Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation | |
Chen，Yang1; Sikorowski，J.1; Zhu，Mengkun2 | |
2019-12-15 | |
Source Publication | Applied Mathematics and Computation |
ISSN | 0096-3003 |
Volume | 363 |
Abstract | We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned Hankel 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 . 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 constitutes 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, supported on [0, ∞) and β > 0. Such weight generates a Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tends to 0 exponentially fast. If β < 1/2, which is the situation where the classical moment problem is indeterminate, then the smallest eigenvalue is bounded from below by a positive number. 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 a fantastic match between the theoretical value of 2/π and the numerical result. |
Keyword | Extremely ill-conditioned Hankel matrices Parallel eigensolver Random matrix Smallest eigenvalue |
DOI | 10.1016/j.amc.2019.124628 |
URL | View the original |
Language | 英语 |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | University of Macau |
Corresponding Author | Zhu，Mengkun |
Affiliation | 1.Department of Mathematics,University of Macau,Taipa,Avenida da Universidade,Macao 2.School of Mathematics and Statistics,Qilu University of Technology (Shandong Academy of Sciences),Jinan,250353,China |
First Author Affilication | University of Macau |
Recommended Citation GB/T 7714 | Chen，Yang,Sikorowski，J.,Zhu，Mengkun. Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation[J]. Applied Mathematics and Computation,2019,363. |
APA | Chen，Yang,Sikorowski，J.,&Zhu，Mengkun.(2019).Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation.Applied Mathematics and Computation,363. |
MLA | Chen，Yang,et al."Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation".Applied Mathematics and Computation 363(2019). |
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