No / An algorithm called VPAStab is given for the acceleration of convergence of a sequence of vectors. It combines a method of vector-Padé approximation with a successful technique for stabilisation. More generally, this algorithm is designed to find the fixed point of the generating function of the given sequence of vectors, analogously to the way in which ordinary Padé approximants can accelerate the convergence of a given scalar sequence. VPAStab is justified in the context of its application to the solution of a large sparse system of linear equations. The possible breakdowns of the algorithm are listed. Numerical experiments indicate that these breakdowns can be classified either as pivot-type (type L) or as ghost-type (type D).

No / The effects of the three principal possible exact breakdowns which may occur using BiCGStab are discussed. BiCGStab is used to solve large sparse linear systems of equations, such as arise from the discretisation of PDEs. These PDEs often involve a parameter, say . We investigate here how the numerical error grows as breakdown is approached by letting tend to a critical value, say c, at which the breakdown is numerically exact. We found empirically in our examples that loss of numerical accuracy due stabilisation breakdown and Lanczos breakdown was discontinuous with respect to variation of around c. By contrast, the loss of numerical accuracy near a critical value c for pivot breakdown is roughly proportional to |¿c|¿1.