In the mid-1980s Parsons (1987), and the author (Gutknecht, 1989) independently had the idea to generalize linear stationary k-step methods to stationary (k, 𝓁)-step methods, which were further generalized to nonstationary and even nonlinear (k, 𝓁)-step methods. Later, conjugate-gradient-type methods that are (k, 𝓁)-step methods of a similar sort were introduced and investigated in the PhD thesis of Barth (1996) under T. A. Manteuffel. Recently, the family of Induced Dimension Reduction (IDR) methods (Sonneveld and van Gijzen, 2008) aroused some interest for the class of linear nonstationary (k, 𝓁)-step methods because IDR(s) fits into it and belongs to a somewhat special subclass; see Gutknecht (2010). In this paper we first reformulate and review the class of nonlinear nonstationary (k, 𝓁)-step methods and a basic theoretical result obtained in the author’s 1989 article. Then we specialize to linear methods and introduce alternative iterations that can be used to implement them and compare them with the iterations suggested and investigated by Barth and Manteuffel.
