A framework for generalized conjugate gradient methods—with special emphasis on contributions by Rüdiger Weiss

Abstract

We discuss a general framework for generalized conjugate gradient methods, that is, iterative methods (for solving linear systems) that are based on generating residuals that are formally orthogonal to each other with respect to some true or formal inner product. This includes methods that generate residuals that are minimal with respect to some norm based on an inner product. A similar, even more general framework was introduced and extensively discussed by Weiss in his work. Weiss also emphasized in his work the relationship between certain pairs of orthogonal and minimal residual methods, where the results of the second method can be generated from those of the first method by applying the minimal residual smoothing process. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMR es) method, as well as the CGNE and CGNR versions of applying CG to the normal equations.