Stable Numerical Algorithms for Equilibrium Systems

An equilibrium system (also known as a KKT system, a saddlepoint system, or a sparse tableau) is a square linear system with a certain structure. G. Strang has observed that equilibrium systems arise in optimization, finite elements, structural analysis, and electrical networks. Recently, G. W. Stewart established a norm bound for a type of equilibrium system in the case that the "stiffness" portion of the system is very ill-conditioned. In this paper we investigate the algorithmic implications of Stewart's result. We show that all standard textbook algorithms for equilibrium systems are unstable. Then we show that a certain hybrid method has the right stability property. 1 Equilibrium systems Recently, Strang [1986] has observed that the problem of solving the structured linear system / D \ A T 0 !/ x y ! = / b c ! (1) This work supported by an NSF Presidential Young Investigator grant, with matching funds received from Xerox Corp. y Department of Computer Scie...