Journal Article10.1137/0215011
Computational complexity : on the geometry of polynomials and a theory of cost: II
Michael Shub,Steve Smale +1 more
TL;DR: Traditional algorithms, Newton's method and higher order generalization due to Euler are dealt with and some understanding of this phenomenon of finding a zero of a complex polynomial is given.
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Abstract: This paper deals with traditional algorithms, Newton’s method and a higher order generalization due to Euler. These iterations schemes and their modifications have had a great success in solving nonlinear systems of equations. We give some understanding of this phenomenon by giving estimates of efficiency. The problem we focus on is that of finding a zero of a complex polynomial.
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Problèmes rencontrés dans mon parcours mathématique : un bilan
TL;DR: In this article, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Newton's method and the Computational Complexity of the Fundamental Theorem of Algebra
TL;DR: Fast factoring methods which yield root-approximations are constructed using some algebraic Newton iteration for initial factor approximations.
Geometry of Polynomials and Root-Finding via Path-Lifting
TL;DR: In this article, a path-lifting algorithm for finding approximate zeros of polynomials is presented. But the complexity of the algorithm does not depend directly on the degree of the polynomial, but only on the geometry of the critical values.
Random polynomials and approximate zeros of Newton's method
TL;DR: The results of this paper show that the set of approximate zeros for Newton’s method is at least $Cd^{(-1.5-\varepsilon)}$ for any positive $\vARpsilon$, with C depending only on $\varpsilon$.
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Applied and Computational Complex Analysis
Henry C. Thacher,Peter Henrici +1 more
- 01 Jan 1974
2.5K