Analysis of the structure of the Krylov subspace in various preconditioned CGS algorithms

An improved preconditioned CGS (PCGS) algorithm has recently been proposed, and it performs much better than the conventional PCGS algorithm. In this paper, the improved PCGS algorithm is verified as a coordinative to the left-preconditioned system; this is done by comparing, analyzing, and executing numerical examinations of various PCGS algorithms, including the most recently proposed one. We show that the direction of the preconditioned system for the CGS method is determined by the operations of αk and βk in the PCGS algorithm. By comparing the logical structures of these algorithms, we show that the direction can be switched by the construction and setting of the initial shadow residual vector.