, Volume 29, Issue 1-3, pp 97-105

The Breakdowns of BiCGStab

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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 γ. We investigate here how the numerical error grows as breakdown is approached by letting γ tend to a critical value, say γc, 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 γ around γc. By contrast, the loss of numerical accuracy near a critical value γc for pivot breakdown is roughly proportional to |γ−γc|−1.