  This paper addresses the question of what exactly is an analogue of the
preconditioned steepest descent (PSD) algorithm in the case of a symmetric
indefinite system with an SPD preconditioner. We show that a basic PSD-like
scheme for an SPD-preconditioned symmetric indefinite system is mathematically
equivalent to the restarted PMINRES, where restarts occur after every two
steps. A convergence bound is derived. If certain information on the spectrum
of the preconditioned system is available, we present a simpler PSD-like
algorithm that performs only one-dimensional residual minimization. Our primary
goal is to bridge the theoretical gap between optimal (PMINRES) and PSD-like
methods for solving symmetric indefinite systems, as well as point out
situations where the PSD-like schemes can be used in practice.
