A Fast Contour-Integral Eigensolver for Non-Hermitian Matrices

Abstract

We present a fast contour-integral eigensolver for finding selected or all of the eigenpairs of a non-Hermitian matrix based on a series of analytical and computational techniques, such as the analysis of filter functions, quick and reliable eigenvalue count via low-accuracy matrix approximations, and fast shifted factorization update. The quality of some quadrature rules for approximating a relevant contour integral is analyzed. We show that a filter function based on the trapezoidal rule has nearly optimal decay in the complex plane away from the unit circle (as the mapped contour) and is superior to the Gauss--Legendre rule. The eigensolver needs to count the eigenvalues inside a contour. We justify the feasibility of using low-accuracy matrix approximations for the quick and reliable count. Both deterministic and probabilistic studies are given. With high probabilities, the matrix approximations give counts very close to the exact one. Our eigensolver is built upon an accelerated FEAST algorithm. Both the eigenvalue count and the FEAST eigenvalue solution need to solve linear systems with multiple shifts and right-hand sides. For this purpose and also to conveniently control the approximation accuracy, we use a type of rank structured approximations and show how to update the factorization for varying shifts. The eigensolver may be used to find a large number of eigenvalues, where a search region is then partitioned into subregions. We give an optimal threshold for the number of eigenvalues inside each bottom level subregion so as to minimize the complexity which is $\mathcal{O}(rn^{2})+\mathcal{O}(r^{2}n)$ to find all the eigenpairs of an order-$n$ matrix with maximum off-diagonal rank or numerical rank $r$. Numerical tests demonstrate the efficiency and accuracy and confirm the benefit of our acceleration techniques.