch of mathematics that develops and analyzes computational methods dealing with problems arising in classical analysis, approximations theory, the theory of equations, and ordinary differential equations.
Some variant of newtons method with third-order convergence
M. Frontini,E Sormani +1 more
TL;DR: A general error analysis providing the higher order of convergence is given, and the best efficiency, in term of function evaluations, of two of this new methods is provided.