A FEAST algorithm with oblique projection for generalized eigenvalue problems

Abstract

The contour integral-based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non-Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non-Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design-efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.