Differentiation Matrices for Fun & Profit
Taking derivatives - the action of differentiation - is a fundamental operation in the mathematics of functions, and because of that it is astonishingly useful in Science and Engineering (maybe the causation goes the other way). There are an equally astonishing number of methods to take derivatives in various situations, including "by hand" symbolic methods, computer-assisted symbolic methods (this pre-dates FORTRAN), finite differences (these pre-date computers), compact finite differences, automatic differentiation (forward and reverse mode), complex methods, and - the subject of the talk - differentiation matrices.
These are useful for solving ODE, PDE and oscillatory integrals. In this talk I will introduce differentiation matrices and contrast their use with other methods. I'll explain some universal properties, and show some niche methods.
I'll end by using them to evaluate some highly oscillatory integrals by Levin-type methods.
This is joint work with Amir Amiraslani (U. Hawaii) and Jeet Trivedi.