Making fast multiplication of polynomials numerically stable


Consider two polynomials and with multiple precision floating point coefficients. Although the product can in principle be computed efficiently using FFT multiplication, this algorithm is numerically stable only if the coefficients of are of the same order of magnitude and similarly for . In this paper, we present a new asymptotically fast multiplication algorithm which has a “better” numerically stability. We also provide some theoretical support for what we feel to be “better”, by introducing the concept of “relative Newton error”.

Keywords: polynomial, floating point number, multiplication, algorithm, FFT

A.M.S. subject classification: 65T50, 42-04, 68W30

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Errata. Unfortunately, an annoying error slipped into the preprint: the bound (3) is wrong. Consequently, the complexity estimates in the paper only work in one direction. Nevertheless, we have implemented a first version of the algorithm in Mathemagix and the approach works well in practice. In particular, we use it in the implementation of a method for computing the roots of complex polynomials using iterated Graeffe transforms and tangent numbers. We observed a good asymptotic complexity for this root finder, although the constant factor turned out to be rather bad.