Breakdown and near-breakdown control in the CGS algorithm using stochastic arithmetic

Abstract

In the Conjugate-Gradient-Squared method, a sequence of residualsr k defined byr k=P k 2 (A)r0 is computed. Coefficients of the polynomialsP 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 ∈ i in some ordering relations like |x≤∈ i . The user has to choose the ∈ 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 ∈ i with the estimation of the accuracy of some intermediate results.