Real root finding for determinants of linear matrices

TL;DR

This paper presents an efficient algorithm for finding points in each connected component of the real determinantal variety defined by the determinant of a linear matrix pencil, with practical implementation and complexity analysis.

Contribution

The paper introduces a new algorithm with nearly quadratic complexity under generic conditions for computing points in each connected component of real determinantal varieties.

Findings

Algorithm performs well in practice, confirming theoretical complexity.

For fixed matrix size, the complexity is polynomial in the number of variables.

Experiments demonstrate the algorithm's efficiency on real-world problems.

Abstract

Let $\A_0, \A_1, \ldots, \A_n$ be given square matrices of size $m$ with rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal variety $\{\X \in\RR^n \: :\: \det(\A_0+x_1\A_1+\cdots+x_n\A_n)=0\}$. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Using standard complexity results this problem can be solved using $m^{O(n)}$ arithmetic operations. Under some genericity assumptions on the coefficients of the matrices, we provide an algorithm solving this problem whose runtime is essentially quadratic in ${{n+m}\choose{n}}^{3}$. We also report on experiments with a computer implementation of this algorithm. Its practical performance illustrates the complexity estimates. In particular, we emphasize that for subfamilies of this problem where $m$ is fixed, the complexity is polynomial in $n$.