A FEAST algorithm with oblique projection for generalized eigenvalue problems

Abstract

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 (SS) 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. In this paper, we extend the FEAST algorithm to non-Hermitian problems. The approach can be summarized as follows: (i) to construct a particular contour integral to form a subspace containing the desired eigenspace, and (ii) to use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. We also address some implementation issues such as how to choose a suitable starting matrix and design good stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

Efficient and scalable calculation of complex band structure using Sakurai-Sugiura method

“…anks to this high-level hierarchical parallelism, complex moment-based eigensolvers achieve higher scalability [24,52]. Today, there are several complex moment-based eigensolvers including direct extensions of the Sakurai-Sugiura's approach [16][17][18][19]21] and FEAST eigensolver [38] and its variants [44,53,54]. For details, we refer [20] and reference therein.…”

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

“…The accuracy of the approximate eigenpairs computed by a Rayleigh–Ritz projection on the subspace created by can be improved by repeating the procedure in , using the most recent approximate eigenvectors as the new matrix V to multiply . If direct solvers are used to solve the complex linear systems in , this approach essentially amounts to subspace iteration with the matrix A replaced by ρ ( A ), that is, the FEAST package ; see also other works . It is also possible to consider contour integrals of other rational functions, for example, the scalar function u ∗ ( ζ I − A ) −1 v , with , as proposed in other works .…”

“…If direct solvers are used to solve the complex linear systems in (3), this approach essentially amounts to subspace iteration with the matrix A replaced by (A), that is, the FEAST package 1,4,14,17 ; see also other works. [18][19][20][21] It is also possible to consider contour integrals of other rational functions, for example, the scalar function u * ( I − A) −1 v, with u, v, ∈ C n , as proposed in other works. 2,3,22 The poles of this scalar function are the eigenvalues of A.…”

A FEAST Algorithm for the Linear Response Eigenvalue Problem

“…In [20], the FEAST algorithm was developed to solve nonlinear eigenvalue problems for eigenvalues that lie in a user-defined region. Some other work for the FEAST method can be found in [21][22][23][24]. Motivated by these facts, in this paper, we will continue the effort by extending the FEAST algorithm to LREP.…”