Accelerated subspace iteration with aggressive shift

The subspace iteration method is a very classical method for solving large general eigenvalue problems, and it is accepted as one of the reliable methods to solve large size eigenvalue problems through 1970–1980s. However, the classical subspace method is less efficient than Lanczos iteration method in terms of CPU time, because its parameters and iteration procedure were selected for today’s small and medium size eigenvalue problems. In the last 30 years, researchers have been trying to accelerate the classical subspace iteration method in different ways, such as, power acceleration, relaxation acceleration, so that it can deal with larger and larger eigenvalue problems arising in finite element analysis. Shifting technique is recognized as an efficient way to speed up the convergence rate for small and medium size eigenvalue problems. However the shifting cost for large size eigenvalue problems is expensive and thus makes it unacceptable. That is why almost all improvements in the last 20 years did not deal with shifts. In this paper, an aggressive shifting strategy is proposed based on a computable convergence criterion involving both eigenvalue and eigenvector instead of eigenvalue only. A wide range of numerical tests shows that the proposed aggressive shifting strategy can greatly decrease CPU time.