Efficient symbolic-numeric algorithms for structured constrained differential polynamial systems

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Symbolic Computation is the study of algorithms and software systems to manipulate mathematical expressions. Until recently the algorithms have processed exact expressions and equations only. Driven by the needs of applications, in which data is approximate, the subject has grown to include symbolic-numeric computation for polynomial equations. This is part of a broad effort to combine the generality of exact symbolic computation with the realism and efficiency of numeric computation. Building on my previous results my proposal extends this subject further to efficient symbolic-numeric algorithms for approximate differential polynomial systems. Such systems express fundamental laws of science and applications have yielded ever more complicated systems involving, for example 100's of differential equations for the currents in complex circuits and the motions of components of medical robots. To analyze and solve such complicated models, computers are required at every stage: from formulation to solution. Differentiation of such systems can reveal hidden constraints which are crucial in their geometric and solution properties. The proposal is to create and analyze new algorithms for such systems which enable: the stable and efficient symbolic-numeric algorithmic determination of their constraints; and the computer exploitation of the resulting geometry in the analysis and solution process. Algorithms will also be developed for exploiting the structure of systems typically arising in applications. This builds on powerful results in differential geometry and differential algebra, combined with the new area Numerical Algebraic Geometry, that gives a numerically stable approach for the first time to the simpler subclass of approximate polynomial equations. Existing symbolic techniques can detect global constraint structure but are limited to exact systems; while existing numeric techniques for more realistic approximate systems detect local, but not global structure. The research focuses on symbolic-numeric algorithms for structured systems, including global structure.