Theoretical and numerical aspects of coalescing of eigenvalues and singular values of parameter dependent matrices

Abstract

In this thesis, we consider real matrix functions that depend on two parameters and study the problem of how to detect and approximate parameters' values where the singular values coalesce. We prove several results connecting the existence of coalescing points to the periodic structure of the smooth singular values decomposition computed around the boundary of a domain enclosing the points. This is further used to develop algorithms for the detection and approximation of coalescing points in planar regions. Finally, we present techniques for continuing curves of coalescing singular values of matrices depending on three parameters, and illustrate how these techniques can be used to locate coalescing singular values of complex-valued matrices depending on three parameters.