arXiv:1803.11322v1 [math-ph] 30 Mar 2018
The Smallest Eigenvalue of Large Hankel Matrices Generated by a Deformed Laguerre Weight Mengkun Zhu∗1 , Niall Emmart†2 , Yang Chen‡1 and Charles Weems§2 1
Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China 2 College of Information and Computer Sciences, University of Massachusetts, Amherst, MA 01003, USA April 2, 2018
Abstract We study the asymptotic behavior of the smallest eigenvalue, λN , of the Hankel (or moments) matrix denoted by HN = (µm+n )0≤m,n≤N , with respect β
to the weight w(x) = xα e−x , x ∈ [0, ∞), α > −1, β > 12 . Based on the research by Szeg¨ o, Chen, etc., we obtain an asymptotic expression of the orthonormal polynomials PN (z) as N → ∞, associated with w(x). Using this, we obtain the specific asymptotic formulas of λN in this paper. Applying the parallel algorithm discovered by Emmart, Chen and Weems, we get a variety of numerical results of λN corresponding to our theoretical calculations.
1
Introduction
Random matrix theory (RMT) originated in multivariate statistics in the work of Hsu, Wishart and others in the 1930s (see the monograph [23]). In 1950s, Wigner put forward similar models for the regularity observed in the energy level distribution of heavy nuclei, where the energy levels are the eigenvalues of large random matrices. From the 1960s to 1970s, through the fundamental work of Dyson, Mehta, Gaudin, des Cloizeaux, Widom, Tracy, Wilf and others, RMT developed into a branch of Mathematical Physics. Its rapid development from the 1990s is due a string of fundamental discoveries of Tracy and Widom on the probability laws ∗
Zhu
[email protected] [email protected] ‡
[email protected] §
[email protected] †
1
governing the largest and smallest eigenvalues of two families of Hermitian random matrices, the Gaussian Unitary Ensembles (GUE) and the Laguerre Unitary Ensembles (LUE). RMT plays an important role in many diverse fields, multivariate statistics, quantum physics, Multi-Input-Multi-Output (MIMO) wireless communication, and stock movements in financial markets, etc. For a variety of theories and applications of RMT, see [2, 6, 7, 14–16, 22, 24, 25] and related references therein. RMT considers the properties, e.g. determinants, eigenvalues, eigenvalue distributions, eigenvectors, spectra, inverse, etc., of matrices whose elements are random variables chosen from a given distribution. The analysis of Hankel matrices, occurs naturally in moment problems, which plays an important role in RMT. On moment problems, please see the monographs by Akhiezer [1] and by Krein [20]. The study of the largest and smallest eigenvalues are important since they provide useful information about the nature of the Hankel matrix generated by a given weight function, e.g. they are related with the inversion of Hankel matrices, where the condition numbers are enormously large. Given {µk } the moment sequence of a weight function w(x)(> 0) with infinite support s, Z (1.1) µk := xk w(x)dx, k = 0, 1, 2, . . . , s
the Hankel matrices, it is known that HN := (µm+n )N m,n=0 , N = 0, 1, 2, . . .
(1.2)
are positive definite, see [19]. Let λN denote the smallest eigenvalue of HN . The asymptotic behavior of λN for large N has been investigated in [5, 8, 12, 13, 17, 26, 28–31]. Also see [4, 21], in which the authors have studied the behavior of the condition number κ (HN ) := ΛλNN , where ΛN denotes the largest eigenvalue of HN . Szeg¨o [26] studied the asymptotic behavior of λN for the Hermite (or Gaussion) 2 weight (w(x) = e−x , x ∈ R) and the Laguerre weight (w(x) = e−x , x ≥ 0). He found∗ √ 1 λN ' AN 4 B N , where A, B are certain constants, satisfying 0 < A, 0 < B < 1. Also, Szeg¨o [26] showed that the largest eigenvalue ΛN corresponding to the Hankel matrih iN N 1 π ces i+j+1 , Γ i+j+1 and [Γ(i + j + 1)]N i,j=0 were approximated by 2 , 2 i,j=0 i,j=0 Γ N + 21 and (2N )! respectively. In [29], Widom and Wilf investigated the case where w(x) is supported in a compact interval [a, b], such that the Szeg¨o condition Z a ∗
b
ln w(x) p dx > −∞, (b − x)(x − a)
In all of this paper, aN ' bN means limN →∞ aN /bN =1.
2
(1.3)
holds, then they obtained
√ λN ' A N B N .
Chen and Lawrence [12] found the asymptotic behavior of λN with the weight β function w(x) = e−x , x ∈ [0, ∞), β > 12 . Berg, Chen and Ismail [5] proved that the moment sequence (1.1) is determinate iff λN → 0 as N → ∞. This is a new criteria for the determinacy of the Hamburger moment problem. Also, in the same paper, they obtained a lower bound of λN for large N . In [13], Chen and α Lubinsky obtained the behavior of λN when w(x) = e−|x| , x ∈ R, α > 1. Berg and Szwarc [8] proved that λN has exponential decay to zero for any measure which with compact support. Zhu, Chen, Emmart and Weems [31] studied the Jacobi case, i.e. w(x) = α x (1 − x)β , x ∈ [0, 1], α > −1, β > −1 and provided a asymptotic behavior of λN , 15 4
λN ' 2 π
3 2
1+2
1 2
−2α
1+2
− 12
−2β
N
1 2
1+2
1 2
−4(N +1)
,
which reduces to Sezg¨o’s result [26], if α = β = 0. The examples above show that the values of λN , N → ∞ are exponentially small, and the asymptotic behavior of λN depends on the w(x) in a non-trivial way. We are motivated by this phenomenon and the purpose of this paper is again to study the asymptotic behavior of λN , here we choose an generalised Laguerre β weight w(x) = xα e−x , x ∈ [0, ∞), α > −1, β > 12 . The remainder of this paper is organized in 5 sections. In section 2 we reproduce some known results (Refs. [5, 12, 13, 26], etc.) that will be applied to find the estimation of λN . In section 3, by adopting a previous result [11], we obtain the asymptotic formula for the polynomials orthonormal with respect to w(x) = β xα e−x , x ∈ [0, ∞), α > −1, β > 21 , which is then employed in sections 4 and 5 for the determination of the large N behavior of λN . And finally, in section 6, we present a comparison of the theoretical results to numeric calculations for the smallest eigenvalue, for various values of α, β and N . The numerical computations were performed using the parallel algorithms developed in [17].
2
Preliminaries Consider the weight 1 β w(x) := xα e−x , x ∈ [0, ∞), α > −1, β > , 2
in this case, the moments are Z ∞ 1 1+α+n n µn := x w(x)dx = Γ , β β 0 and the positive Hankel matrix is HN := (µm+n )N m,n=0 . 3
The focus of this paper is to derive the asymptotic behavior of the smallest eigenvalue λN of HN . It is well known that the smallest eigenvalue λN can be found using the classical Rayleigh quotient ( PN ) x µ x m,n=0 m m+n n T λN = min X := (x0 , x1 , . . . , xN ) ∈ CN +1 \ {0} . (2.1) PN 2 n=0 |xn | Let PN be the orthogonal polynomials associated with w(x), and denote by PN (z) :=
N X
xn z n ,
n=0
then ∞
Z
2
|PN (x)| w(x)dx = 0
N X
xm µm+n xn ,
(2.2)
m,n=0
If we denote the orthonomal polynomials associated with the weight w(x) by PN (x), through p PN (z) = hN PN (z), where hN is the square of the L2 norm of PN (z), such that Z ∞ |PN (x)|2 w(x)dx = 1,
(2.3)
0
then the expression for λN , (2.1), can be recast as ( ) 2π λN = min R π . iθ )|2 dθ |P (e N −π
(2.4)
If we define PN (z) :=
N X
Z ξn Pn (z),
π
and Kmn :=
Pm eiθ Pn e−iθ dθ,
−π
n=0
we can see that Z
π
N X PN eiθ 2 dθ = ξ m Kmn ξn ,
−π
m,n=0
Hence, the formula (2.4) will be equivalent to ( ) N X 2π λN = min PN : |ξn |2 = 1 . m,n=0 ξ m Kmn ξn n=0 Based on the Cauchy-Schwarz inequality, we will find that 4
(2.5)
N X m,n=0
ξ m Kmn ξn ≤
N X
1
1
2 2 Kmm Knn |ξm | |ξn | ≤
m,n=0
N X
Kmm ·
m=0
N X n=0
|ξn |2 ≤
N X
Knn .
n=0
Therefore, a lower bound for the smallest eigenvalue of λN is given by 2π λN ≥ PN . n=0 Knn
3
(2.6)
The orthonomal polynomials with respect to β the weight w(x) = xα e−x .
The purpose of this section is to find the asymptotics of the orthonomal polyβ nomials {PN (z)} with respect to the weight w(x) = xα e−x , x ∈ [0, ∞), α > −1, β > 21 . Based on the Coulomb fluid linear statistics method, it has been proved in [11], for N → ∞, that the monic orthogonal polynomials PN (z) associated with w(x) = e−v(x) can be approximated by PN (z) ' exp [−S1 (z) − S2 (z)] ,
(3.1)
where " √ 2 # √ z−a− z−b 1 16(z − a)(z − b) √ , z∈ / [a, b], S1 (z) = ln √ 4 (b − a)2 z−a+ z−b √ √ 2 z−a+ z−b S2 (z) = −N ln 2 "p # Z b (z − a)(z − b) 1 v(x) p + + 1 dx, z ∈ / [a, b]. 2π a x−z (b − x)(x − a) Chen and his co-authors [9] also gave an equivalent representation for S1 : " 14 14 # 1 z − b z − a + , z∈ / [a, b]. e−S1 (z) = 2 z−a z−b Consequently, we have, Theorem 3.1. For N → ∞, the orthonomal polynomials associated with the weight β w(x) = xα e−x , x ∈ [0, ∞), α > −1, β > 21 are approximated by √ √ (−1)N exp −I(z) + (2N + 1 + α) log η + η + 1 −α PN (z) ' (−z) 2 √ , 1 2πb [η(1 + η)] 4 5
with 2N + α p (−z)β 3 I(z) : = − · sec(πβ) η(1 + η) · 2 F1 1, 1 − β; − β; −η − 2β − 1 2 2 (3.2) r 2N + α η 1 1 =− · 2 F1 1, ; 1 + β; , 2β 1+η 2 1+η where z ∈ / [0, b] and η := − zb , whilst b := C(2N + α)
1 β
and
Γ(β + 1)Γ(β) C = C(β) := 4 Γ(2β + 1)
β1 .
Proof. For our problem, a = 0, whilst b(N, α, β) follows from the supplementary condition [10, 11] Z b xv 0 (x) p = 2πN, (b − x)(x − a) a where v(x) = − ln w(x) = −α ln x + xβ . Hence we have b = C(2N + α)
1 β
Γ(β + 1)Γ(β) with C := 4 Γ(2β + 1)
β1 .
Let η := − zb , z ∈ / [0, b], by taking the branch −bη = bηeiπ , −bη − b = b(1 + η)eiπ we have h p √ i − 14 −1 −S1 (z) = ln 2 · (η(η + 1)) η+1+ η , "p # √ √ 2 Z b z(z − b) −α ln x + xβ −bη + −bη − b 1 p −S2 (z) = N ln − + 1 dx 2 2π 0 x−z (b − x)x 2 √ √ −b η + η + 1 = N ln − f (z) − K, 4 where Z b 1 −α ln x + xβ α 2N + α p K := dx = − ln b + α ln 2 + , 2π 0 2 2β (b − x)x and f (z) is defined by p Z z(z − b) b −α ln x + xβ p dx, z ∈ / [0, b]. f (z) := 2π x(b − x)(x − z) 0
(3.3)
Next, we will focus on the explicit formula of f (z). From (3.3), we have f (z) := I1 (z) + I(z) p p Z Z α z(z − b) b z(z − b) b ln x xβ p p dx + dx. =: − 2π 2π x(b − x)(x − z) x(b − x)(x − z) 0 0 6
With the aid of the integral identities in the Appendix, we get p Z p α z(z − b) b ln x α √ p I1 (z) = − dx = ln(−z) − α ln η+1+ η . 2π 2 x(b − x) 0 (x − z) From the definition and basic properties of the Hypergeometric function [18], p Z z(z − b) 1 (by)β p I(z) = bdy 2π by(b − by) 0 (by − z) r z(z − b) bβ Γ 21 + β 1 1 =− · · · 2 F1 1, + β; 1 + β; − π 2z Γ(1 + β) 2 η p (−z)β 2N + α 3 · sec(πβ) =− η(1 + η) · 2 F1 1, 1 − β; − β; −η − 2β − 1 2 2 r 2N + α η 1 1 =− · 2 F1 1, ; 1 + β; . 2β 1+η 2 1+η Consequently, by (3.1), the monic orthogonal polynomials can be obtained as follows: i h 2N +α N √ 2N +α+1 exp − 2β − I(z) p b (−1)N · η+ 1+η PN (z) ' α+1 · · . 2α+1 1 2 4 η 4 (1 + η) 4 Thus the orthonomal polynomials PN (z) of Theorem 3.1 can be obtained using the standard method, stated as the below Lemma. Lemma 3.1. [11] The orthonomal polynomials PN (z) with respect to the weight w(x), i.e. Z b [PN (x)]2 w(x)dx = 1, a
can be given by: s PN (z) = where Z A := 2 a
b
2 A exp PN (z), π(b − a) 2
v(x)dx 2π
p
(b − x)(x − a)
− 2N log
b−a 4
,
and the orthogonal polynomials PN (z) is approximated by (3.1). Remark 3.1. Apparently, the first representation in (3.2) is more convenient for sufficiently large N , where |η| 1. However, it cannot be used for β = N + 12 , N = 1, 2, . . . by the nature of the Hypergeometric function, that is why the second expression in (3.2) is needed.
7
To make further progress, we will be continuing to simplify the representation of PN (z). Using the inverse hyperbolic sine and the formula in [18] (cf. 9.121. 26), the following identity holds √ p 1 1 3 √ √ log η + η + 1 = arcsinh η = η · 2 F1 , ; ; −η . (3.4) 2 2 2 According to this, if we denote E[β − 12 ]∗ by Eβ , we have Lemma 3.2. The asymptotic expression of the polynomials for z ∈ / [0, ∞), |η| 1, is, Eβ N 41 β X 1 (−1) η (−z) PN (z) ' p · exp −I(z) + √ β (−1)k · ak · η k−β+ 2 , (3.5) 2 πC k=0 2π(−z)α+1 where I(z) is given in (3.2) and ak is defined as Γ k + 12 ak := . k + 12 Γ (k + 1)
(3.6)
Proof. By (3.4), we find √ p 1 1 3 √ η + 1 + η ' (2N + α) η · 2 F1 , ; ; −η (2N + α + 1) log 2 2 2 ∞ (−z)β 1 √ X 21 k 21 k (−η)k = η· 3 ηβ C β k! 2 k k=0 Eβ Γ k + 21 (−z)β X k k−β+ 12 ' √ β (−1) · , · η 2 πC k=0 k + 12 Γ (k + 1) where, the Pochhammer symbol (also called the shifted factorial) reads (x)k :=
Γ (k + x) = x(x + 1) · · · (x + k − 1). Γ(x)
Hence the Lemma 3.2 is obtained immediately. In sections 4 and 5, we will follow the techniques of [26] and [12] to show that using an appropriate selection of vectors {ξm }, that the lower bound given by (2.6) is actually an asymptotic estimate of λN for sufficiently large N P.NTaking full advantage of the Laplace method, we can obtain an estimation of n=0 Knn . Consequently, the asymptotic behavior of λN follows. As mentioned in the Remark 3.1, our problem will be discussed in two different cases. ∗
Throughout this paper, E[x] denotes the integer part of x.
8
4
The approximation of λN for β 6= n+ 12 , n ∈ {1, 2, 3, . . .}
P 1 To find the asymptotic estimate of N n=0 Knn for β 6= n + 2 , n ∈ {1, 2, 3, . . .}, we will first deal with the term I(z) in (3.5) by using the first form in equation (3.2). Lemma 4.1. For β 6= n + 21 , n ∈ {1, 2, 3, . . .}, we have h i 1 β 1 −4 (−z) (−1)N · C (2N + α) β · exp sec πβ PN (z) ' q 2 α+ 21 2π(−z) 1 Eβ 1 1− 2β k+ X (−z) 2 (2N + α) √ · exp (−1)k Ak k , 1 2 πC k=0 C (2N + α) β
(4.1)
here Γ 21 − β bk Ak := ak + 2Γ(1 − β)
with
k X Γ j − 12 Γ (k − j + 1 − β) , bk := 3 Γ (j + 1) Γ k − j + − β 2 j=0
where ak is same as that given in (3.6). Specially, √ 4 πβ A0 = . 2β − 1
(4.2)
Proof. For |η| < 1, the hypergeometric function 2 F1 1, 1 − β; 23 − β; −η has the below series expansion ∞ Γ 32 − β X Γ (k + 1 − β) k 3 η . − β; −η = (−1)k 2 F1 1, 1 − β; 2 Γ (1 − β) k=0 Γ k + 23 − β Applying the formula ( [18], p1015)
then for |η| < 1,
√
2 F1
(−n, β; β; −z) = (1 + z)n ,
1 + η may be written as p
∞ 1 X Γ k − 1 2 1+η = (−1)k ηk . Γ (k + 1) Γ − 12 k=0
(4.3)
So as η → 0, the expansion for I(z) is Eβ β Γ 12 − β X 1 1 −z (−z)β I(z) ' − √ (−1)k bk · η k−β+ 2 − sec πβ, Γ(1 − β) k=0 2 4 π C where
(4.4)
k X Γ j − 21 Γ (k − j + 1 − β) . bk := 3 Γ (j + 1) Γ k − j + − β 2 j=0 1
Substituting (4.4) into (3.5), and bear in mind η = −zC −1 (2N + α)− β , then the Lemma 4.1 follows. 9
√ Remark 4.1. Letting α = 0, β = 1, we find C = 2 and A0 = 4 π. Consequently, the classical result for Laguerre polynomials due to Perron [27] is recovered, i hz √ (−1)N − 14 + 2 −zN , z ∈ PN (z) ' √ (−zN ) exp / [0, ∞). 2 2 π Remark 4.2. The Laplace method [3] gives, s Z b
f (t)e−λg(t) dt ' e−λg(c) f (c)
a
2π , λg 00 (c)
as λ → ∞,
where g assumes a strict minimum over [a, b] at an interior critical point c, such that g 0 (c) = 0,
g 00 (c) > 0
f (c) 6= 0.
and
An alternative expression for Laplace method may be stated as: If for x ∈ [a, b], the real continuous function g(x) has as its maximum the value g(b), then as N → ∞ Z b f (b)eN g(b) . (4.5) f (x)eN g(x) dx ' N g 0 (b) a Theorem 4.1. For β 6= n + 12 , n ∈ {1, 2, 3, . . .}, the smallest eigenvalue λN of the HN can be approximated by λN ' 5
5
1
1
1
1
2 2 π 4 C − 4 A02 (2N + α) 2 − 4β exp − sec πβ −
1 1− 2β
(2N + α) √ πC
Eβ X k Ak (−1)k k (2N + α)− β , C k=0 (4.6)
where
Γ(β + 1)Γ(β) C := 4 Γ(2β + 1)
β1
Γ 12 − β , Ak := ak + bk , 2Γ(1 − β)
1 β> , 2
with k X Γ j − 21 Γ (k − j + 1 − β) . bk := Γ (j + 1) Γ k − j + 23 − β j=0
Γ k + 12 ak := , k + 12 Γ (k + 1)
Proof. Note that A0 > 0 for β > 12 by (4.2), so the essential contribution to Kµν comes from a small neighborhood of z = −1 as µ → ∞ and ν → ∞. Let ω > 0 be a fixed number and restrict the values of µ and ν to satisfy 1
N − ωN 2β ≤ µ, ν ≤ N, N → ∞, 10
(4.7)
thus we have
Z
ε
Pµ −eiθ Pν −e−iθ dθ.
Kµν '
(4.8)
−ε
Expanding the integrand for |θ| 1, we obtain Kµν
" Eβ Z ε 1 (−1)µ+ν esec πβ 1 X Ak − 2β √ ' exp √ · (2N + α) · (−1)k k C 2π C 2 πC k=0 −ε 1 1 k k (2k + 1)2 θ2 · 1− · (2µ + α)1− 2β − β + (2ν + α)1− 2β − β 8 # 1 k 1 k (2k + 1)iθ + (2µ + α)1− 2β − β − (2ν + α)1− 2β − β dθ. 2 1
1
k
(4.9)
k
Note that (2µ + α)1− 2β − β − (2ν + α)1− 2β − β remains bounded because of restricting µ and ν as in (4.7), so we can get rid of the linear term in (4.9) for θ 1. As mentioned above, contributions to the integral (4.9) from (−∞, ε) and (ε, ∞) are small enough compared with those from [−ε, ε] as µ → ∞ and ν → ∞. Therefore, we can extend the integration interval to R but without affecting the approximation of Kµν . Using the Laplace method given by Remark 4.2, we obtain r 1 1 2 (−1)µ+ν (2N + α)− 2 − 4β esec πβ Kµν ' 1 A0 (πC) 4 (4.10) Eβ X 1 k 1 k 1 A k 1− − 1− − k · exp √ (−1) k (2µ + α) 2β β + (2ν + α) 2β β . C 2 πC k=0 Observing (4.10), we can find that when µ and ν satisfied (4.7) and large enough, 1
1
2 2 Kµν ' (−1)µ+ν Kµµ Kνν .
(4.11)
Using the approach of [26] and [12] with the following choices of {ξν }, allows us to determine the asymptotic behavior of λN for large N , 1 ν 2 , if N0 ≤ ν ≤ N, (−1) σK νν ξν = h i 1 0, if ν < N0 := E N − ωN 2β , and the positive number σ is determined by the condition N X
2
|ξν | = σ
ν=0
2
N X
Kνν = 1.
(4.12)
ν=N0
It follows from (4.11) and (4.12) that N X µ,ν=0
Kµν ξµ ξ ν =
N X
µ+ν
(−1)
2
1 2
1 2
σ Kµν Kµµ Kνν ' σ
2
N X ν=N0
µ,ν=N0
!2 Kνν
=
N X
Kνν .
ν=N0
(4.13) 11
This means the minimum value in equation (2.5) can be approximated by (2.6), following (4.13), because of the arbitrariness of ω, i.e. 2π λN ' PN . K νν ν=0 It follows that λN ' R N 0
2π Kνν dν
.
(4.14)
Substituting (4.10) into (4.14), with a simple calculation by applying the Laplace method, see Remark 4.2, then the asymptotic behavior of λN , for β 6= n + 21 , n ∈ {1, 2, 3, . . .}, is obtained. Example 4.1. If we take α = − 21 , β = 74 , then 5 145 1 7 1 √ 2N − 5 5 1 1 − 2 √ λN ' 2 2 π 4 C 4 A02 2N − exp − 2 − 2 πC where C = 4
Γ( 11 Γ 7 4 ) (4) Γ(
9 2
)
74 , A0 =
√ 14 π 5
and A1 =
− 47 ! A1 1 , A0 − 2N − C 2
√ 7 π . 3
Corollary 4.1. For the classical Laguerre weight xα e−x , x ∈ [0, ∞), α > −1, i.e. taking β = 1 for our weight w(x), we have h 3 i 1 1 3 13 4 2 4 2 2 λN ' 2 π e (2N + α) exp −2 (2N + α) . Remark 4.3. When α = 0, β = 1, Szeg¨o’s [26]∗ classical result for the Laguerre weight e−x is recovered: h i 3 1 1 7 λN ' 2 2 π 2 eN 4 exp −4N 2 . Remark 4.4. With the restriction of α = 0, β 6= n + 12 , n ∈ {1, 2, 3, . . .}, Chen and β Lawrence’s result on the weight e−x , x ∈ [0, ∞) is also recovered: 1 Eβ 1− 2β X e 1 r 5 Ar e− 14 A e02 N 12 − 4β1 exp − sec πβ − N p λN ' 8π 4 C (−1)r N − β , er e C πC r=0
e0 , A er , C, e please see [12]. for details of A From (4.6) we find that λN is exponentially small for large N and tends to 0 as N → ∞. ∗
The original formula of λN in the last equation on page 461 missed a factor of 4.
12
5
The approximation of λN for β = n+ 21 , n ∈ {1, 2, 3, . . .}
Our goal for this section is to find the approximation of λN for the cases where β = n + 12 , n ∈ {1, 2, 3, . . .}. Such cases, as was illustrated in Remark 3.1, require the second representation of I(z) in (3.2). Before obtaining the asmptotic behavior of λN , we first establish the following lemma for P(z). Lemma 5.1. For |η| 1, then as N → ∞, β− 1 1 1 4 (−z)β X2 (−1)β+ 2 (−z)β log + (−1)k δβ− 1 −k η k−β+ 2 I(z) ' 3 2 2π η 4π 2 k=0
where k Lk := C k (k), π with γk :=
β− 21
and
δk :=
X γj−k , Lj− 1 j=1 2
(k− 12 ) ΓΓ(k+1) , if k ≥ 0, 0,
if k < 0.
Proof. Based on the Gauss’ recursion relation [18], see (7.5) in the Appendix, Chen and Lawrence [12] built the following version formula: n + 32 (z − 1) 1 5 1 3 1 1 = − 2 F1 1, ; n + ; z 2 F1 1, ; n + ; z 2 F1 1, ; n + ; z 2 2 (n + 1)z 2 2 2 2 3 n n+ 2 3 1 F 1, ; n + ; z , + 1 2 1 2 2 (n + 1) n + 2 together with the fact that √ 1 5 3 (z − 1) 1+ z 3 √ + z, = log 2 F1 1, ; ; z 3 2 2 4 z2 2 1− z we can get 1 k √ β− 2 β− 12 X √ 1 (z − 1) 1 z z log 1 + √z + , = Lβ 2 F1 1, ; β + 1; z 1 1 2 L z − 1 1− z z β+ 2 k− k=1 2
where β = n + 12 , n = 1, 2, 3, . . . and Lk is given by Lk :=
k k C (k). π
Consequently, we have I(z) =
β+ 21
(−1) 2π
β− 12 √ −k X p 1+η+1 η (−z)β log √ + 1+η (−1)k . Lk− 1 1+η−1 k=1 2
13
By using (4.3) again, we find 1 β− 2 β− 12 β j −j X 4 (−1) (−1) (−z) (−z) (−1) η log + I(z) ' 3 2π η Lj− 1 4π 2 j=1 2 1 β− 2 1 1 X − 2 k+j−(β− 1 ) 1 Γ k + j − β − 2 2 η · (−1)k+j−(β− 2 ) 1 Γ k + j − β − + 1 2 k=0 Γ(k+j−(β− 12 )− 21 ) β− 12 β− 12 1 X Γ(k+j−(β− 12 )+1) 1 (−1)β+ 2 (−z)β 4 X = η k+ 2 −β . log + (−1)k 2π η Lj− 1 j=1 k=0 β+ 12
β
2
With an easy simplification, the Lemma is obtained immediately. 1
Substituting η = −zC −1 (2N + α)− β , together with a simple calculation gives the following strong asymptotics of PN (z) for z ∈ / [0, ∞), Lemma 5.2. For z ∈ / [0, ∞), we have 1
" 1 − 14 1 α 1 (−1)N (−1)β− 2 (−z)β − 2 −4 β PN (z) ' √ (−z) exp log C(2N + α) 2π 2π 1 1 β− 1 2 1− 2β X (−z)k+ 2 (2N + α) √ (−1)k Bk · exp k , 1 2 πC k=0 C(2N + α) β where Bk := ak − In particularly,
4C(2N + α) β −z
Lβ δ 1 , 2β β− 2 −k
√ 4 πβ B0 = . 2β − 1
Theorem 5.1. For λN , we have 1
5 2
5 4
λN ' 2 π C
− 14
1 2
B0 (2N + α)
1 1 − 4β 2
1 1 β− 1− 2β X2
(2N + α) √ · exp − πC
k=0
4C (2N + α)
1 β
(−1)πβ+ 2
Bk (−1)k k C
(2N + α)
− βk
(5.1)
.
Proof. Since B0 > 0 and by an argument like that in the Section 4, again we find that the dominant contribution to Kµν is from the arc of the unit circle around 1 k z = −1. Restricting µ, ν to the same range given by (4.7), then (2µ + α)1− 2β − β −
14
!#
1
k
(2ν + α)1− 2β − β and log [(2µ + α)/(2ν + α)] remain bounded and (4.8) will also be true at here. By the Laplace method, we have Z ∞ (5.2) Pµ −eiθ Pν −e−iθ dθ. Kµν ' −∞
As previously, we expand the exponential in the integrand for |θ| 1, reserving terms up to the second order. We obtain β− 1 r 2 (−1) µ+ν 1 1 1 (−1) 2 π − 2 − 4β β Kµν ' (2N + α) 4C (2N + α) 1 B0 (πC) 4 β− 12 X 1 1 k k 1 Bk · exp √ (−1)k k (2µ + α)1− 2β − β + (2ν + α)1− 2β − β , C 2 πC k=0 For large enough µ and ν, restricted by (4.7), again we will have 1
1
2 2 Kµν ' (−1)µ+ν Kµµ Kνν .
As per the discussion in the previous section, it follows that 2π λN ' R N . K dν νν 0 Taking an application of the Laplace method and doing the same argument as before, we get the asymptotic behavior for the integration, 1
Z
N
− 23
Kνν dν ' 2
− 14
1 4
−1 B0 2
1 − 12 + 4β
1 β
(−1)πβ− 2
4C (2N + α) π C (2N + α) 1 1 β− 2 1− 2β X k (2N + α) Bk √ · exp (−1)k k (2N + α)− β , C πC k=0
0
(5.3)
which completes the proof of this theorem. Example 5.1. If we take α = − 43 , β = 32 , then 2 13 + 3π2 3 3 1 2N − 2 5 1 1 5 3 λN ' 2 2 + π π 4 C π − 4 B02 2N − exp − √ 4 4 πC where C =
π 2
23
− 23 ! B1 3 B0 − 2N − C 4
√ √ , B0 = 3 π and B1 = − 6π .
Remark 5.1. Putting α = 0, Chen and Lawrence’s result for β = n + 12 , n = 1, 2, 3, . . . is recovered. 1 β+ 1 1 β− 2 (−1) 2 1− ek 1 k 1 2N 2β X B π e− 41 B e02 N 12 − 4β1 4CN e β1 exp − p (−1)k N − β , λN ' 8π 4 C ek e k=0 C πC where
e = 2− β1 C C
ek = Bk . and B 15
Comparing (4.6) with (5.1), we note that the essential difference between them is 1 h i (−1)πβ+ 2 1 the term exp(− sec πβ) becomes 4C (2N + α) β . The alternating behavior of the second term depends on whether β + 21 is even or odd. Anyway, λN → 0 as N → ∞. On the basis of the standard theory [1], the moment problem with respect to w(x), x ∈ [0, ∞) is indeterminate if Z ∞ log w(x) √ dx > −∞. x(1 + x) 0 β
For our weight w(x) = xα e−x , x ∈ [0, ∞), Z ∞ Z ∞ xβ −α log x + xβ √ √ dx = dx = π sec (πβ) , x(1 + x) x(1 + x) 0 0 Therefore, β = 12 is the critical point at which the moment problem becomes indeterminate. If we assume the approximation of PN (z) given in (3.5) holds, we see that √ 1 (−1)N −z 2π (2N + α) − 21 −α − √ PN (z) ' √ (−z) 2 4 (2N + α) exp +1 . log π −z 2π (5.4) We assume that both µ and ν are large, however µ−ν is bounded by a constant, and thus the asymptotic expression holds. Again, we see that the main contributions to Kµν come from the arc of the unit circle around z = −1. However, for |η| 1, it follows the behavior of PN given by (5.4): Z ε Kµν ' Pµ (−eiθ )Pν (−e−iθ )dθ −ε
(−1)µ+ν 2 + 2 log 2π − 12 + π1 − 21 + π1 π π e (2µ + α) (2ν + α) ' 2 Z ε 2π 2 − 2 log 2π − log(2µ + α) − log(2ν + α) 2 · 1+ θ dθ. 8π −ε Quite obviously, |Kµν | decreases as µ and ν increase, which invalidates the argument of the previous section. However, it is possible to get an approximative lower bound for the smallest eigenvalue using (2.6). Using the Christoffel-Darboux formula ( [19], Theorem 2.2.2), which reads: N −1 X j=0
Pj (x)Pj (y) PN (x)PN −1 (y) − PN (y)PN −1 (x) = , hj hN −1 (x − y)
and is valid for monic orthogonal polynomials {Pn (z)}, where hj is the square of the L2 norm of PN (z), and the result presented in [10] for large N off-diagonal
16
recurrence coefficients, we have N X
Z Kνν =
π
N X
Pν −eiθ Pν −e−iθ dθ
−π ν=0
ν=0
π (2N + α)2 ' 4 2
Z
π
−π
PN −eiθ PN +1 −e−iθ − PN −e−iθ PN +1 −eiθ dθ. eiθ − e−iθ
As a result, applying the Laplace method, gives N X
2
Kνν
ν=0
[2π(2N + α)e] π . ' p 4 log [2π(2N + α)e]
We found the smallest eigenvalue for β = exponentially, since λN ' PN2πK . ν=0
6
1 2
decreases algebraically rather than
νν
Numerical results
It is well known that Hankel matrices (moment matrices) of this form are extremely ill-conditioned. This can be observed directly from the terms on the main diagonal of HN as follows. The condition number, κ(HN ) = ΛλNN where ΛN and λN are the largest and smallest eigenvalues of HN respectively. By applying the Rayleigh quotient, we know ΛN is greater than all elements on main diagonal and λN is less than all elements on the main diagonal. Thus, for some constant c > 0: κ(HN ) ≥
2N ) Γ( α+1+2N µ2N β > c Γ , = µ0 β Γ( α+1 ) β
2N β
if
2N 3 ≥ . β 2
We note that the condition number is of order Γ . Due to the ill-conditioned nature of these matrices, standard eigensolver packages based double precision floating values can solve only small instances, i.e. N < 20, before they exhaust the available precision (53 bit in the mantissa, 11 bits in the exponent). In [17], Emmart, Chen and Weems developed an efficient parallel algorithm based on arbitrary precision arithmetic and the Secant method that can handle the extreme ill-conditioning and we employ their algorithms here for our numerical computations. We use the numerical results to test the convergence of our asymptotic formulas to the actual smallest eigenvalues for various N and several values of the parameters α and β. Even with efficient software, the computation times for the largest size, N = 1000, require almost 10 hours of CPU time on a modern Core i7 processor.
17
Figure 1: The percentage error of the theoretical values of λN vs. those obtained numerically, for various α and β.
Table 1: List of numerical results vs. theoretical values, for α = 0, β = 21 . Size N Numerical λN Theoretical λN error 100 0.27397 0.40360 47.32% 300 0.15837 0.21365 34.91% 500 0.12047 0.15855 31.61% 1000 0.082087 0.10555 28.58% 1500 0.065295 0.083130 27.33% 2000 0.055431 0.070108 26.48% 2500 0.048788 0.061430 25.91% 3000 0.043940 0.055135 25.48% Remark 6.1. In Figure 1 and Tables 1-3, Theoretical λN − Numerical λN % error := × 100 . Numerical λN
18
(6.1)
Table 2: List of numerical α β Size N 100 − 12 1 200 300 400 500 600 700 800 900 1000 1 1 100 200 300 400 500 600 700 800 900 1000 − 12 74 100 200 300 400 500 600 700 800 900 1000 2 2 100 200 300 400 500 600 700 800 900 1000
results vs. theoretical values, for β Numerical λN Theoretical λN 2.2434 × 10−15 2.4171 × 10−15 1.7127 × 10−22 1.8059 × 10−22 5.7241 × 10−28 5.9779 × 10−28 1.3647 × 10−32 1.4170 × 10−32 1.1453 × 10−36 1.1845 × 10−36 2.3530 × 10−40 2.4265 × 10−40 9.5340 × 10−44 9.8098 × 10−44 6.6167 × 10−47 6.7956 × 10−47 7.1306 × 10−50 7.3123 × 10−50 1.1110 × 10−52 1.1378 × 10−52 2.0119 × 10−15 2.0845 × 10−15 1.5845 × 10−22 1.6258 × 10−22 5.3703 × 10−28 5.4855 × 10−28 1.2911 × 10−32 1.3153 × 10−32 1.0898 × 10−36 1.1081 × 10−36 2.2486 × 10−40 2.2831 × 10−40 9.1416 × 10−44 9.2716 × 10−44 6.3614 × 10−47 6.4462 × 10−47 6.8707 × 10−50 6.9572 × 10−50 1.0725 × 10−52 1.0853 × 10−52 2.0753 × 10−45 2.1203 × 10−45 4.6281 × 10−76 4.7027 × 10−76 1.7181 × 10−102 1.7412 × 10−102 1.6945 × 10−126 1.7144 × 10−126 7.6149 × 10−149 7.6955 × 10−149 5.9500 × 10−170 6.0077 × 10−170 4.4336 × 10−190 4.4735 × 10−190 2.0973 × 10−209 2.1149 × 10−209 4.7024 × 10−228 4.7398 × 10−228 4.0152 × 10−246 4.0455 × 10−246 1.5626 × 10−54 1.3738 × 10−54 4.0862 × 10−93 3.7101 × 10−93 4.6575 × 10−127 4.2866 × 10−127 2.7728 × 10−158 2.5723 × 10−158 1.2618 × 10−187 1.1769 × 10−187 1.5155 × 10−215 1.4191 × 10−215 2.4610 × 10−242 2.3117 × 10−242 3.4223 × 10−268 3.2228 × 10−268 2.9306 × 10−293 2.7654 × 10−293 1.2053 × 10−317 1.1394 × 10−317
19
6= n + 12 , n ∈ N+ error 0.07745 0.05445 0.04435 0.03835 0.03427 0.03126 0.02893 0.02705 0.02549 0.02417 0.03610 0.02605 0.02145 0.01867 0.01676 0.01534 0.01423 0.01333 0.01258 0.01195 0.02168 0.01613 0.01344 0.01177 0.01059 0.00970 0.00899 0.00842 0.00794 0.00753 0.12082 0.09204 0.07964 0.07230 0.06729 0.06358 0.06067 0.05831 0.05635 0.05467
Table 3: List of numerical α β Size N 100 − 34 32 200 300 400 500 600 700 800 900 1000 100 − 12 52 200 300 400 500 600 700 800 900 1000 5 2 100 2 200 300 400 500 600 700 800 900 1000 7 0 100 2 200 300 400 500 600 700 800 900 1000
results vs. theoretical values, for β Numerical λN Theoretical λN 7.8618 × 10−36 8.1371 × 10−36 3.3759 × 10−58 3.4638 × 10−58 5.3913 × 10−77 5.5084 × 10−77 8.6803 × 10−94 8.8455 × 10−94 3.1289 × 10−109 3.1825 × 10−109 1.1225 × 10−123 1.1402 × 10−123 2.4313 × 10−137 2.4668 × 10−137 2.2724 × 10−150 2.3035 × 10−150 7.2147 × 10−163 7.3079 × 10−163 6.5115 × 10−175 6.5914 × 10−175 1.7527 × 10−68 1.8456 × 10−68 1.6233 × 10−121 1.6918 × 10−121 1.9900 × 10−169 2.0635 × 10−169 2.8604 × 10−214 2.9564 × 10−214 5.9391 × 10−257 6.1240 × 10−257 5.5030 × 10−298 5.6642 × 10−298 1.0780 × 10−337 1.1080 × 10−337 2.6691 × 10−376 2.7400 × 10−376 5.7436 × 10−414 5.8901 × 10−414 8.0870 × 10−451 8.2859 × 10−451 3.5614 × 10−69 3.5767 × 10−69 3.9629 × 10−122 4.0207 × 10−122 5.3740 × 10−170 5.4660 × 10−170 8.2704 × 10−215 8.4184 × 10−215 1.8070 × 10−257 1.8397 × 10−257 1.7433 × 10−298 1.7748 × 10−298 3.5306 × 10−338 3.5937 × 10−338 8.9908 × 10−377 9.1495 × 10−337 1.9823 × 10−414 2.0168 × 10−414 2.8512 × 10−451 2.9002 × 10−451 7.0602 × 10−89 6.8217 × 10−89 9.5989 × 10−164 9.3315 × 10−164 1.5672 × 10−233 1.5286 × 10−233 3.4926 × 10−300 3.4140 × 10−300 1.3836 × 10−364 1.3546 × 10−364 3.0405 × 10−427 2.9804 × 10−427 1.7488 × 10−488 1.7160 × 10−488 1.5603 × 10−548 1.5322 × 10−548 1.4700 × 10−607 1.4446 × 10−607 1.0901 × 10−665 1.0719 × 10−665
20
= n + 12 , n ∈ N+ error 0.03502 0.02605 0.02172 0.01902 0.01713 0.01571 0.01459 0.01368 0.01292 0.01227 0.05303 0.04223 0.03693 0.03356 0.03114 0.02928 0.02779 0.02655 0.02550 0.02460 0.00428 0.01460 0.01711 0.01790 0.01809 0.01803 0.01787 0.01766 0.01743 0.01719 0.03379 0.02786 0.02464 0.02251 0.02096 0.01976 0.01879 0.01798 0.01729 0.01669
7
Appendix
The integral identities listed below, which are relevant to our derivation and can be found in [14], [18] and [15]. Z b dx p = π. (7.1) (b − x)(x − a) a Z b dx π p =p . (7.2) (b − x)(x − a) (t + a)(t + b) a (x + t) √ √ Z b t+a+ t+b log(x + t) p dx = 2π log . (7.3) 2 (b − x)(x − a) a √ 2 ! √ (t+1)2 − (t+a)(t+b)− (1−a)(1−b) log √ √ 2 Z b ( t+a+ t+b) log(1 − x) p p dx = π . (7.4) (b − x)(x − a)(x + t) (t + a)(t + b) a Gauss’ recursion functions ( [18], P1019, 9.1377 , 1): γ [γ − 1 − (2γ − α − β − 1)z] F (α, β; γ; z) + (γ − α)(γ − β)zF (α, β; γ + 1; z) + γ(γ − 1)(z − 1)F (α, β; γ − 1; z) = 0. (7.5)
8
Acknowledgements
The financial support of the Macau Science and Technology Development Fund under grant number FDCT 130/2014/A3 and FDCT 023/2017/A1 are gratefully acknowledged. We would also like to thank the National Science Foundation (NSF): CCF-1525754 and the University of Macau for generous support: MYRG 201400011 FST, MYRG 2014-00004 FST.
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