Reliable tracking algorithms for principal and minor eigenvector computations

Many problems in control and signal processing require the tracking of certain eigenvectors of a time-varying matrix; the eigenvectors associated with the largest eigenvalues are called the principal eigenvectors and those with the smallest eigenvalues the minor eigenvectors. This paper presents a novel algorithm for tracking minor eigenvectors. One interesting feature, inherited from a recently proposed minor eigenvector flow upon which part of this work is based, is that the algorithm can be used also for tracking principal eigenvectors simply by changing the sign of the matrix whose eigenvectors are being tracked. The other key feature is that the algorithm has a guaranteed accuracy. Indeed, the algorithm is based on a flow which can be interpreted as the combination of a homotopy method and a Newton method, the purpose of the latter to compensate for discretisation errors.