The NumericalCertification package in Macaulay2
Kisun Lee
TL;DR
The NumericalCertification package in Macaulay2 provides tools for certifying numerical solutions to polynomial systems, including methods for regular and singular solutions, enhancing the reliability of numerical algebraic geometry computations.
Contribution
It introduces a comprehensive Macaulay2 package implementing Smale's alpha-theory, Krawczyk method, and deflation techniques for certifying solutions of polynomial systems.
Findings
Successfully certifies regular solutions using alpha-theory and Krawczyk method.
Effectively verifies singular solutions through iterative deflation.
Integrates with Macaulay2's numerical solvers for improved solution validation.
Abstract
The package \texttt{NumericalCertification} implements methods for certifying numerical approximations of solutions for a given system of polynomial equations. For certifying regular solutions, the package implements Smale's -theory and Krawczyk method. For a singular solution, we implement soft verification using the iterative deflation method. We demonstrate the functionalities of the package focusing on interaction with current numerical solvers in \texttt{Macaulay2}.
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Taxonomy
TopicsPolynomial and algebraic computation · Numerical Methods and Algorithms · Numerical methods for differential equations