On spectral clustering of HSS preconditioner for generalized saddle-point matrices

Abstract

For the nonsingular generalized saddle-point matrix of a Hermitian positive definite or semidefinite leading block, we rigorously analyze clustering property for the eigenvalues of the corresponding preconditioned matrix with respect to the Hermitian and skew-Hermitian splitting preconditioner. The result shows that these eigenvalues are clustered around 0+, 2−, and a few points located on the unit circle centered at 1, as the iteration parameter is close to 0.