Research Project
Counting and computing all isolated solutions of systems of nonlinear equations.
(FWO project G.0261.96 )

• Ronald Cools
• Ann Haegemans
• Peter Kravanja
• Tom Michiels
• Jan Verschelde
• Pierre Verlinden

1. Aim
   The aim of this research project is to develop reliable numerical methods and to construct software tools for counting and computing all isolated solutions of nonlinear systems in a certain (bounded) domain.
2. Objectives
   The objectives can be summarized in the following items:
   1. Developing accurate (real) root counts for a system over a (bounded) domain.
   2. Extending the homotopy continuation methods from polynomial to nonlinear systems.
   3. Developing a workbench where the craft of various techniques can be combined.
   4. Solving efficiently interesting systems coming from applications of practical significance.
3. Design and methodology
   Each of the objectives will be realized by working according to the design and methodology described next:
   1. The theoretical background can be found in the theory of `fewnomials'. The founder of this theory, Khovanskii, developed upper bounds for the number of real solutions not only for polynomial systems, but also for nonlinear systems. The bounds take advantage of the sparseness of the systems as not the degrees, but the number of terms is taken into account. By application of polyhedral homotopy continuation, it is expected that the bounds can be improved significantly, which yields an efficient generalized rule of Descartes.
   2. Information about the number and the location of solutions of general nonlinear systems can be computed using numerical integration. This information enables us to construct a polynomial system that has the same solutions as the original more general nonlinear system. Finally, this polynomial system is to be solved using homotopy continuation methods.
   3. Managing large complex information systems can be done by object oriented programming. The choice of Ada provides a very effective way to obtain this goal. The existence of a GNU Ada compiler helps in providing executables for a lot of various computer configurations.
   4. The novel methods will be applied to symmetric problems and other special structured applications. One possible field of applications includes kinematics. The equations stated there contain often trigonometric functions. Efficiency gains are to be expected when these can be solved directly, without transforming them first into a polynomial system.