
 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.
 Objectives
The objectives can be summarized in the following items:
 Developing accurate (real) root counts for a system over a (bounded)
domain.
 Extending the homotopy continuation methods from polynomial to nonlinear
systems.
 Developing a workbench where the craft of various techniques can be
combined.
 Solving efficiently interesting systems coming from applications of
practical significance.
 Design and methodology
Each of the objectives will be realized by working according to the design and
methodology described next:
 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.
 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.
 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.
 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.
