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We propose a novel parallel numerical algorithm for calculating the smallesteigenvalues 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 ofthe original $N \times N$ Hankel matrix $H_N^{-1}$ . The computation involvesextremely high precision arithmetic, message passing interface, and sharedmemory parallelisation. We demonstrate that this approach achieves goodscalability on a high performance computing cluster (HPCC) which constitute amajor improvement of the earlier approaches. We use this method to study afamily of Hankel matrices generated by the weight $w(x)={\rm e}^{-x^{\beta}},$supported on $[0,\infty)$ and $\beta>0.$ Such weight generates Hankeldeterminant, a fundamental object in random matrix theory. In the situationwhere $\beta>1/2,$ the smallest eigenvalue tend to 0, exponentially fast as $N$gets large. If $\beta<1/2,$ the situation where the classical moment problem isindeterminate, the smallest eigenvalue is bounded from below by a positivenumber for all $N$, including infinity. If $\beta=1/2,$ it is conjectured thatthe smallest eigenvalue tends to 0 algebraically, with a precise exponent. Thealgorithm run on the HPCC producing fantastic match between the theoreticalvalue of $2/\pi$ and the numerical result.

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