2018/09/24 by Yang Chen, Chen, Yang, Jakub Sikorowski +3
Chemistry · Computer Science · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Molecular spectroscopy and chirality #Numerical Analysis (math.NA) #Random Matrices and Applications #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1810.01478
openalex publication_date 2018/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 × k sub-matrix of the inverse of the original N × N Hankel matrix HN-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-xβ, 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.