Avoiding breakdown in variants of the BI-CGSTAB algorithm

The BI-CG method and its variants such as CGS, BI-CGSTAB, and BI-CGSTAB2 for solving nonsymmetric linear systems of equations can suffer from breakdown. By using the theory of the formal orthogonal polynomials we present a breakdown-free BI-CGSTAB algorithm and a breakdown-free BI-CGSTAB2 algorithm. Only exact breakdowns are cured accurately.