The theory of Newton's method
type: Item
instance of: scholarly article
title: The theory of Newton's method
zbMATH Open document ID: 0965.65080
DOI: 10.1016/S0377-0427(00)00435-0
published in: Journal of Computational and Applied Mathematics
publication date: 2001-01-25T00:00:00Z
review text: The author deals with the iterative solution of a nonlinear equation in some Banach space via Newton's method. Convergence results are provided for both, smooth and non-smooth equations. Afterwards, error estimates are presented, and the method is studied for underdetermined equations and singular Jacobians. Finally, the question of computational complexity is discussed.
reviewed by: Etienne Emmrich
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 1556981
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: convergence
zbMATH Keywords: Banach space
zbMATH Keywords: Newton's method
zbMATH Keywords: error estimates
zbMATH Keywords: underdetermined equations
zbMATH Keywords: computational complexity
author: Aurél Galántai
MaRDI profile type: Publication
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cites work: On Newton's method and nondiscrete mathematical induction
cites work: On Newton's method under mild differentiability conditions and applications
cites work: A Newton-Raphson method for the solution of systems of equations
cites work: Über die Einzugsbereiche der Nullstellen von Polynomen beim Newton-Verfahren
cites work: Hybrid Krylov Methods for Nonlinear Systems of Equations
cites work: Convergence of Newton's Method for Singular Smooth and Nonsmooth Equations Using Adaptive Outer Inverses
cites work: On a numerical treatment for the curve-tracing of the homotopy method
cites work: On the Continuous Realization of Iterative Processes
cites work: Q4000274
cites work: Q3702408
cites work: Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods
cites work: Q5565773
cites work: Über die Lösung der im Banachschen Raume definierten nichtlinearen Gleichungen
cites work: Comparison of Brown's and Newton's method in the monotone case
cites work: Verallgemeinerte Diagonaldominanz bei nichtlinearen Funktionen. I
cites work: Optimal Error Bounds for the Newton–Kantorovich Theorem
cites work: Starlike domains of convergence for Newton's method at singularities
cites work: On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions
cites work: On Algorithms for Solvingf(x)=0
cites work: Q5566058
cites work: Qualitative Analysis of Newton Flow
cites work: Q4524235
cites work: Q5332416
cites work: Newton's method under mild differentiability conditions
cites work: Q4841246
cites work: Optimal Order and Efficiency for Iterations with Two Evaluations
cites work: Error bounds of Newton type process on Banach spaces
cites work: Error analysis for the Newton-Raphson method
cites work: Q3835692
cites work: The Kantorovich Theorem with Optimal Error Bounds
cites work: Unified error analysis for Newton-type methods
cites work: Majorizing Sequences and Error Bounds for Iterative Methods
cites work: An updated version of the Kantorovich theorem for Newton's method
cites work: Q5795687
cites work: Convergence of Newton-like methods for singular operator equations using outer inverses
cites work: Q3954739
cites work: The Newton-Kantorovich Theorem
cites work: Q5652137
cites work: An upper and a lower bound for the distance of a manifold to a nearby point
cites work: Nonsmooth Equations: Motivation and Algorithms
cites work: Sharp error bounds for Newton's process
cites work: The rate of convergence of Newton's process
cites work: A nonsmooth version of Newton's method
cites work: Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations
cites work: Convergence of the Newton process to multiple solutions
cites work: A Note on the Convergence of Newton’s Method
cites work: On Newton’s Method for Singular Problems
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cites work: Variational Analysis
cites work: Newton's method under mild differentiability conditions with error analysis
cites work: Q4187244
cites work: The fundamental theorem of algebra and complexity theory
cites work: On the efficiency of algorithms of analysis
cites work: Q4720691
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cites work: Continuous Newton-Raphson method for solving and underdetermined system of nonlinear equations

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