Related papers: The NumericalCertification package in Macaulay2
In the present paper we are concerned with a numerical algorithm for the approximation of the two-dimensional neural field equation with delay. We consider three numerical examples that have been analysed before by other authors and are…
We report on a package of routines for the computer algebra system Maple which supports the explicit determination of the geometric quantities, field equations, equations of motion, and conserved quantities of General Relativity in the…
Probabilistic pushdown automata (pPDA) are a standard model for discrete probabilistic programs with procedures and recursion. In pPDA, many quantitative properties are characterized as least fixpoints of polynomial equation systems. In…
The objective of this research is the development of a practical system to manipulate and validate software package specifications. The validation process developed is based on consistency checks. Furthermore, by means of scenarios, the…
We present a data-driven approach to the quantitative verification of probabilistic programs and stochastic dynamical models. Our approach leverages neural networks to compute tight and sound bounds for the probability that a stochastic…
Assessing non-negativity of multivariate polynomials over the reals, through the computation of {\em certificates of non-negativity}, is a topical issue in polynomial optimization. This is usually tackled through the computation of {\em…
We present a new package ZpL for the mathematical software system SM. It implements a sharp tracking of precision on p-adic numbers, following the theory of ultrametric precision introduced in [4]. The underlying algorithms are mostly based…
Numerical and symbolic methods for optimization are used extensively in engineering, industry, and finance. Various methods are used to reduce problems of interest to ones that are amenable to solution by such software. We develop a…
This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation,…
Accuracy certificates for convex minimization problems allow for online verification of the accuracy of approximate solutions and provide a theoretically valid online stopping criterion. When solving the Lagrange dual problem, accuracy…
We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
The problem of mechanically formalizing and proving metatheoretic properties of programming language calculi, type systems, operational semantics, and related formal systems has received considerable attention recently. However, the dual…
This paper discusses the algorithms and implementations of three Mathematica packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically…
The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. Despite the reputation of learned NN models to behave as black boxes and…
We present the Macaulay2 package Resultants, which provides commands for the effective computation of multivariate resultants, discriminants, and Chow forms. We provide some background for the algorithms implemented and show, with a few…
Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular…
Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. Bertini, a popular software package for computational applications of this field, includes…