The application of Leja points to Richardson iteration and polynomial preconditioning

AbstractLet Ax=b be a linear system of algebraic equations with a large nonhermitian matrix A, and let σ(A) denote the spectrum of A. Assume that there is an explicitly known compact set K in the complex plane, such that σ(A)⊂K and 0∉K. We introduce sequences of Leja points {zj}∞j=0 for K and discuss convergence and stability properties of the Richardson iteration method with relaxation parameters δj≔1zj. By replacing K with a finite set Km and using reciprocal values of the Leja points for Km as relaxation parameters, we obtain a practical scheme for determining relaxation parameters for Richardson iteration. With a suitable choice of Km this scheme can be used to order any given sequence of relaxation parameters so as to avoid large amplification of roundoff errors. We also show how Leja points can be used to determine polynomial preconditioners