A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Methods for Solving Systems of Nonlinear Equations
Methods for Solving Systems of Nonlinear Equations
Solving nonlinear polynomial systems in the barycentric Bernstein basis
The Visual Computer: International Journal of Computer Graphics
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Optimal bounding cones of vectors in three dimensions
Information Processing Letters
A New Approach for Solving Nonlinear Equations Systems
IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans
Computer Aided Geometric Design
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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.