Abstract
In the Conjugate-Gradient-Squared method, a sequence of residualsr k defined byr k=P 2k (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.
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Chesneaux, JM., Matos, A.C. Breakdown and near-breakdown control in the CGS algorithm using stochastic arithmetic. Numer Algor 11, 99–116 (1996). https://doi.org/10.1007/BF02142491
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DOI: https://doi.org/10.1007/BF02142491