On matrices with displacement structure: generalized operators and faster algorithms

TL;DR

This paper extends the class of matrices with displacement structure using generalized operators and develops faster algorithms for matrix operations like multiplication and inversion, improving computational efficiency especially for large matrices.

Contribution

It introduces generalized displacement operators based on block diagonal matrices and designs faster algorithms for structured matrix multiplication and inversion.

Findings

Algorithms achieve complexity from $O( ext{alpha}^{ ext{omega}-1} imes ext{M}(n))$ to $O( ext{alpha}^{ ext{omega}-1} imes ext{M}(n) ext{log}(n))$

Faster Las Vegas algorithms for structured inversion and linear system solving

Generalization of classical displacement operators for broader structured matrices

Abstract

For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product $\mathsf{A}\mathsf{B}$, where $\mathsf{A}$ is a structured $n \times n$ matrix of displacement rank $\alpha$, and $\mathsf{B}$ is an arbitrary $n\times\alpha$ matrix. Given $\mathsf{B}$ and a so-called "generator" of $\mathsf{A}$, this product is classically computed with a cost ranging from $O(\alpha^2 \mathscr{M}(n))$ to $O(\alpha^2 \mathscr{M}(n)\log(n))$ arithmetic operations, depending on the type of structure of $\mathsf{A}$; here, $\mathscr{M}$ is a cost function for polynomial multiplication. In this paper, we first generalize classical displacement operators, based on block diagonal matrices with companion diagonal blocks, and then design fast algorithms to perform the task above for this extended class of structured matrices. The cost of these algorithms ranges from $O(\alpha^{\omega-1} \mathscr{M}(n))$ to $O(\alpha^{\omega-1} \mathscr{M}(n)\log(n))$, with $\omega$ such that two $n \times n$ matrices over a field can be multiplied using $O(n^\omega)$ field operations. By combining this result with classical randomized regularization techniques, we obtain faster Las Vegas algorithms for structured inversion and linear system solving.