Jan Verschelde

Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation

Polynomial systems occur in a wide variety of application domains. Homotopy continuation methods are reliable and powerful methods to compute numerically approximations to all isolated complex solutions. During the last decade considerable progress has been accomplished on exploiting structure in a polynomial system, in particular its sparsity. In this article the structure and design of the software package PHC is described. The main program operates in several modes, is menu driven, and is file oriented. This package features great variety of root-counting methods among its tools. The outline of one black-box solver is sketched, and a report is given on its performance on a large database of test problems. The software has been developed on four different machine architectures. Its portability is ensured by the gnu-ada compiler.

Homotopies exploiting Newton polytopes for solving sparse polynomial systems

This paper is concerned with the problem of finding all isolated solutions of a polynomial system. The BKK bound, defined as the mixed volume of the Newton polytopes of the polynomials in the system, is a sharp upper bound for the number of isolated solutions in

Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

\mathbb{C}_0^n ,\mathbb{C}_0 = \mathbb{C} \backslash \{ 0\}

Newton's method with deflation for isolated singularities of polynomial systems

, of a polynomial system with a sparse monomial structure. First an algorithm is described for computing the BKK bound. Following the lines of Bernshtei˘n’s proof, the algorithmic construction of the cheater’s homotopy or the coefficient homotopy is obtained. The mixed homotopy methods can be combined with the random product start systems based on a generalized Bezout number. Applications illustrate the effectiveness of the new approach.

Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials.

Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems

We present a modification of Newtons method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution as a regular root. Using standard bases, a tool for the symbolic computation of multiplicities, we show that the number of deflation stages is bounded by the multiplicity of the isolated root. Our implementation performs well on a large class of applications.

Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components

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 procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem.

Advances in Polynomial Continuation for Solving Problems in Kinematics

Many polynomial systems have solution sets comprised of multiple irreducible components, possibly of different dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using floating-point numerical processes, into its components. Prior work has shown how to generate sets of generic points guaranteed to include points from every component. Furthermore, we have shown how monodromy can be used to efficiently predict the partition of these points by membership in the components. However, confirmation of this prediction required an expensive procedure of sampling each component to find an interpolating polynomial that vanishes on it. This paper proves theoretically and demonstrates in practice that linear traces suffice for this verification step, which gives great improvement in both computational speed and numerical stability. Moreover, in the case that one may still wish to compute an interpolating polynomial, we show how to do so more efficiently by building a structured grid of samples, using divided differences, and applying symmetric functions. Several test problems illustrate the effectiveness of the new methods.

Mixed-volume computation by dynamic lifting applied to polynomial system solving

To decompose solution sets of polynomial systems into irreducible components, homotopy continuation methods generate the action of a natural monodromy group which partially classifles generic points onto their respective irreducible components. As illustrated by the performance on several test examples, this new method achieves a great increase in speed and accuracy, as well as improved numerical conditioning of the multivariate interpolation problem. 2000 Mathematics Subject Classiflcation. Primary 65H10; Secondary 13P05, 14Q99, 68W30.

Higher-Order Deflation for Polynomial Systems With Isolated Singular Solutions

For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.