Iterative processes in Krylov–Sonneveld subspaces

The paper presents a generalized block version of the Induced Dimension Reduction (IDR) methods in comparison with the Multi–Preconditioned Semi-Conjugate Direction (MPSCD) algorithms in Krylov subspaces with deflation and low-rank matrix approximation. Common and individual orthogonality and variational properties of these two methodologies are analyzed. It is demonstrated, in particular, that for any sequence of Krylov subspaces with increasing dimensions there exists a sequence of the corresponding shrinking subspaces with decreasing dimensions. The main conclusion is that the IDR procedures, proposed by P. Sonneveld and other authors, are not an alternative to but a further development of the general principles of iterative processes in Krylov subspaces.