Stable Pivoting for the Fast Factorization of Cauchy-Like Matrices

Recent work by Sweet and Brent on the fast factorization of Cauchy-like matrices through a fast version of Gaussian elimination with partial pivoting has uncovered a potential stability problem which is not present in ordinary Gaussian elimination: excessive growth in the generators used to represent the matrix and its Schur complements can lead to large errors. A natural way to x this problem has been proposed by Gu. The idea is to re-factor the generators at each stage of Gaussian elimination so that one of the generator matrices remains well-conditioned. This factorization of the generators permits an eecient pivoting scheme with properties which are similar to complete pivoting. As the algorithm was presented, the additional computation required is substantial relative to the computation involved in the unmodiied algorithm. In this paper, we introduce a pivoting scheme which is more eecient than that of Gu and modify the analysis of Sweet and Brent to show that, with such a pivoting scheme, stability is guaranteed if the generators do not grow much larger than the original Cauchy matrix. This seems to have been implicit in the analysis of Sweet and Brent and in that of Gu, but in both cases the bounds make it appear that it is also important that the generators remain well-conditioned so as to avoid cancellations in inner products used to compute elements of the Schur complements. Experimental evidence and analogy with the occurence of element growth in Gaussian elimination with partial pivoting suggests that the pivoting scheme should be suucient in most cases to prevent growth in the generators. Applying the new pivoting scheme without computing new generators represents a substantial simpliication which should not compromise stability.