高斯消元中的元素增长
Entry growth in Gaussian elimination
- Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究高斯消元中增长因子的最坏情况行为，确定了完全与 rook 选主元下最大增长因子为准多项式，证明部分选主元的指数增长在稀疏矩阵中仍存在且随机部分选主元不稳定，同时指出最优行置换的NP难性。
AI中文摘要：
高斯消元是数学中最古老的算法之一，也是求解非结构化线性方程组最常用的方法。其有限精度下的稳定性由增长因子控制，该因子衡量消元过程中产生的元素可达到的最大规模。自20世纪40年代以来，理解该量的最坏情况行为一直是数值分析的核心问题。本文在这一理解上取得重大突破，解决了多个开放问题：具体而言，确定了完全选主元与 rook 选主元下最大增长因子的渐近行为，证明二者在维度上均为准多项式；还表明部分选主元下的指数增长在稀疏矩阵中依然存在，且随机部分选主元存在同样的不稳定性；相比之下，证明每个矩阵都存在使增长为多项式的行置换，但寻找最优行置换是NP难问题。
英文摘要：
Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s. Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. In contrast, we show that every non-singular matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard.
