🤖 AI Summary
This study addresses the challenge of simultaneously achieving numerical stability and computational efficiency in floating-point polynomial multiplication by proposing a novel algorithm with significantly simplified structure. The method integrates Newton iteration error analysis, divide-and-conquer strategies, and floating-point stability control theory to realize efficient and stable polynomial multiplication while preserving the optimal $O(np \log(np))$ time complexity and maintaining tight relative error bounds. Furthermore, the core ideas are extended to the near-convex Min-plus convolution problem, successfully optimizing the Bringmann–Cassis algorithm and yielding a logarithmic-factor speedup.
📝 Abstract
In the preprint [vdH08], van der Hoeven considers the problem of multiplying two polynomials with floating-point coefficients, and give an algorithm to compute the product with small relative Newton error in time $O(np \log(np))$, where $n$ is the degree and $p$ is the required precision. In this paper, we describe a significantly simpler algorithm with the same time complexity and error bound.
Independently, Bringmann and Cassis considered the near-convex min-plus convolution problem in [BC23a], and presented an algorithm to solve that problem. We observe that our algorithm can be adapted to that problem to speed up Bringmann and Cassis' algorithm by a logarithmic factor.