alphaCertified: certifying solutions to polynomial systems

Details alphaCertified: certifying solutions to polynomial systems alphaCertified: certifying solutions to polynomial systems

2010-11-04

arxiv.org/abs/1011.1091v2

Mathematics

Numerical Analysis

21 pages

Scientific paper

Smale's alpha-theory uses estimates related to the convergence of Newton's method to give criteria implying that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on alpha-theory to certify solutions to polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements an algorithm to certify whether a given point corresponds to a real solution to a real polynomial system, as well as algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.

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