Hybrid Bi-CG methods with a Bi-CG formulation closer to the IDR approach

The Induced Dimension Reduction(s) (IDR(s)) method has recently been developed. Sleijpen et al. have reformulated the Bi-Conjugate Gradient STABilized (BiCGSTAB) method to clarify the relationship between BiCGSTAB and IDR(s). The formulation of Bi-Conjugate Gradient (Bi-CG) part used in the reformulated BiCGSTAB is different from that of the original Bi-CG method; the Bi-CG coefficients are computed by a formulation that is closer to the IDR approach. In this paper, we will redesign variants of the Conjugate Gradient Squared method (CGS) method, BiCGSTAB and the Generalized Product-type method derived from Bi-CG (GPBiCG)/BiCG×MR2 by using the Bi-CG formulation that is closer to the IDR approach. Although our proposed variants are mathematically equivalent to their counterparts, the computation of one of the Bi-CG coefficients differs, and the recurrences of the variants are also partly different from those of the original hybrid Bi-CG methods. Numerical experiments show that the variants of BiCGSTAB and GPBiCG/BiCG×MR2 are more stable and lead to faster convergence typically for linear systems for which the methods converge slowly (long stagnation phase), and that the variants of CGS attain more accurate approximate solutions.