Duality in conjugate gradient methods (Q1969406)

The authors present the reverse algorithms of \textit{Cs. J. Hegedüs} [Comput. Math. Appl. 21, No. 1, 71-85 (1991; Zbl 0727.65023)] in a new perspective and show how they are related to the more conventional algorithms if the latter is regarded as solving problems involving the original preconditioning matrices. The dual algorithms obtained are shown to be equivalent to the reverse algorithms of Hegedüs and are subsequently generalized to their block forms. It is shown how these may be used to construct dual (or reverse) algorithms for solving equations with nonsymmetric coefficient matrices using only short recurrences. Several reasons why some of these algorithms may be more numerically stable than their primal counterparts are presented.