Locating coalescing singular values of large two-parameter matrices

Consider a matrix valued function A(x) ∈ R, m ≥ n, smoothly depending on parameters x ∈ Ω ⊂ R, where Ω is simply connected and bounded. We consider a technique to locate parameter values where some of the q dominant (q ≤ n) singular values of A coalesce, in the specific case when A is large and m > n ≫ q. Notation. An m × n real matrix is indicated with A ∈ Rm×n. We always consider the 2-norm for vectors and matrices. A matrix valued function A : R → Rm×n, continuous with its first l derivatives (l ≥ 0), is indicated as A ∈ Cl(R,Rm×n). If l = 0, we also simply write A ∈ C. If A ∈ Cl(R,Rm×n) is periodic of (minimal) period τ > 0, we write it as A ∈ C τ (R,R m×n). With Ω ⊂ R2 we indicate an open and bounded simply connected planar region, and x = (x1, x2) will be coordinates in Ω. For a function A(x), x ∈ Ω, we will write A ∈ Cl(Ω,Rm×n) as appropriate.