A control-theoretic view of diagonal preconditioners
Eugenius Kaszkurewicz, Amit Bhaya, Patricia Vanessa de Ramos · International Journal of Systems Science · 1995
The condition number k(S) of a matrix S is the ratio of the largest singular value of S to the smallest, and is a very important quantity in the sensitivity and convergence analysis of many problems in numerical linear algebra. The optimal condition number of a matrix S is the minimum, over all positive diagonal matrices P, of K; (PS). In this paper we interpret the problem of finding the optimal preconditioner P that minimizes k( PS) as the equivalent problem of maximally clustering the poles of a suitably defined dynamical system by the choice of a positive diagonal stabilizing feedback matrix F ( = P2). This allows us: to give a control-theoretic proof of a characterization of perfect preconditioners, thereby making connections between various geometric inequalities and the condition number; and to use results on constrained linear quadratic optimal control to give an interpretation for optimal preconditioners.