🤖 AI Summary
This work proposes the first fast deterministic algorithm for linear systems with arbitrary rectangular displacement structures—including Toeplitz-like, Vandermonde-like, and Cauchy-like matrices—addressing the limitation of existing fastest methods that rely on randomization and lack deterministic guarantees. The approach reformulates the structured linear system as a univariate polynomial modular equation and solves it via three deterministic steps: computing a vector M-Padé approximant basis, eliminating extraneous variables by solving a linear system over the polynomial ring, and recovering the solution through a synchronized M-Padé approximation. The algorithm deterministically computes both the solution to the linear system and a basis for its nullspace in $\tilde{O}(\alpha^{\omega-1}(m+n))$ operations over the base field, offering broader applicability and stronger reliability than prior randomized methods.
📝 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 $\tilde{O}(α^{ω-1} n)$ for solving linear systems and $\tilde{O}(α^2 n)$ for computing the nullspace. Here $n \times n$ is the size of the square matrix, $α$ is its displacement rank, $ω> 2$ 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 $\tilde{O}(α^{ω-1} (m+n))$ 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-Padé 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-Padé approximation.