Efficient preconditioner updates for unsymmetric shifted linear systems

In this paper, we introduce a preconditioning strategy for unsymmetric shifted linear systems (A+αI)x=b(A+αI)x=b, which is a generalization of the scheme proposed by Bellavia et al. (2011). By modifying the nonzero entries in the LDULDU factorization of AA, we give a series of preconditioners with the same sparsity pattern as the seed   preconditioner. Theoretical analyses show that when α→0α→0 or ∞∞, the eigenvalues of the preconditioned systems will cluster about 1. Some practical examples, including non-Hermitian eigenvalue problems arising from the convection diffusion equation, are given to illustrate the efficiency of the preconditioners.