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Simultaneous point estimates for Newton's method
Publikationstyp
Journal Article
Date Issued
2002-09-01
Sprache
English
Author(s)
Institut
TORE-URI
Journal
Volume
42
Issue
3
Start Page
467
End Page
476
Citation
BIT Numerical Mathematics 42 (3): 467-476 (2002-09-01)
Scopus ID
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.
Subjects
Convergence theorems
Newton iteration
Point estimates
Polynomial roots
Practical conditions for convergence
Simultaneous methods
DDC Class
510: Mathematik