This paper presents an adaptive randomized algorithm for computing the butterfly factorization of a $m\times n$ matrix with $m\approx n$ provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The resulting factorization is composed of $O(\log n)$ sparse factors, each containing $O(n)$ nonzero entries. The factorization can be attained using $O(n^{3/2}\log n)$ computation and $O(n\log n)$ memory resources. The proposed algorithm applies to matrices with strong and weak admissibility conditions arising from surface integral equation solvers with a rigorous error bound, and is implemented in parallel.

中文翻译：

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通过随机矩阵向量乘法进行蝴蝶分解

本文提出了一种自适应随机算法，用于计算具有 $m\approx n$ 的 $m\times n$ 矩阵的蝶式分解，前提是矩阵及其转置都可以快速应用于任意向量。结果分解由 $O(\log n)$ 稀疏因子组成，每个包含 $O(n)$ 非零条目。可以使用 $O(n^{3/2}\log n)$ 计算和 $O(n\log n)$ 内存资源来实现分解。所提出的算法适用于由具有严格误差界限的表面积分方程求解器产生的具有强和弱可容许条件的矩阵，并并行实现。