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We establish a new perturbation theory for orthogonal polynomials using aRiemann--Hilbert approach and consider applications in numerical linear algebraand random matrix theory. This new approach shows that the orthogonalpolynomials with respect to two measures can be effectively compared using thedifference 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 deriveexplicit and accurate expansion formulae for the perturbed orthogonalpolynomials.  In contrast to other approaches, a key strength of the methodology is thatestimates can remain valid as the degree of the polynomial grows. The resultsare applied to analyze several numerical algorithms from linear algebra,including the Lanczos tridiagonalization procedure, the Cholesky factorizationand the conjugate gradient algorithm. As a case study, we investigate thesealgorithms applied to a general spiked sample covariance matrix model byconsidering the eigenvector empirical spectral distribution and its limits. Forthe first time, we give precise estimates on the output of the algorithms,applied to this wide class of random matrices, as the number of iterationsdiverges. In this setting, beyond the first order expansion, we also derive anew mesoscopic central limit theorem for the associated orthogonal polynomialsand other quantities relevant to numerical algorithms.

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