Related papers: PHCpack in Macaulay2
We describe the package "IncidenceCorrespondenceCohomology" for the computer algebra system Macaulay2. The main feature concerns the computation of characters and dimensions for the cohomology groups of line bundles on the incidence…
In this chapter, we describe the Parallel Sparse Computation Toolkit (PSCToolkit), a suite of libraries for solving large-scale linear algebra problems in an HPC environment. In particular, we focus on the tools provided for the solution of…
POCP is a new Matlab package running jointly with GloptiPoly 3 and, optionally, YALMIP. It is aimed at nonlinear optimal control problems for which all the problem data are polynomial, and provides an approximation of the optimal value as…
This note describes a Macaulay2 package for handling divisors. Group operations for divisors are included. There are methods for converting divisors to reflexive or invertible sheaves. Additionally, there are methods for checking whether…
We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in…
Systems of polynomial equations arise frequently in computer vision, especially in multiview geometry problems. Traditional methods for solving these systems typically aim to eliminate variables to reach a univariate polynomial, e.g., a…
This article highlights the ToricHigherDirectImages package in Macaulay2. The central feature is a method for computing (higher) direct images of line bundles under surjective toric morphisms.
Numerical algebraic geometry provides a number of efficient tools for approximating the solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A…
Algebraic methods have a long history in statistics. The most prominent manifestation of modern algebra in statistics can be seen in the field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to…
We introduce the ForeignFunctions package for Macaulay2, which uses libffi to provide the ability to call functions from external libraries without needing to link against them at compile time. As examples, we use the library FFTW to…
Solving a system of $m$ multivariate quadratic equations in $n$ variables over finite fields (the MQ problem) is one of the important problems in the theory of computer science. The XL algorithm (XL for short) is a major approach for…
We introduce the Brackets package for the computer algebra system Macaulay2, which provides convenient syntax for computations involving the classical invariants of the special linear group. We describe our implementation of bracket rings…
Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions…
{\tt AbstractSimplicialComplexes.m2} is a computer algebra package written for the computer algebra system {\tt Macaulay2} \cite{M2}. It provides new infrastructure to work with abstract simplicial complexes and related homological…
The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary…
Homotopy methods to solve polynomial systems are well suited for parallel computing because the solution paths defined by the homotopy can be tracked independently. Both the static and dynamic load balancing models are implemented in C with…
In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that…
We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…