When: Feb 29 2024 @ 1:30 PM
Where: Olin 305

Location:Olin 305

When: February 29th at 1:30 p.m.

Title: The Lanczos method, matrix functions, and the quest for optimality

Abstract: First introduced in 1950, the Lanczos method is a backbone of numerical linear algebra, underlying our fastest algorithms for solving linear systems, computing eigenvectors, and much more. It is easy to implement, works well in finite precision, and most importantly, is remarkably fast. In fact, the Lanczos method typically converges much faster than our best theoretical results are able to predict, a convenient yet puzzling mystery that has endured for decades. In this talk, I will discuss recent progress on better explaining the power of the Lanczos method for the important problem of computing matrix functions like the matrix exponential, square root, matrix log, etc. In particular, we prove that, for a wide class of functions, the Lanczos method performs near optimally among *all possible* Krylov subspace methods. Previously, such a result was only known for the matrix inverse. In addition to our main result, I will discuss a number of open problems, including questions about numerical stability and connections to recent work on faster linear system solves for sparse matrices.

Based on joint work with Noah Amsel, Tyler Chen, Anne Greenbaum, and Cameron Musco.

Zoom link: https://wse.zoom.us/j/94601022340