Algorithms for computer algebra applied to industrial modelling

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In science, engineering and in industrial modelling, computers can be used both for numeric and symbolic calculations. Symbolic calculation is often called computer algebra, and the Maple program, developed in Waterloo, is one of several systems widely used for such calculations. In this proposal, new or improved algorithms will be developed which will allow computer algebra systems, specifically Maple, to solve mathematical problems more effectively. The research will emphasize mathematical problems known to arise in industry, but more general problems will also be addressed. The first set of problems concern "large-expression management". One of the important differences between symbolic computing and numerical computing is the way computer memory is handled. For some important problems, symbolic computations generate equations that would take very many pages to print if someone chose to try. Not only are the equations too large for a person to comprehend, but even the computer cannot continue to work with them, and all progress slows to a halt. By implementing new data structures and algorithms, these problems can be alleviated. Another set of problems concerns the analysis of the equations of typical industrial models when they include switching behaviour. Handling the discontinuous behaviour in a general algorithmic way requires software development and mathematical development. Other topics of more general mathematical interest include algorithms for solving nonlinear matrix equations, and further investigations of the properties of the Lambert W function.