Condition and Convergence of Linear Iterative Schemes
by
Ray Zahar
Simon Fraser University, Vancouver, British Columbia, Canada

A theory of condition for linear, non-stationary, iterative schemes is developed. Perturbations in all the data, including the defining matrices, is considered. It is demonstrated that many iterative systems are exponentially ill-conditioned - and thus the obvious numerical methods for their solution are unstable - because their desired solutions are asymptotically subdominant with respect to some complementary solutions of their functional equations. An ill-conditioned iterative problem, however, can often be transformed to a replacement problem: one which is well-conditioned and which shares the same desired solution, but which requires no additional defining values. In this talk we show that an ill-conditioned initial value problem can be recast as a well-conditioned boundary value problem expressed in asymptotic form, and we present a theorem for asymptotic convergence. Stable numerical algorithms are presented, and we demonstrate that they can be expressed uniformly as a method of matrix decomposition. Applications to the numerical solution of differential equations are discussed.i

Date received: April 25, 2001