I am co-founder of the Maple research project in the Symbolic
Computation Group at the University of Waterloo and co-founder
of the software company Maplesoft.
Contributions of code which has been incorporated into the Maple computer
algebra system represents my most significant contribution to research and
to practical applications.
Documentation for this work is represented by
the Maple reference books:
The Maple system has achieved an influential position worldwide as a research and teaching tool in engineering, mathematics, and science.
For some presentations on the history of the Maple project, see Maple Archives.
This book develops the mathematical foundations for the fundamental algorithms used by computer algebra systems.
Two papers published in 1983 and 1984 set out the design criteria for Maple and present some performance measurements.
This work exploits the increased problem-solving power which can be realized by a hybrid approach combining traditional numerical computation with automated symbolic analysis.
The latter paper is derived from Tom Robinson's Masters thesis.
The following paper, joint with Masters student Wei Wei Zheng, considers numerical computations where a very high precision result is desired. We show that, rather than performing the complete computation in high precision, it can be significantly more efficient to exploit the speed of hardware precision and to use an iterative refinement scheme to build up the desired high precision result.
In the following paper, we develop a new hybrid symbolic-numeric method for the fast and accurate evaluation of multiple integrals. The new method is shown to be effective both in high dimensions and with high accuracy.
This work exploits the theory of tensor product series developed by Frederick Chapman in his 2003 Ph.D thesis, and the paper is derived from Orlando Carvajal's Masters thesis.
Work on this topic aims to advance the algorithmic knowledge incorporated into computer algebra systems for computing closed-form solutions of indefinite and definite integrals.
In the above-referenced textbook, I present a detailed development of the Risch integration algorithm for computing indefinite integrals (anti-derivatives) for elementary functions.
The following paper, joint with Masters student L.Y. Stefanus, studies a variation of the Risch algorithm known as Risch-Norman (or the parallel Risch algorithm).
Below are three selected papers dealing with various aspects of the integration problem. Note that for the case of a definite integral when the corresponding indefinite integral is not known in closed form, the technique of "differentiation under the integral sign" leads to a solution for some classes of integrands.
In joint work with Ph.D student Ha Le, Sergei Abramov of Moscow State University, and Jacques Carette of Maplesoft (now at McMaster University), we develop some new algorithms and a comprehensive library of Maple routines for computing closed-form solutions of summation problems.
The following paper, joint with former Ph.D student Frederick Chapman, presents an algorithm for generating and proving various mathematical function identities.
F.W. Chapman and K.O. Geddes, An improved algorithm for the automatic derivation and proof of tensor product identities via computer algebra. Proceedings of Calculemus'06 (Genoa, Italy, Jul 2006), Anna Bigatti and Silvio Ranise (ed.), 2006, pp. 52—67. (To appear in Electronic Notes in Theoretical Computer Science, Elsevier B.V., Amsterdam, 2007.)
The following two papers present the design of some algorithms for polynomial GCD computation based on probabilistic approaches. The significance of these algorithms is that their practical implementations achieve a significant speedup compared with traditional algorithms for polynomial GCD computation.
Ph.D student Trevor Smedley wrote a thesis on the latter topic.
The problem of computing limits of mathematical functions is the topic of the following paper, which presents a new algorithm based on hierarchical series. (For more recent results on this topic, see the work of Bruno Salvy and his co-authors.)
The following two journal papers deal with approximation by rational functions, specifically Padé and Chebyshev-Padé approximants.
K.O. Geddes, Symbolic computation of Padé approximants.
ACM Trans. Math. Software, 5 (2),
Jun 1979, pp. 218—233.
In the following two journal papers, the topic is near-minimax polynomial approximation in regions of the complex plane.
K.O. Geddes and J.C. Mason, Polynomial approximation by projections on the unit circle. SIAM J. Numer. Anal., 12 (1), Mar 1975, pp. 111—120.