On nonsymmetric saddle point matrices that allow conjugate gradient iterations

Abstract

Linear systems in saddle point form are usually highly indefinite,which often slows down iterative solvers such as Krylov subspace methods. It has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix ${{\mathcal A}}$ whose spectrum is entirely contained in the right half plane. In this paper we study conditions so that ${{\mathcal A}}$ is diagonalizable with a real and positive spectrum. These conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form,with respect to which ${{\mathcal A}}$ is symmetric. In case the latter conditions are satisfied, there exists a well defined conjugate gradient (CG) method for solving linear systems with ${{\mathcal A}}$. We give an efficient implementation of this method, discuss practical issues such as error bounds, and present numerical experiments.