Simultaneous point estimates for Newton's method

Simultaneous point estimates for Newton's method

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Article ID: iaor20032074
Country: Netherlands
Volume: 42
Issue: 3
Start Page Number: 467
End Page Number: 476
Publication Date: Sep 2002
Journal: BIT
Authors:
Abstract:

Beside the classical Kantorovich theory there exist convergence criteria for the Newton iteration which only involve data at one point, i.e. point estimates. Given a polynomial P, these conditions imply the point evaluation of n = deg(P) functions (from a certain Taylor expansion). Such sufficient conditions ensure quadratic convergence to a single zero and have been used by several authors in the design and analysis of robust, fast and efficient root-finding methods for polynomials. In this paper a sufficient condition for the simultaneous convergence of the one-dimensional Newton iteration for polynomials will be given. The new condition involves only n point evaluations of the Newton correction and the minimum mutual distance of approximations to ensure ‘simultaneous’ quadratic convergence to the pairwise distinct n roots.

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