Dear GMS,
I hope this email finds you well.
We will have a seminar on March 9, Monday (tomorrow) between 11:30 and 12:30 at 225 St. Paul's.
Our speaker will be @Gabriel Ogulumailto:Gabriel.Ogulu@umanitoba.ca. You can find more information about his talk below.
Title: Matrix Analysis of the Ultraspherical Spectral Method
Abstract: Spectral methods are prized in scientific computing for their exponential convergence rates, yet classical collocation approaches suffer from severe algebraic bottle necks: they generate dense differentiation matrices with condition numbers scaling as O(N4). This research project investigates the Ultraspherical Spectral Method, a structured approach that overcomes these limitations by utilizing a hierarchy of orthogonal polynomial bases to represent differentiation as a sparse, banded operation. We perform a rigorous matrix analysis of the infinite-dimensional operators arising from this discretization, specifically constructing the banded differentiation (D(2)), multiplication (X), and basis conversion (R(2)) matrices. A key focus is the solution of ”almost-banded” linear systems structures arising from the imposition of boundary conditions on banded operators. We implement a stable QR factorization algorithm utilizing Givens rotations to solve these systems while minimizing fill-in. Numerical experiments performed in Julia validate the theoretical claims on two canonical problems: the Airy boundary value problem and the Harmonic Oscillator eigenvalue problem. Our benchmarks confirm that the ultraspherical solver achieves O(N) computational complexity, a significant improvement over the O(N3) scaling of dense spectral methods, while maintaining bounded or slowly growing condition numbers. This work demonstrates that exploiting structured linear algebra is essential for unlocking the full potential of spectral discretizations in modern differential equation solvers.
See you all in the seminar! GMS Websitehttps://sites.google.com/view/umgradmathsociety/ / Instagramhttps://www.instagram.com/umgradmathsociety/
Sincerely, Berkanthttps://cnnk.xyz GMS Executive