A Note on Preconditioning for Indefinite Linear Systems

Preconditioners are often conceived as approximate inverses. For nonsingular indeenite 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 eeective even though they are in no sense approximate inverses. In many areas there arise matrix systems of the form Au = " A B T C 0 # " x y # = " f g # (1) where A 2 < nn and B; C 2 < mn with n m (see for example 3]). When A arises from a constrained variational or optimisation problem it is usual that B = C. There are also many problems where additionally A is symmetric, so that A is symmetric (see for example 2], 4], 9], 11]). Whether symmetric or not, the matrix A is generally indeenite (i.e. it has eigenvalues with both positive and negative real parts). Iterative solution of systems of the form (1) can be achieved by any of a number of methods: in particular Krylov subspace methods such as MINRES 10] or GMRES 12] are applicable in the symmetric and non-symmetric cases respectively. It is often advantageous (and in many situations necessary) to employ a preconditioner, P, with such iterative methods. The role of P is to reduce the number of iterations required for convergence whilst not increasing signiicantly the amount of computation required at each iteration. Intuitively if P can be chosen so that P ?1 is an inexpensive approximate inverse of A, then this might make a good preconditioner, however it is not necessary for a good preconditioner to have that P ?1 be an approximate inverse of A. A suucient condition for a good preconditioner is that the preconditioned matrix T = P ?1 A has a low degree minimum polynomial. This condition is more usually expressed in terms of T having only a few distinct eigenvalues: in this form we must insist that T is not degenerate (derogatory) or at least that it's Jordan canonical form has Jordan blocks of only small dimension. In this note we show how preconditioners can be derived for systems of the form (1) based upon anèxact' preconditioner which yields a preconditioned matrix with exactly