Broughton, Gary John (2013) Symbolic algorithms for the local analysis of systems of pseudo-linear equations. (PhD thesis), Kingston University, .
Abstract
This thesis is concerned with the design and implementation of algorithms in Computer Algebra - a discipline which pursues a symbolic approach to solving mathematical equations and problems in contrast to computing solutions numerically. More precisely, we study sys¬tems of pseudo-linear equations, which unify the classes of linear differential, difference and q-difference systems. Whilst the classical mathematical theory of asymptotic expansions and the notion of formal solutions of this type of solutions are well established for all these indi-vidual cases, no unifying theoretical framework for pseudo-linear systems was known prior to our work. From an algorithmic point of view, the computation of a complete fundamental system of formal solutions is implemented by the formal reduction process. The formal reduction of linear differential systems had been treated in the past, and linear difference systems were also investigated and partly solved. In the case of linear q-difference systems, the structure of the formal solution is much easier which results in an alleviated formal reduction. However, no satisfying algorithm had been published that would be suitable to compute the formal solutions. We place ourselves in the generic setting and show that various algorithms that are known to be building blocks for the formal reduction in the differential case can be extended to the general pseudo-linear setting. In particular, the family of Moser- and super-reduction algorithms as well as the Classical Splitting Lemma and the Generalised Splitting Lemma are amongst the fundamental ingredients that we consider and which are essential for an effective formal reduction procedure. Whereas some of these techniques had been considered and adapted for systems of difference or q-difference equations, our novel contribution is to show that they can be extended and formulated in such a way that they are valid generically. Based on these results, we then design our generic formal reduction method, again in-spired by the differential case. Apart from the resulting unified approach, this also yields a novel approach to the formal reduction of difference and q-difference systems. Together with a generalisation of an efficient algorithm for computing regular formal solutions that was devised for linear differential systems, we finally obtain a complete and generic algorithm for computing formal solutions of systems of pseudo-linear equations. We show that we are able to compute a complete basis of formal solutions of large classes of linear functional systems, using our formal reduction method. The algorithms presented in this thesis have been implemented in the Computer Algebra System Maple as part of the Open Source project ISOLDE.
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