Exploiting BiCGstab(ℓ) Strategies to Induce Dimension Reduction

IDR(s) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062] and BiCGstab( ) [G. L. G. Sleijpen and D. R. Fokkema, Electron. Trans. Numer. Anal., 1 (1993), pp. 11–32] are two of the most efficient short-recurrence iterative methods for solving large nonsymmetric linear systems of equations. Which of the two is best depends on the specific problem class. In this paper we describe IDRstab, a new method that combines the strengths of IDR(s) and BiCGstab( ). To derive IDRstab we extend the results that we reported on in [G. L. G. Sleijpen, P. Sonneveld, and M. B. van Gijzen, Appl. Numer. Math., (2009), DOI: 10.1016/j.apnum.2009.07.001], where we considered Bi-CGSTAB as an induced dimension reduction (IDR) method. We will analyze the relation between hybrid Bi-CG methods and IDR and introduce the new concept of the Sonneveld subspace as a common framework. Through numerical experiments we will show that IDRstab can outperform both IDR(s) and BiCGstab( ).