Iterative Methods with k-Part Splittings

Given linear invertible A:H → H where Ax = b. In the “classical” one-part iterative stationary schemes, we write A0xn +1 − A1 Xn = b to define xn + 1 in terms of the previous xn, once we write A = A0 − A'1 with A0-1 easy to find. In our k-part schemes, we write A0xn + k − A1xn + k − 1 − … − Ak xn = b to define Xn + k in terms of the previous xn + k − 1, … , Xn, once we write A = A0 − A1 − A2 − … − Ak 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).