# na5

Treatment of near-breakdown in the CGS algorithm. Lanczos’ method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x k ) such thatr k =b−Ax k =P k (A)r 0 wherer 0=b−Ax 0.P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr k =P k 2 (A)r0. In the recurrence relation forP k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.

## References in zbMATH (referenced in 24 articles , 1 standard article )

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