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Generalized Differentiability Conditions for Newton's Method. [PDF]
, 2002 The use of majorizing sequences is the usual way to prove the convergence of Newton's method. An alternative technique to majorizing sequences is provided in this paper, in which three scalar sequences are used, so that the analysis of convergence is ...
Ezquerro, J.A. [0000-0001-8120-167X] +3 more
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A modification of the classical Kantorovich conditions for Newton's method [PDF]
, 2001 In the classical Kantorovich theorem on Newton's method it is assumed that the second Fréchet derivative of the involved operator satisfies the condition ||F″(x)||⩽K in an appropiate domain.
Hernández, null [0000-0001-5478-2958] +1 more
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A new semilocal convergence theorem for Newton's method [PDF]
, 1997 A new semilocal convergence theorem for Newton's method is established for solving a nonlinear equation F(x) = 0, defined in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable, and F″ satisfies a Lipschitz type condition ...
JoséM Gutiérrez +2 more
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Computers & Mathematics with Applications, 2015 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Owe Axelsson, Stanislav Sysala
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A generalized multivariable Newton method [PDF]
Fixed Point Theory and Algorithms for Sciences and Engineering, 2021 AbstractIt is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution.
Regina S. Burachik +2 more
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Chaotic Convergence of Newton’s method
IEEE Transactions on Signal Processing, 2023 <p>Newton's method is considered to be unstable, or worse, chaotic. In this manuscript we show the source of this instability, and present a method to fix the probelm.</p> <p>We also investigate the eigen roots of a modified black-body radiation formula, that seems to explain a connection to quantum mechanics.
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WIREs Computational Statistics, 2010 AbstractNewton's method is one of the most powerful techniques for solving systems of nonlinear equations and minimizing functions. It is easy to implement and has a provably fast rate of convergence under fairly mild assumptions. Because of these and other nice properties, Newton's method is at the heart of many solution techniques used to solve real ...
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Majorizing sequences for Newton's method from initial value problems [PDF]
, 2012 The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton's method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot.
Hernández, null [0000-0001-5478-2958] +2 more
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Newton's method in Riemannian manifolds
Journal of Numerical Analysis and Approximation Theory, 2008 Using more precise majorizing sequences than before [1], [8], and under the same computational cost, we provide a finer semilocal convergence analysis of Newton's method in Riemannian manifolds with the following advantages: larger convergence domain ...
Ioannis K. Argyros
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Newton's Method in Mixed Precision
SIAM Review, 2022 Abstract: We investigate the use of reduced precision arithmetic to solve the linear equation for the Newton step. If one neglects the backward error in the linear solve, then well-known convergence theory implies that using single precision in the linear solve has very little negative effect on the nonlinear convergence rate.
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