Global Solutions of Well-Constrained Transcendental Systems Using Expression Trees and a Single Solution Test

Abstract

We present an algorithm which is capable of globally solving a well-constrained transcendental system over some sub-domain \(D\subset \mathbb R^n\), isolating all roots. Such a system consists of n unknowns and n regular functions, where each may contain non-algebraic (transcendental) functions like sin, \(\exp\) or log. Every equation is considered as a hyper-surface in \(\mathbb R^n\) and thus a bounding cone of its normal field can be defined over a small enough sub-domain of D. A simple test that checks the mutual configuration of these bounding cones is used that, if satisfied, guarantees at most one zero exists within the given domain. Numerical methods are then used to trace the zero. If the test fails, the domain is subdivided. Every equation is handled as an expression tree, with polynomial functions at the leaves, prescribing the domain. The tree is processed from its leaves, for which simple bounding cones are constructed, to its root, which allows to efficiently build a final bounding cone of the normal field of the whole expression. The algorithm is demonstrated on curve-curve and curve-surface intersection problems.KeywordsGlobal SolutionNormal ConePolynomial SystemInterval ArithmeticTranscendental FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.