Avoiding breakdown in the CGS algorithm (Q1186617)

In Lanczos-type methods a sequence of vectors \(x\) is constructed such that \(r_ k=Ax_ k-b=P_ k(A)r_ 0\) where \((P_ k)\) is some family of orthogonal polynomials. \textit{P. Sonneveld} [SIAM J. Sci. Stat. Comput. 10, No. 1, 36-52 (1989; Zbl 0666.65029)] suggested the conjugate gradient squared (CGS) method where the polynomials \(P_ k\) are computed by means of some recurrence relations whose coefficients are given as ratios of some scalar products. Since the scalar product in the denominator can vanish, this can lead to the process breakdown in the considered method. In order to avoid the breakdown of the normal CGS process a generalization of the CGS method suggested by the authors for Lanczos- type algorithms is proposed. All the necessary formulae are supplied as well as numerical examples.