A Symbolic-Numerical Branch and Prune Algorithm for Solving Non-linear Polynomial Systems
Laurent Granvilliers (LIFO - Univ. Orléans - IIIA - Rue L. de Vinci BP 6759 -- 45067 ORLÉans Cedex 2, France)
Abstract: This paper discusses the processing of non-linear polynomial systems using a branch and prune algorithm within the framework of constraint programming. We propose a formalism for a kind of branch and prune algorithm implementing symbolic and numerical methods to reduce the systems with respect to a relation defined from both inclusion of variable domains and inclusion of sets of constraints. The second part of the paper presents an instantiation of this general scheme. The pruning step is implemented as a cooperation of factorizations, substitutions and partial computations of Groebner bases to simplify the systems, and interval Newton methods address the numerical, approximate solving. The branching step creates a partition of domains or generates disjunctive constraints from equations in factorized form. Experimental results from a prototype show that interval methods generally benefit from the symbolic processing of the initial constraints.
Keywords: Branch and prune algorithm, Groebner basis, cooperative constraint solvers, interval Newton methods, non-linear constraint solving, polynomial system., symbolic simplification