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Description

Newton was a master of computation. Indeed, many modern computational techniques find their roots in Newton's work. In the first half of this talk, I will recount Newton's beautifully simple trick for computing the roots of a perturbed polynomial as a function of the perturbation. The trick is called the Newton polygon. This part of the talk is accessible to students having a working knowledge of College Algebra. After introducing the method, I will use Newton's notes to give some indication of why it works.

The second half of the talk will discuss how Newton's polygon can be used to derive the Lidskii-Vishik-Lyusternik perturbation theory for the eigenvalues of matrices with arbitrary Jordan structure. I then show how this approach through the Newton polygon can be used to extend Lidskii's results to some non-generic cases where the standard theory does not apply. This half of the talk is accessible to senior level undergraduates and graduate students having a working knowledge of the Jordan form for matrices over the complex numbers.

This talk is based on a collaboration with Julio Moro of the Universidad Carlos III, Madrid, and Michael Overton of the Courant Institute.