The Lanczos Tau Framework for Time-Delay Systems: Padé Approximation and Collocation Revisited

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2529-2548, December 2024.
Abstract. We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency domain. We illustrate that Lanczos tau methods straightforwardly give rise to sparse, self-nesting discretizations. Equivalence is also demonstrated with pseudospectral collocation, where the nonzero collocation points are chosen as the zeros of orthogonal polynomials. The importance of such a choice manifests itself in the approximation of the [math]-norm, where, under mild conditions, supergeometric convergence is observed and, for a special case, superconvergence is proved, both of which are significantly faster than the algebraic convergence reported in previous work.