A Riemann--Hilbert approach to Jacobi operators and Gaussian quadrature

2013/11/22 by Trogdon, Thomas, Olver, Sheehan
#33C47 #65E99 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1311.5838

Abstract

The computation of the entries of Jacobi operators associated with orthogonal polynomials has important applications in numerical analysis. From truncating the operator to form a Jacobi matrix, one can apply the Golub--Welsh algorithm to compute the Gaussian quadrature weights and nodes. Furthermore, the entries of the Jacobi operator are the coefficients in the three-term recurrence relationship for the polynomials. This provides an efficient method for evaluating the orthogonal polynomials. Here, we present an \mathcal O(N) method to compute the first N rows of Jacobi operators from the associated weight. The method exploits the Riemann--Hilbert representation of the polynomials by solving a deformed Riemann--Hilbert problem numerically. We further adapt this computational approach to certain entire weights that are beyond the reach of current asymptotic Riemann--Hilbert techniques.

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