Control-theoretic design of iterative methods for symmetric linear systems of equations

Abstract

Iterative methods for linear systems with a symmetric positive definite coefficient matrix are designed from a control-theoretic viewpoint. In particular, it is shown that a control-theoretic approach loosely based on m-step dead beat control of the error or residual system, with a suitable definition of error norm can be utilized to design new iterative methods that are competitive with the popular Barzilai-Borwein method, that is well known to be an efficient method with low computational cost. Numerical experiments are reported on to confirm the claimed results.