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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 PublicationApplied Mathematics and Computation
ISSN0096-3003
Volume363
AbstractWe 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.
KeywordExtremely ill-conditioned Hankel matrices Parallel eigensolver Random matrix Smallest eigenvalue
DOI10.1016/j.amc.2019.124628
URLView the original
Language英语
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Cited Times [WOS]:2   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionUniversity of Macau
Corresponding AuthorZhu,Mengkun
Affiliation1.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 AffilicationUniversity 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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