# PhD Defence • Symbolic Computation • Fast Algorithms for Computing with Integer Matrices: Normal Forms and Applications

Friday, September 17, 2021 — 10:00 AM EDT

## Please note: This PhD defence will be given online.

Stavros Birmpilis, PhD candidate
David R. Cheriton School of Computer Science

Supervisors: Professors George Labahn, Arne Storjohann

The focus of this thesis is on fundamental computational problems in exact integer linear algebra. Specifically, for a nonsingular integer input matrix A ∈ Zn×n, we consider problems such as linear system solving and computing integer matrix normal forms.

Our goal is to design algorithms that have complexity about the same as the cost of multiplying together two integer matrices of the same dimension and size of entries as the input matrix A. If 2 ≤ ω ≤ 3 is a valid exponent for matrix multiplication, that is, if two n × n matrices can be multiplied in O(nω) basic operations from the domain of entries, then our target complexity is

(nω log ||A||)1+o(1)

bit operations. Here ||A|| = maxij |Aij| denotes the largest entry in absolute value, and the exponent 1 + o(1) indicates some missing log n and loglog ||A|| factors.

The first contribution is solving the problem of computing the Smith normal form S ∈ Zn×n of a nonsingular matrix A ∈ Zn×n along with computing unimodular matrices U, V ∈ Zn×n such that

AV = US

in time (nω log ||A||)1+o(1). The algorithm we give is a Las Vegas probabilistic algorithm which means that we are able to verify the correctness of its output.

The second contribution of the thesis is with respect to linear system solving. We present a deterministic reduction to matrix multiplication for the problem of linear system solving: given as input a nonsingular A ∈ Zn×n and b ∈ Zn×1, compute A−1b. The system solution is computed in (nω log ||A||)1+o(1) bit operations.

To join this PhD defence on Zoom, please go to https://us02web.zoom.us/j/86413126350?pwd=QlJWUzdyVktVeC8zc3drdGlZWlYwdz09.

Location
Online PhD defence
200 University Avenue West

Waterloo, ON N2L 3G1
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