Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Abstract

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T n ( f ), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f . A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where f is an s × s matrix-valued trigonometric polynomial with s ≥ 1, and T n (f) is the block Toeplitz matrix generated by f, whose size is N (n, s) = sn. The case s = 1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrixless. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the Q p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

“…According to Remark 2.9, if we have {A n } n ∼ λ f (and hence also {A n } n ∼ λ φ), then, for n large enough, the eigenvalues of A n , with the possible exception of o(d n ) outliers, are approximately equal to the samples of φ over a uniform grid in [0, 1]. Precise error estimates as well as an analysis under suitable assumptions may be produced by following [19] and the references therein, possibly also considering the numerics in [44]. REMARK 2.11 (Canonical rearranged version and quantile function).…”

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

“…The numerical results are extremely precise, even compared with the already good performances described in [1,12,13,14,15], since it is not difficult to reach machine precision, and the complexity is still linear.…”

“…The previous works [8,13,14,15] used (2.3) as the basic expansion. We noticed that (2.3) is absorbing all the almost non-increasining consequences of f at {0, π} and all the related troubles that the derivatives of f can produce.…”

“…In the case where g = 1 identically, Conjecture 1.1 was originally formulated and supported through numerical experiments in [15]. Then the algorithmic proposal was extended and refined in [1,12,13,14]. When g ≡ 1 and l satisfies further technical additional assumptions, those of the simple-loop method, Conjecture 1.1 was formally proved by Bogoya, Böttcher, Grudsky, and Maximenko in a series of papers [5,6,9].…”

A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

“…In [7, pp. 1329], [8], and then [18], an algorithm was proposed to approximate the functions c k (θ ), which was subsequently extended and studied, see [2][3][4][15][16][17], to cover also other types of Toeplitz-like matrices A n , possessing an asymptotic expansion such as (8). We call this type of methods matrix-less, since they do not need to construct the large matrix A n to approximate its eigenvalues; indeed, they approximate the functions c k (θ ) from α small matrices A n 1 , .…”