Numerical Solution of Riemann-Hilbert Problems: Random Matrix Theory and Orthogonal Polynomials

Recently, a general approach to solving Riemann-Hilbert problems numerically has been developed. We review this numerical framework and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with the approach of Bornemann to compute Fredholm determinants, we are able to calculate spectral densities and gap statistics for a broad class of finite-dimensional unitary invariant ensembles. We show that the accuracy of the numerical algorithm for approximating orthogonal polynomials is uniform as the degree grows, extending the existing theory to handle g-functions. As another example, we compute the Hastings-McLeod solution of the homogeneous Painlev, II equation.