*Title*: **Divide and Conquer Algorithms and Software for Large Hermitian Eigenvalue Problems**

*Abstract*: Divide-and-conquer paradigms can lead to efficient and flexible
techniques for solving large Hermitian eigenvalue problems. This talk
will discuss how these techniques can be put to work to implement
`spectrum slicing' strategies, i.e., strategies that extract slices of
the spectrum independently. The presentation will begin with an
overview of polynomial filtering, a general approach that can be quite
efficient in the situation where the matrix-vector product operation
is inexpensive and when a large number of eigenvalues is sought. We
will present a few techniques based on the Lanczos algorithm with and
without restarts, as well as subspace iteration. An alternative to
polynomial filtering that is generating a growing interest is a class
of methods that rely on rational functions. Good representatives of
this general approach are the FEAST eigensolver and the
Sakurai-Sugiura algorithm. We will argue that the standard Cauchy
integral--based approach can be substantially improved upon --
especially when iterative solvers are involved. Finally, the talk
will discuss our recently released code named EVSL (for eigenvalues
slicing library) that implements these ideas.

Smith Colloquium Spring 2018