We analyze several versions of Jacobi's method for the symmetric eigenvalue problem. Our goal is to reduce the asymptotic cost of the algorithm as much as possible, as measured by the number of arithmetic operations performed and associated (serial or parallel) communication, i.e., the amount of data moved between slow and fast memory or between processors in a network. The first half of this effort, which considers the serial setting, is presented here; this paper contains rigorous complexity bounds for a variety of serial Jacobi algorithms, built on both classic $O(n^3)$ matrix multiplication and fast, Strassen-like $O(n^{ω_0})$ alternatives. In the classical case, we show that a blocked implementation of Jacobi's method attains the communication lower bound for $O(n^3)$ matrix multiplication (and is therefore expected to be communication optimal among $O(n^3)$ eigensolvers). In the fast setting, we demonstrate that a recursive version of blocked Jacobi can go further, reaching essentially optimal complexity in both measures. We also derive analogous complexity bounds for (one-sided) Jacobi SVD algorithms. A forthcoming sequel to this paper will extend our complexity analysis to the parallel case.

翻译：本论文分析求解对称特征值问题的若干版本雅可比法。我们的目标是在不增加渐进成本的前提下，尽可能降低算法代价——该代价通过算术运算次数及其（串行或并行）通信量（即慢速与快速内存之间或网络中处理器之间传输的数据量）来衡量。本文侧重于串行场景，给出了严谨的复杂度下界，涵盖多种基于经典$O(n^3)$矩阵乘法与快速类斯特拉森$O(n^{ω_0})$替代算法的串行雅可比算法。在经典情形下，我们证明分块实现的雅可比法可达到$O(n^3)$矩阵乘法的通信下界（因此有望成为$O(n^3)$特征值求解器中通信最优的算法）。在快速设置中，我们展示递归分块雅可比法能进一步突破，在两个度量上均达到近乎最优的复杂度。此外，我们还推导了（单边）雅可比SVD算法的相似复杂度界。本文的后续续篇将把复杂度分析扩展至并行情形。