Abstract

The main obstacle to efficient computation of these forms is the potential for excessive growth of intermediate expressions. Most of our work here focuses on the difficult case of polynomial matrices: matrices with entries univariate polynomials having rational number coefficients. One first result is a fast Las Vegas probabilistic algorithm to compute the Smith normal form of a polynomial matrix for those cases where pre- and post-multipliers are not also required. For computing Hermite normal forms of polynomial matrices, and for computing pre- and post-multipliers for the Smith normal form, we give a new sequential deterministic algorithm. We present our algorithms for the special case of square, nonsingular input matrices. Generalizations to the nonsquare and/or singular case are provided via a fast Las Vegas probabilistic preconditioning algorithm that reduces to the square, nonsingular case. In keeping with our main impetus, which is practical computation of these normal forms, we show how to apply homomorphic imaging schemes to avoid computation with large integers and polynomials. Bounds for the running times of our algorithms are given that demonstrate significantly improved complexity results over existing methods. 1 Acknowledgements My first acknowledgment must go to my supervisor George Labahn. In addition to providing vision, funding, and encouragement, George gave me unlimited freedom to explore new avenues for research, yet was always available to discuss new ideas. My thanks also goes to the other members of the Symbolic Computation Group, who provided a relaxed and friendly working environment. Second, I would like to thank the members of my examining committee, William Gilbert and Jeffrey Shallit, for their time spent in reviewing this thesis.

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