The smallest eigenvalue of large Hankel matrices

Zhu, Mengkun; Chen, Yang; Emmart, Niall; Weems, Charles C.

doi:10.1016/j.amc.2018.04.012

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…Berg and Szwarc [4] proved that λ 1 has exponential decay to zero for any measure which has compact support. Recently, Zhu et al [19] studied the Jacobi weight, w(x) = x α (1 − x) β , x ∈ [0, 1], α > −1, β > −1 and derived a approximation formula of λ 1 , which reduces to Sezgö's result [15] if α = β = 0. This paper is concerned with a numerical computation that is motivated by random matrix theory.…”

Section: Background and Motivationmentioning

confidence: 99%

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Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation

Chen

Sikorowski

Zhu

2019

Applied Mathematics and Computation

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We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned 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 −1 N . 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) = e −x β , 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.

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Section: Background and Motivationmentioning

confidence: 99%

“…The asymptotic behavior of λ 1 for large N has broad interest, see e.g. [3,4,6,7,9,[15][16][17][18][19]. The authors in [2,13] have studied the behavior of the condition number cond (H N ) := λ N λ 1 , where λ N denotes the largest eigenvalue of H N .…”

Section: Background and Motivationmentioning

confidence: 99%

Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation

Chen

Sikorowski

Zhu

2019

Applied Mathematics and Computation

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“…Let λ N denote the smallest eigenvalue of

H_{N}

. The asymptotic behavior of λ N for large N has been investigated . Also see Beckermann and Lubinsky,in which the authors have studied the behavior of the condition number

κ ((), H_{N}) : = \frac{Λ_{N}}{λ_{N}}

, where Λ N denotes the largest eigenvalue of

H_{N}

.…”

Section: Introductionmentioning

confidence: 99%

“…Zhu et al studied the Jacobi weight (or Beta density), ie, w ( x ) = x α (1 − x ) β , x ∈ [0,1], α > −1, β > −1, and derived a approximation formula of λ N ,

λ_{N} ≃ 2^{\frac{15}{4}} π^{\frac{3}{2}} {(1 + 2^{\frac{1}{2}})}^{- 2 α} {(1 + 2^{- \frac{1}{2}})}^{- 2 β} N^{\frac{1}{2}} {(1 + 2^{\frac{1}{2}})}^{- 4 (N + 1)},

which reduces to Sezgö's result, if α = β = 0.…”

Section: Introductionmentioning

confidence: 99%

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