Two nondeterministic positive definiteness tests for unidiagonal integral matrices

Abstract

For standard algorithms verifying positive definiteness of a matrix $A\in\mathbb{M}_n(\mathbb{R})$ based on Sylvester's criterion, the computationally pessimistic case is this when $A$ is positive definite. We present two algorithms realizing the same task for $A\in\mathbb{M}_n(\mathbb{Z})$, for which the case when $A$ is positive definite is the optimistic one. The algorithms have pessimistic computational complexities $\mathcal{O}(n^3)$ and $\mathcal{O}(n^4)$ and they rely on performing certain edge transformations, called inflations, on the edge-bipartite graph (=bigraph) $\Delta=\Delta(A)$ associated with $A$. We provide few variants of the algorithms, including Las Vegas type randomized ones with precisely described maximal number of steps. The algorithms work very well in practice, in many cases with a better speed than the standard tests. Moreover, the algorithms yield some additional information on the properties on the quadratic form $q_A:\mathbb{Z}^n\to\mathbb{Z}$ associated with a matrix $A$. On the other hand, our results provide an interesting example of an application of symbolic computing methods originally developed for different purposes, with a big potential for further generalizations in matrix problems. This is an extended version of the article [A. Mr\'oz, Effective nondeterministic positive definiteness test for unidiagonal integral matrices, Proceedings SYNASC 2016, IEEE Computer Society CPS (2016), 65-71] in which we discussed the algorithm of the complexity $\mathcal{O}(n^4)$.