Polynomial Preconditioning for Indefinite Matrices

Abstract: Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices that make polynomial preconditioning more generally applicable. The new techniques include balancing the polynomial so that it produces a definite spectrum. Then a stability approach is given that is specialized for the indefinite case. Also, very complex spectra are examined. Then convergence estimates are given for polynomial preconditioning of real, indefinite spectra. Finally, tests are preformed of finding interior eigenvalues.