2010/11/04 by Jonathan D. Hauenstein, Hauenstein, Jonathan D., Frank Sottile +1
Computer Science · Engineering · #65G20 #65H05 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Software (cs.MS) #Numerical Analysis (math.NA) #Numerical Methods and Algorithms #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1011.1091
openalex publication_date 2010/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.