Related papers: Implementing real polyhedral homotopy
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…
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which…
Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…
We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients…
Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate options for performing…
A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of…
Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach…
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types…
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…
We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…
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…
The paper aims to propose a suitable method in finding the solution of tensor complementarity problem. The tensor complementarity problem is a subclass of nonlinear complementarity problems for which the involved function is defined by a…
Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a…
We describe, for the first time, a completely rigorous homotopy (path--following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial…
The basis of the identity representation of a polyhedral group is able to describe functions with symmetries of a platonic solid, i.e., 3-D objects which geometrically obey the cubic symmetries. However, to describe the dynamic of assembles…
We combine tools from homotopy continuation solvers with the methods of analytic combinatorics in several variables to give the first practical algorithm and implementation for the asymptotics of multivariate rational generating functions…
The double-direction orthogonalization algorithm is applied to construct sequences of polynomials, which are orthogonal over the interval [0,1]with the weighting function 1. Functional and recurrent relations are derived for the sequences…
Polytopal methods provide a flexible framework for the numerical approximation of partial differential equations on general meshes. Their convergence analysis raises specific challenges due to their inherently non-conforming nature and, in…