Divide and Conquer (D&C) is a widely used algorithmic strategy for symmetric eigenvalue decomposition. Its natural parallelism makes D&C attractive on modern multicore CPUs and GPUs, but existing eigenvalue-only routines often default to QR-based methods because conventional D&C still materializes or replays large transformation matrices during the conquer phase. This paper proposes a boundary-row D&C algorithm for eigenvalue-only computation. The key observation is that the conquer phase only needs selected boundary rows/columns rather than the full accumulated eigenvector matrix. By propagating these boundary rows directly through the recursion, the proposed algorithm reduces the memory requirement from quadratic to linear space while also eliminating unnecessary matrix-vector work in the conventional lazy-replay formulation. We provide the algorithm, its time and space complexity analysis, correctness and stability arguments, optimized CPU and GPU implementations, and an evaluation against QR and D&C routines in standard numerical libraries.

翻译：分治策略是广泛应用于对称特征值分解的算法框架。其天然并行性使其在现代多核CPU和GPU上极具吸引力，但现有仅计算特征值的程序常默认采用基于QR的方法，因为传统分治算法在合并阶段仍需生成或重放大型变换矩阵。本文提出一种面向仅特征值计算的边界行分治算法。核心发现是：合并阶段仅需选取的边界行/列，而非完整的累积特征向量矩阵。通过直接在递归中传播这些边界行，所提算法将存储需求从平方级降至线性空间，同时消除了传统惰性重放公式中不必要的矩阵-向量计算。本文提供算法实现、时间与空间复杂度分析、正确性与稳定性论证、优化的CPU与GPU实现，以及针对标准数值库中QR和分治程序的性能评估。