Condition numbers of Hankel matrices for exponential weights

Lubinsky, D. S.

doi:10.1016/j.jmaa.2005.03.101

Cited by 6 publications

(5 citation statements)

References 13 publications

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“…The large N behaviour of λ N has been studied in the papers [4,8,9,18,22,24]. See also results in [2,14] about the behaviour of the condition number κ(H N ) = Λ N /λ N , where Λ N denotes the largest eigenvalue of H N . Widom and Wilf [22] found the asymptotic behaviour…”

Section: Introductionmentioning

confidence: 99%

The Smallest Eigenvalue of Hankel Matrices

Berg

Szwarc

2010

Constr Approx

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Let H N = (s n+m ), n, m ≤ N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue λ N of H N . It is proved that λ N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λ N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where λ N is known to be bounded below, we prove that the limit of the n'th smallest eigenvalue of H N for N → ∞ tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

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Section: Introductionmentioning

confidence: 99%

The Smallest Eigenvalue of Hankel Matrices

Berg

Szwarc

2010

Constr Approx

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“…[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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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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“…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%

“…The asymptotic behavior of N for large N has been investigated. [5][6][7][8][9][10][11][12][13][14] Also see Beckermann 15 and Lubinsky, 16 in which the authors have studied the behavior of the condition number ( N ) ∶=…”

Section: Introductionmentioning

confidence: 99%

The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight

Zhu

Emmart

Chen

et al. 2019

Math Methods in App Sciences

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We study the asymptotic behavior of the smallest eigenvalue, λN, of the Hankel (or moments) matrix denoted by HN=()μm+n0≤m,n≤N, with respect to the weight wfalse(xfalse)=xαnormale−xβ,3.0235ptx∈false[0,∞false),3.0235ptα>−1,3.0235ptβ>12. An asymptotic expression of the polynomials orthogonal with w(x) is established. Using this, we obtain the specific asymptotic formulas of λN in this paper. Applying a parallel numerical algorithm, we get a variety of numerical results of λN corresponding to our theoretical calculations.

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Condition numbers of Hankel matrices for exponential weights

Cited by 6 publications

References 13 publications

The Smallest Eigenvalue of Hankel Matrices

The Smallest Eigenvalue of Hankel Matrices

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

The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight

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