Singular Values of Two-Parameter Matrices: An Algorithm To Accurately Find Their Intersections

Consider the Singular Value Decomposition (SVD) of a two-parameter function $A(x)$, $x\in \Omega\subset \R^2$, where $\Omega$ is simply connected and compact, with boundary $\Gamma$. No matter how differentiable the function $A$ is (even analytic), in general the singular values lose all smoothness at points where they coalesce. In thiswork, we propose and implement algorithms which locate points in $\Omega$ where the singular values coalesce. Our algorithms are based on the interplay between coalescing singular values in $\Omega$, and the periodicity of the SVD-factors as one completes a loop along $\Gamma$