Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation

Abstract

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the LDLT decomposition and involves finding a k × k sub-matrix of the inverse of the original N × N Hankel matrix H −1 N . The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitute a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w(x) = e −x β , supported on [0, ∞) and β > 0. Such weight generates Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tend to 0, exponentially fast as N gets large. If β < 1/2, the situation where the classical moment problem is indeterminate, the smallest eigenvalue is bounded from below by a positive number for all N , including infinity. If β = 1/2, it is conjectured that the smallest eigenvalue tends to 0 algebraically, with a precise exponent. The algorithm run on the HPCC producing fantastic match between the theoretical value of 2/π and the numerical result.

The Problem of Moments: A Bunch of Classical Results with Some Novelties

“…and that the positive numerical sequence {λ 1 (H n,p )} ∞ n=1 is decreasing as n → ∞. The smallest eigenvalues of Hankel matrices are fundamentally involved when studying (in)determinacy of measures; see the historical paper by Hamburger [14][15][16] and the more recent works by Chen-Lawrence [17], Berg-Chen-Ismail [13], Berg-Szwarc [18] and Chen-Sikorowski-Zhu [19].…”

An inexact Krylov subspace method for large generalized Hankel eigenproblems

“…A fast eigenvalue algorithm for Hankel matrices was proposed [3] based on the Lanczos-type tridiagonalization and QR-type diagonalization methods. Some studies [4][5][6][7] have focused specifically on the smallest eigenvalue of large scale Hankel matrices. The sensitivity of the nonlinear application [2] mapping the vector of Hankel entries to its generalized eigenvalues was studied.…”

“…The sensitivity of the nonlinear application [2] mapping the vector of Hankel entries to its generalized eigenvalues was studied. The parallel algorithm and asymptotic behavior of the smallest eigenvalue of a Hankel matrix were studied [4][5][6].…”

The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight

“…; as well as the weight function w(x) = e −x β , x ∈ [0, ∞) at the critical point β = 1 2 , see [21]. In this paper, we choose the singularly perturbed Laguerre weight w α (x; t) = x α e −x− t x , x ∈ [0, ∞), α > −1, t ≥ 0, which was motivated in part by an integrable quantum field theory at finite temperature.…”