A CONTROL-THEORETIC VIEW OF DIAGONAL PRECONDITIONERS

The condition number kappa(S) of a matrix S is the ratio of the larges t singular value of S to the smallest, and is a very important quantit y in the sensitivity and convergence analysis of many problems in nume rical linear algebra. The optimal condition number of a matrix S is th e minimum, over all positive diagonal matrices P, of kappa(PS). In thi s paper we interpret the problem of finding the optimal preconditioner P that minimizes kappa(PS) as the equivalent problem of maximally clu stering the poles of a suitably defined dynamical system by the choice of a positive diagonal stabilizing feedback matrix F(=P-2). This allo ws us: to give a control-theoretic proof of a characterization of perf ect preconditioners, thereby making connections between various geomet ric inequalities and the condition number; and to use results on const rained linear quadratic optimal control to give an interpretation for optimal preconditioners.