Numerical Experiments on Circular Ensembles and...

Numerical Experiments on Circular Ensembles and JackPolynomials with Julia

N. Kemal UreAerospace Controls Lab, MIT

May 15, 2013

Ure () Numerical Experiments on Circular Ensembles and Jack Polynomials with JuliaMay 15, 2013 1 / 12

Introduction

Overview

▶Brief background on circular ensembles and Jack polynomials

▶How can we efficiently compute averages of Jack polynomials oncircular ensembles ?

▶Numerical Results

▶ Julia Language


Background

Circular Ensembles

▶Measures on space of unitary matrices (U such that UU∗ = I)

1 Circular Orthogonal Ensemble (COE), β = 1, symmetric unitary matrices2 Circular Unitary Ensemble (CUE), β = 2, unitary matrices3 Circular Symplectic Ensemble (CSE), β == 4, unitary quaternion matrices

▶ Eigenvalues are on the unit circle λk = eiθk , k = 1, ..., n.

▶ Joint Density is given by,

p(θ) = 1

Zn,β∏

1≤k<j≤n∣eiθk − eiθj ∣β. (1)


Background

Jack Polynomials

▶Multivariate polynomials parametrized by a parameter β > 0.

▶ Serve as a homogeneous base for space of kth order symmetricpolynomials in n variables.

▶ Jack Polynomials orthogonalize the β-circular ensembles ! [3]

∫[0,2π]n Jβκ (eiθ1 , ..., eiθn), J

βλ (eiθ1 , ..., eiθn)∏

j<k∣eiθj−ejθk ∣βdθ1...dθn = δκ,λ.


Background

The Experiment

▶ Jack polynomials with matrix arguments, Jβκ (U) = Jβκ (λ1, ..., λn), λkare the eigenvalues.

▶We want to numerically verify that

E[Jβκ (U)Jβλ (U)] = 0, (2)

▶U is a random unitary matrix, drawn according to p(θ)

▶We need two subroutines

∎ Sample a random unitary matrix and find its eigenvalues

∎ Evaluate the Jack function on these eigenvalues


Algorithms

Sampling Unitary Matrices

▶Hessenberg matrices: ”almost” upper triangular entries with all zeroentries above below it’s first superdiagonal.

▶ Example:

⎡⎢⎢⎢⎢⎢⎣

0.5693 + 0.094i −0.0042 + 0.0132i 0.2468 + 0.7785i0.8168 0.0014 − 0.0097i −0.2614 + 0.5153i

0 0.9999 0.9999

⎤⎥⎥⎥⎥⎥⎦▶Ammar et. al [4] showed that there is a one to one correspondence

between the Schur parameters γj ∈ C, j = 1, ..., n and n × n upperunitary Hessenberg matrices.

▶ Schur parameters: ∣γj ∣ ≤ 1,1 ≤ j < n, ∣γn∣ = 1.

H({γ}) = G1(γ1)...Gn−1(γn−1)G̃n(γn),

▶Hence Hessenberg matrices are parametrized by 2n− 1 real numbers !


Algorithms

Evaluation of Jack Polynomials

▶ Jack polynomials can be expressed in the monomial basis,

Jβλ = ∑T−SSY T

fT (β)xT ,

▶ Sum over SSYT: Semi-standard Young Tableaux. Numerically veryinefficient.

▶Demmel and Koev developed a recursive algorithm based on theprinciple of dynamic programming

Jβλ (x1, ..., xn) = ∑µ≤λ

Jβµ (x1, ..., xn−1)x∣λ/µ∣n cµλ,

▶The basic idea is to represent a polynomial in n variables in terms ofpolynomials in (n − 1) variables.

▶MATLAB implementation available.


Results

Sampling Unitary Hessenberg Matrices

▶Generate random Schur parameters and then construct the unitaryupper Hessenberg matrix (Implemented in Julia)

▶ Plots of eigenvalues for different sample sizes (n = 3):

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1t= 10

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1t= 100

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1t= 1000

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1t= 10000


Results

Averaging Jack Polynomials

▶Koev’s algorithm implemented on Julia and averaged overeigenvalues of random unitary matrices

▶ Results are also averaged over different partitions (also selectedrandomly), β = 7, n = 3.

1 2 3 4 5 6 7 8 9 10

x 104

0

0.005

0.01

0.015

0.02

0.025

Abs

olut

e va

lue

of A

vera

ges

of J

ack

Pol

ynom

ials

Number of samples


Results

Averaging Jack Polynomials

▶ Increasing β results in faster convergence

▶ Plot shows the number of samples required to converge withinε = 10−4 as a function of β.

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3x 10

5

Num

ber

of s

ampl

es r

equi

red

to c

onve

rge

with

in ε

β


Results

Julia Language

▶ Scientific computing languages: Trade-off between ease of codingand computational power

▶ Julia: Easy to learn and code as in MATLAB and comparable to Cin performance∎ It is also very easy to parallelize !

▶ Computing time (in seconds) across MATLAB, Julia and ParallelJulia (4 cores) for the averaging Jack polynomials on circularensemble:

Number of Samples MATLAB Julia Parallel Julia500 5.1 1.7 1.31000 10.2 1.9 1.82000 20.3 2.4 1.95000 51.8 3.6 2.150000 516.4 25.6 3.9

▶MATLAB users: convert your code to Julia !▶Users of other languages (C,C++, Python, Java): Discover Julia’s

parallel computing power.Ure () Numerical Experiments on Circular Ensembles and Jack Polynomials with JuliaMay 15, 2013 11 / 12

Results

References

[1] F.M. Dyson ”The threefold way. Algebraic structure of symmetry groupsand ensembles in quantum mechanics”. J. Math. Phys. 3: 1199. (1962).

[2] Jack, Henry, ”A class of symmetric polynomials with a parameter”,Proceedings of the Royal Society of Edinburgh, Section A. Mathematics69: 118, (1971).

[3] Macdonald, I. G., Symmetric functions and Hall polynomials, OxfordMathematical Monographs (2nd ed.), New York: Oxford University Press,ISBN 0-19-853489-2, MR 1354144 (1995)

[4] Ammar, Gregory, William Gragg, and Lothar Reichel. ”Constructing aunitary Hessenberg matrix from spectral data.” In Numerical linearalgebra, digital signal processing and parallel algorithms, pp. 385-395.Springer Berlin Heidelberg, 1991.

[5] Demmel, James, and Plamen Koev. ”Accurate and efficient evaluation ofSchur and Jack functions.” Mathematics of computation 75, no. 253(2006): 223-239.

[6] Forrester, Peter J., and Eric M. Rains. ”Jacobians and rank 1perturbations relating to unitary Hessenberg matrices.” arXiv preprintmath/0505552 (2005).


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