Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters
Luca Dieci, Alessandra Papini, Alessandro Pugliese · SIAM Journal on Matrix Analysis and Applications · 2013
We consider a Hermitian matrix valued function $A(x)\in \mathbb{C}^{n\times n}$, smoothly depending on parameters $x\in \Omega\subset \mathbb{R}^3$, where $\Omega$ is an open bounded region of ${\mathbb R}^3$. We develop an algorithm to locate parameter values where the eigenvalues of $A$ coalesce: conical intersections of eigenvalues. The crux of the method requires one to monitor the geometric phase matrix of a Schur decomposition of $A$, as $A$ varies on the surface $S$ bounding $\Omega$. 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 relative to the case of locating conical intersections for a few of $A$'s dominant eigenvalues. Several examples illustrate the effectiveness of our technique.