Smallest Eigenvalue of Large Hankel Matrices at Critical Point: Comparing Conjecture With Parallelised Computation

TL;DR

This paper introduces a parallel numerical algorithm to compute the smallest eigenvalues of large, ill-conditioned Hankel matrices, confirming conjectures about their asymptotic behavior with high precision computations.

Contribution

A novel parallel algorithm based on LDLT decomposition for eigenvalue computation of large Hankel matrices, enabling high-precision analysis and validation of theoretical conjectures.

Findings

Algorithm achieves good scalability on high performance clusters.

Numerical results match theoretical predictions for eigenvalue behavior.

Confirmed conjecture about eigenvalue tending to zero for specific matrix families.

Abstract

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the {\it LDLT} decomposition and involves finding a $k \times k$ sub-matrix of the inverse of the original $N \times N$ Hankel matrix $H_N^{-1}$ . 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 $w(x)={\rm e}^{-x^{\beta}},$ supported on $[0,\infty)$ and $\beta>0.$ Such weight generates Hankel determinant, a fundamental object in random matrix theory. In the situation where $\beta>1/2,$ the smallest eigenvalue tend to 0, exponentially fast as $N$ gets large. If $\beta<1/2,$ 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 $\beta=1/2,$ 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 $2/\pi$ and the numerical result.