A Note on Preconditioning for Indefinite Linear Systems
Malcolm F. Murphy, Gene Howard Golub, Andy Wathen · SIAM Journal on Scientific Computing · 2000
Preconditioners are often conceived as approximate inverses. For nonsingular indefinite matrices of saddle-point (or KKT) form, we show how preconditioners incorporating an exact Schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues. Thus approximations of the Schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses.