Revisiting (k, ℓ)-step methods

Abstract

Abstract. In the mid-1980s B.N. Parsons [15] and the author independently had the idea to generalize linear stationary k-step methods to stationary (k, )-step methods, which were further generalized to nonstationary and even nonlinear (k, )-step methods [3]. Later, conjugate-gradienttype methods that are (k, )-step methods were introduced and investigated in the thesis of Teri Barth [1]. Recently, the family of Induced Dimension Reduction (IDR) methods [18] aroused some interest for the class of linear nonstationary (k, )-step methods because IDR(s) fits into it and belongs to a somewhat special subclass [6]. In this paper we reformulate and review the class of linear nonstationary (k, )-step methods and a basic theoretical result obtained in [3].