Iterative Methods with k-Part Splittings

Abstract

Given linear invertible A:H → H where Ax = b. In the “classical” one-part iterative stationary schemes, we write A₀x_{n +1} − A₁ X_n = b to define x_{n + 1} in terms of the previous x_n, once we write A = A₀ − A'₁ with $A0-1$ easy to find. In our k-part schemes, we write A₀x_{n + k} − A₁x_{n + k − 1} − … − A_k x_n = b to define X_{n + k} in terms of the previous x_{n + k − 1}, … , X_n, once we write A = A₀ − A₁ − A₂ − … − A_k with $A0-1$ easy to find. To obtain convergence rates for k-part splittings, a theorem on the spectrum of a general operator-entried companion matrix is proved (Section 2). Then, we compare rates of convergence of k-part splittings with 1-part splittings. Among the results is an asymptotic recapturing of the Chebyshév semi-iterative method when A is positive definite, a favorable comparison with SOR without property A assumptions (cf. Remarks, Section 4).