DeepAI AI Chat
Log In Sign Up

A Riemann–Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory

by   Xiucai Ding, et al.

We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. We show that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. The leading error terms can be fully characterized by the difference of the Stieltjes transforms on the contour. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization and the conjugate gradient algorithm (CGA). As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limit, allowing for precise estimates on the algorithms as the number of iterations diverges. For this concrete random matrix model, beyond the first order expansion, we derive a mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.


A Riemann–Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals

We develop a numerical method for computing with orthogonal polynomials ...

Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices

We introduce and develop a theory of orthogonality with respect to Sobol...

On generating Sobolev orthogonal polynomials

Sobolev orthogonal polynomials are polynomials orthogonal with respect t...

The conjugate gradient method with various viewpoints

Connections of the conjugate gradient (CG) method with other methods in ...

FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups

We give two algebro-geometric inspired approaches to fast algorithms for...

Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory

We consider quasi-compact linear operator cocycles ℒ^n_ω:=ℒ_σ^n-1ω∘⋯∘ℒ_σ...

Gossip of Statistical Observations using Orthogonal Polynomials

Consider a network of agents connected by communication links, where eac...