2D Eigenvalue Problem II: Rayleigh Quotient Iteration and Applications

Abstract

In Part I of this paper, we introduced a 2D eigenvalue problem (2DEVP) and presented theoretical results of the 2DEVP and its intrinsic connetion with the eigenvalue optimizations. In this part, we devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The 2DRQI performs $2\times$ to $8\times$ faster than the existing algorithms for large scale eigenvalue optimizations arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.