Breakdown and near-breakdown control in the CGS algorithm using stochastic arithmetic
Résumé
In the Conjugate-Gradient-Squared method, a sequence of residualsr k defined by $r_k=P_{k}^2 (A)r_0$ is computed. Coefficients of the polynomials $P_k$ may be computed as a ratio of scalar products from the theory of formal orthogonal polynomials. When a scalar product in a denominator is zero or very affected by round-off errors, situations of breakdown or near-breakdown appear. Using floating point arithmetic on computers, such situations are detected with the use of $\varepsilon_i$ in some ordering relations like $|x| \leq \varepsilon_i$. The user has to choose the $\varepsilon_i$ himself and these choices condition entirely the efficient detection of breakdown or near-breakdown. The subject of this paper is to show how stochastic arithmetic eliminates the problem of the $\varepsilon_i$ with the estimation of the accuracy of some intermediate results.