This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well-known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Segupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

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...

Relative Orientation is the recovery of the position and orientation of one imaging system relative to another from correspondences between five or more ray pairs. It is one of four core problems in photogrammetry and is of central importance in binocular stereo, as well as in long range motion vision. While five ray correspondences are sufficient to yield a finite number of solutions, more than five correspondences are used in practice to ensure an accurate solution using least squares methods. Most iterative schemes for minimizing the sum of squares of weighted errors require a good guess as a starting value. The author has previously published a method that finds the best solution without requiring an initial guess. In this paper an even simpler method is presented that utilizes the representation of rotations by unit quaternions. 1. See also: ``Relative Orientation,'' {\it International Journal of Computer Vision}, Vol.~4, No.~1, pp.~59--78, January 1990.

. Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root is contained in a box in one of the sets, the second set may have several boxes in clusters near a single root. Thus, the output is of higher quality if there are relatively more boxes in the first set. In contrast to previously implemented similar techniques, a box expansion technique in this paper, based on using an approximate root finder, ffl-inflation and exact set complementation, decreases the size of the second set, increases the size of the first set, and never loses roots. In addition to the expansion technique, use of second-order extensions to eliminate small boxes that do not contain roots, and interval slopes versus interval d...

Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (C ) n , with C = C n f0g. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (C ) n . In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with approximations of its solutions. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach. keywords : homotopy continuation, polynomial systems, Newton polytopes, Bernshtein bound, cycle number. AMS(MOS) Classification : 14Q99, 52A39, 52B20, 65H10. 1 Introduction All isolated complex solutions to polynomial systems can be approximated numerically by homotopy continuation methods. The strategy is to set up a collection of implicitly d...

. 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 of a generic system that is used to start up the deformations. Software and applications are discussed. AMS Subject Classification. 14N10, 14M15, 52A39, 52B20, 52B55, 65H10, 68Q40. Keywords. polynomial system, numerical algebraic geometry, homotopy, continuation, deformation, path following, dense, sparse, determinantal, B'ezout bound, Newton polytope, mixed volume, root count, enumerative geometry, numerical Schubert calculus. Contents 1. Introduction 1 2. Three Classes of Polynomial Systems 3 3. The Principles of Polynomial Homotopy Continuation Methods 5 4. The Geometry of the Deformations 8 5. Root Counts and Start Systems 10 5.1. Dense Polynomials modeled by Highest Degrees 10 5.2. Mixed Subdivisions of Newton Polytopes to compute Mixed Vo...

This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed, it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or re-run paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.

This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analysis, such as the interval Newton method. Its key conceptual idea is to introduce the notion of box-consistency, which approximates arc-consistency, a notion well-known in articial intelligence. Box-consistency achieves an eective pruning at a reasonable computation cost and generalizes some traditional interval operators. Newton has been applied to numerous applications in science and engineering, including nonlinear equation-solving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equation-solving benchmarks and outperforms the interval-based methods we are aware of on optimization problems. Key words: Constraint Programming, Nonlinear Programming, Interval Reasoning 1 Introduction Many applications in science and engineering (e.g., chemistry, robotics, economics, mechanics) require nding all isolated solutions to a system of nonlinear real constraints or nding the minimum value of a nonlinear function subject to nonlinear constraints. These problems are dicult due to their inherent computational complexity (i.e., they are NP-hard) and due to the numerical issues involved to guarantee correctness (i.e., nding all solutions or the global optimum) and to ensure termination. Preprint submitted to Elsevier Preprint 11 June 2001 Newton is a constraint programming language designed to support this class of applications. It originates from an attempt to reconcile the declarative nature of CLP(Intervals) languag...

: A new method for determining the real solutions to a set of polynomial equations is presented. It is based on the theory of multi-resultants. The inherently unstable calculation of the determinant is replaced by a stable minimization procedure which is able to take advantage of the sparseness of the resultant matrix. Two numerical examples illustrate the method. The paper contains preliminary work which demonstrates the feasibility of the given approach. Keywords: roots, polynomial systems of equations, resultant, conjugate gradient method, Lanczos method AMSMOS: 65H10, 65H20, 26C10, 65F15, 15-04, 15A15 1 Introduction In recent years, a number of authors have considered the task of numerically determining all of the zero points of polynomial systems of equations. In particular, we mention the resultant method of Collins [5] and the homotopy methods, see, e.g., [3, 4, 10, 17, 21, 22, 28]. Further references can be found in the recent books [1, 20]. Since the calculation of the deter...