Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory

TL;DR

This paper introduces an improved matrix-less algorithm for fast eigenvalue computation of Toeplitz matrices, leveraging a change of variable and asymptotic expansion to achieve higher precision and efficiency, especially for extreme eigenvalues.

Contribution

It adapts the simple-loop theory-based algorithm with a variable change, enhancing accuracy and performance in eigenvalue computations of Toeplitz matrices.

Findings

Higher precision up to machine accuracy achieved.

Same linear computational complexity maintained.

Significant improvement in computing extreme eigenvalues.

Abstract

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function $f$. Independently and under the milder hypothesis that $f$ is even and monotonic over $[0,\pi]$, matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is extremely better for the computation of the extreme eigenvalues.