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On Newton’s method and nondiscrete mathematical induction. (English) Zbl 0642.65043
The author considers the equation (1) \(F(x)=0\) where F is a nonlinear operator from a Banach space E into itself. The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton’s method. It is assumed that the operator F has Hölder continuous derivatives. When the Fréchet-derivative of the operator F satisfies a Lipschitz condition the obtained results reduced to the ones obtained by Ptak and Potra in 1972.
Reviewer: O.Hadžić

MSC:
65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
Software:
PITCON
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References:
[1] Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (1970) · Zbl 0241.65046
[2] DOI: 10.1137/0705003 · Zbl 0155.46701
[3] Rheinboldt, Numerical Analysis of Parametrized Nonlinear Equations (1986)
[4] Davis, Introduction to Nonlinear Differential and Integral Equations (1962)
[5] Potra, Nondiscrete Induction and Iterative Processes (1984)
[6] Kantorovich, Functional Analysis in Normed Spaces (1964)
[7] DOI: 10.1007/BF01463998 · Zbl 0434.65034
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