On the Evaluation of the Eigendecomposition of the Airy Integral Operator

04/27/2021
by   Zewen Shen, et al.
0

The distributions of random matrix theory have seen an explosion of interest in recent years, and have been found in various applied fields including physics, high-dimensional statistics, wireless communications, finance, etc. The Tracy-Widom distribution is one of the most important distributions in random matrix theory, and its numerical evaluation is a subject of great practical importance. One numerical method for evaluating the Tracy-Widom distribution uses the fact that the distribution can be represented as a Fredholm determinant of a certain integral operator. However, when the spectrum of the integral operator is computed by discretizing it directly, the eigenvalues are known to at most absolute precision. Remarkably, the integral operator is an example of a so-called bispectral operator, which admits a commuting differential operator that shares the same eigenfunctions. In this manuscript, we develop an efficient numerical algorithm for evaluating the eigendecomposition of the integral operator to full relative precision, using the eigendecomposition of the differential operator. With our algorithm, the Tracy-Widom distribution can be evaluated to full absolute precision everywhere rapidly, and, furthermore, its right tail can be computed to full relative precision.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
10/01/2019

Monotonically Decreasing Sequence of Divergences

Divergences are quantities that measure discrepancy between two probabil...
research
06/04/2021

The sine kernel, two corresponding operator identities, and random matrices

In the present paper, we consider the integral operator, which acts in H...
research
06/02/2020

Computing spectral measures of self-adjoint operators

Using the resolvent operator, we develop an algorithm for computing smoo...
research
12/16/2019

On the Convergence of Numerical Integration as a Finite Matrix Approximation to Multiplication Operator

We study the convergence of a family of numerical integration methods wh...
research
09/19/2019

A numerical method for Hadamard finite-part integrals with an integral power singularity at an endpoint

In this paper, we propose a numerical method for computing Hadamard fini...
research
02/06/2017

Characteristic polynomials of p-adic matrices

We analyze the precision of the characteristic polynomial of an n× n p-a...
research
06/03/2020

Computable structural formulas for the distribution of the β-Jacobi edge eigenvalues

The Jacobi ensemble is one of the classical ensembles of random matrix t...

Please sign up or login with your details

Forgot password? Click here to reset