Look-ahead in Bi-CGSTAB and other product methods for linear systems

The Lanczos method for solvingAx = b consists in constructing the sequence of vectorsxk such thatrk =b − Axk =Pk(A)r0 wherePk is the orthogonal polynomial of degree at mostk with respect to the linear functionalc whose moments arec(ξi) =ci = (y, Air0).In this paper we discuss how to avoid breakdown and near-breakdown in a whole class of methods defined byrk =Qk(A)Pk(A)r0,Qk being a given polynomial. In particular, the case of the Bi-CGSTAB algorithm is treated in detail. Some other choices of the polynomialsQk are also studied.

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