Eigenvalue superposition expansion for Toeplitz matrix-sequences, generated by linear combinations of matrix-order dependent symbols, and applications to fast eigenvalue computations

TL;DR

This paper extends the asymptotic eigenvalue expansion for Toeplitz matrices to linear combinations of symbols, enabling faster eigenvalue computations for large matrices arising in differential equations and numerical analysis.

Contribution

It proves that the eigenvalue expansion holds for superpositions of symbols under mild conditions, facilitating efficient eigenvalue algorithms for large Toeplitz matrices.

Findings

Asymptotic expansion applies to linear combinations of Toeplitz matrix symbols.

Numerical evidence suggests broader applicability and potential for fast eigenvalue algorithms.

Relevance to spectral analysis in numerical methods for differential equations.

Abstract

The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[ \lambda_{j}(T_{n}(f))=f(d_{j,n})+c_{1}(d_{j,n})h+c_{2}(d_{j,n})h^{2}+O(h^{3}), \] where $h=\frac{1}{n+1}$, $d_{j,n}=\pi j h$, and $c_k$ are some bounded coefficients depending only on $f$. The numerical results presented in the literature suggests that the effective conditions for the expansion to hold are weaker and reduce to an even character of $f$, to a fixed smoothness, and to its monotonicity over $[0,\pi]$. \\ In this note we investigate the superposition caused over this expansion, when considering a linear combination of symbols that is \[ \lambda_{j}\big(T_{n}(f_0)+\beta_{n}^{(1)} T_{n}(f_{1}) + \beta_{n}^{(2)} T_{n}(f_{2}) +\cdots\big), \] where $ \beta_{n}^{(t)}=o\big(\beta_{n}^{(s)}\big)$ if $t>s$ and the symbols $f_{j}$ are either simple loop or satisfy the weaker conditions mentioned before. We prove that the asymptotic expansion holds also in this setting under mild assumptions and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type. The problem is of concrete interest in particular in the case where the coefficients of the linear combination are functions of $h$, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, Isogeometric Analysis etc.