Computing the Smith Forms of Integer Matrices and Solving Related Problems

The Smith form of an integer matrix plays an important role in the study of algebraic group theory, homology group theory, systems theory, matrix equivalence, Diophantine systems, and control theory. Asymptotic complexity of the Smith form computation has been steadily improved in the past four decades. A group of algorithms for computing the Smith forms is available now. The best asymptotic algorithm may not always yield the best practical run time. In spite of their different asymptotic complexities, different algorithms are favorable to different matrices in practice. In this thesis, we design an algorithm for the efficient computation of Smith forms of integer matrices. With the computing powers of current computer hardware, it is feasible to compute the Smith forms of integer matrices with dimension in the thousands, even ten thousand. Our new “engineered” algorithm is designed to attempt to combine the best aspects of previously known algorithms to yield the best practical run time for any given matrix over the integers. The adjective “engineered” is used to suggest that the structure of the algorithm is based on both previous experiments and the asymptotic complexity. In this thesis, we also present lots of improvements for solving related problems such as determining the rank of a matrix, computing the minimal and characteristic polynomials of a matrix, and finding the exact rational solution of a non-singular linear system with integer coefficients.