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 ∈ Ω ⊂ R2, where Ω is simply connected and compact, with boundary Γ. No matter how differentiable the function A is (even analytic), in general the singular values lose all smoothness at points where they coalesce. In this work, we propose and implement algorithms which locate points in Ω where the singular values coalesce. Our algorithms are based on the interplay between coalescing singular values in Ω, and the periodicity of the SVD-factors as one completes a loop along Γ.