Matrices with displacement structure: a deterministic approach for linear systems and nullspace bases

TL;DR

This paper presents a fast, deterministic method for solving structured linear systems and nullspace bases with displacement structure, improving over randomized algorithms by removing the generic rank profile constraint.

Contribution

It introduces a deterministic approach that efficiently handles classical displacement structures for arbitrary matrices, extending applicability beyond generic rank profile cases.

Findings

Achieves $ ilde{O}( ext{displacement rank}^{ ext{exponent}} (m+n))$ complexity for rectangular matrices.

Provides explicit algorithms for Toeplitz-like, Vandermonde-like, and Cauchy-like matrices.

Reformulates structured linear systems as polynomial modular equations and solves them deterministically.

Abstract

The fastest known algorithms for dealing with structured matrices, in the sense of the displacement rank measure, are randomized. For handling classical displacement structures, they achieve the complexity bounds $\tilde{O}(\alpha^{\omega-1} n)$ for solving linear systems and $\tilde{O}(\alpha^2 n)$ for computing the nullspace. Here $n \times n$ is the size of the square matrix, $\alpha$ is its displacement rank, $\omega > 2$ is a feasible exponent for matrix multiplication, and the notation $\tilde{O}(\cdot)$ counts arithmetic operations in the base field while hiding logarithmic factors. These algorithms rely on an adaptation of Strassen's divide and conquer Gaussian elimination to the context of structured matrices. This approach requires the input matrix to have generic rank profile; this constraint is lifted via pre- and post-multiplications by special matrices generated from random coefficients chosen in a sufficiently large subset of the base field. This work introduces a fast and deterministic approach, which solves both problems within $\tilde{O}(\alpha^{\omega-1} (m+n))$ operations in the base field for an arbitrary rectangular $m \times n$ input matrix. We provide explicit algorithms that instantiate this approach for Toeplitz-like, Vandermonde-like, and Cauchy-like structures. The starting point of the approach is to reformulate a structured linear system as a modular equation on univariate polynomials. Then, a description of all solutions to this equation is found in three steps, all using fast and deterministic operations on polynomial matrices. Specifically, one first computes a basis of solutions to a vector M-Pad\'e approximation problem; then one performs linear system solving over the polynomials to isolate away unwanted unknowns and restrict to those that are actually sought; and finally the latter are found by simultaneous M-Pad\'e approximation.