A Note On Symmetric Positive Definite Preconditioners for Multiple Saddle-Point Systems

We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, 1 and -1, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, and specify the computational work associated with applying the new preconditioner inexactly. Numerical results validate our theoretical findings.