Approximating coalescing points for eigenvalues of Hermitian matrices of three parameters

We consider a complex Hermitian matrix valued function A=A(x), smoothly depending on parameters x in an open bounded region of R^3. We develop an algorithm to locate conical intersections of eigenvalues, that is parameter values where the eigenvalues of A coalesce. The crux of the method requires to monitor the geometric phase matrix of a Schur decomposition of A, as A varies on the surface S bounding the domain. We develop (adaptive) techniques to find the minimum variation decomposition of A along loops covering S and show how this can be used to detect conical intersections. Further, we give implementation details of a parallelization of the technique, as well as details relatively to the case of locating conical intersections for a few of A's dominant eigenvalues. Several examples illustrate the effectiveness of our technique.