Structure of the polynomials in preconditioned BiCG algorithms and the   switching direction of preconditioned systems
  We present a theorem that defines the direction of a preconditioned system for the bi-conjugate gradient (BiCG) method, and we extend it to preconditioned bi-Lanczos-type algorithms. We show that the direction of a preconditioned system is switched by construction and by the settings of the initial shadow residual vector. We analyze and compare the polynomial structures of four preconditioned BiCG algorithms. 
Shoji Itoh and Masaaki Sugihara
math.NA cs.NA