Abstract by Hayden Ringer
Eigenvalue Methods for Bounded and Unbounded Regions
Spectral theory is ubiquitous in applied mathematics, and thus algorithms for finding eigenvalues of matrices are central to many applied methods. In some applications, it is sufficient to find only some eigenvalues (perhaps in a region of interest in the complex plane). For problems involving large matrices, it can be undesireable to use standard methods: those which find all eigenvalues of a matrix, but at a severe computational cost. Therefore, algorithms which locate only the eigenvalues that are needed by the user, while being more computationally efficient, are valuable.
Here, we present a meta-algorithm based on shifted Arnolidi iteration, which can locate real eigenvalues in an unbounded interval of the real line. Also, we discuss an algorithm called FEAST, developed at UMass Amherst, which can use complex numerical integration techniques to find eigenvalues which are located inside of any closed Jordan curve in the complex plane.