alphaCertified: certifying solutions to polynomial systems

Smale's -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 -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.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Jonathan D. Hauenstein, Frank Sottile
• Keywords: Algorithms, CORR 2010, Education, Polynomial, Solutions