Mathematics Seminar - 11/5/21

Abstract: The aim of this talk is to give an overview of the relationships between matrices, moments, orthogonal polynomials, quadrature rules, and Krylov subspace methods. The underlying goal is to obtain efficient numerical methods for estimating quadratic or bilinear forms involving matrix functions. An obvious application is the computation of some elements of a matrix function. Other applications from numerical linear algebra and numerical PDEs will also be presented. The main idea is to treat the quadratic or bilinear form as a Riemann-Stieltjes integral, and then to apply Gauss quadrature rules to approximate the integral. The nodes and weights of these quadrature rules are given by the eigenvalues and eigenvectors, respectively, of tridiagonal matrices derived from the three-term recurrence satisfied by the orthogonal polynomials associated with the measure of the integral. Beautifully, these orthogonal polynomials can be generated by the Lanczos algorithm applied to the matrix. As the various applications featured in this talk will illustrate, it can be quite fruitful to blend old ideas from classical analysis and numerical linear algebra in new ways. The resulting algorithms can also be of interest to scientists and engineers who are solving problems in which computation of quadratic or bilinear forms arises naturally.