Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We desc...

We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in “near-colliding” situations. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recently discovered “decimation” technique, we present an algorithm which appl...

The kinematic analyses, of manipulators and other robotic devices composed of mechanical links, usually depend on the solution of sets of nonlinear equations. There are a variety of both numerical and algebraic techniques available to solve such systems of equations and to give bounds on the number of solutions. These solution methods have also led to an understanding of how special choices of ...

We consider Hölder smoothness classes of surfaces for which we construct piecewise polynomial approximation networks, which are graphs with polynomial pieces as nodes and edges between polynomial pieces that are in ‘good continuation’ of each other. Little known to the community, a similar construction was used by Kolmogorov and Tikhomirov in their proof of their celebrated entropy results for ...

HOM4PS-2.0 is a software package in FORTRAN 90 which implements the polyhedral homotopy continuation method for solving polynomial systems. It leads in speed over the existing software packages in the same category by huge margins. This article details the description of the parallel version of HOM4PS2.0, named HOM4PS-2.0para. Excellent scalability in the numerical results shows that the parall...

Many problems in kinematic synthesis of mechanisms and robots lead naturally to systems of bilinear equations in design parameters. Such systems are typically solved using well-known methods for solutions of systems of polynomial equations, e.g., Dialytic Elimination, Grobner bases, and Polynomial Continuation. This paper presents a new nullspace analysis method for solving a system of bilinear...

Let C = {V1, V2, . . . , Vk} be a finite collection of nontrivial subspaces of a finite dimensional real vector space V . Let L denote a one dimensional subspace of V and let θ(L, Vi) denote the principal (or canonical) angle between L and Vi. We are interested in finding all lines that maximize the function F (L) = ∑k i=1 cos θ(L, Vi). Conceptually, this is the line through the origin that bes...

We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in “near-colliding” situations. We consider a case when the structure of the system is a-priori fixed. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recentl...

Bertini real is a command line program for numerically decomposing the real portion of a oneor two-dimensional complex irreducible algebraic set in any reasonable number of variables. Using numerical homotopy continuation to solve a series of polynomial systems via regeneration from a witness set, a set of real vertices is computed, along with connection information and associated homotopy func...