AI 中文总结
研究对称特征值问题，基于谱隙信息提出新选主元策略，结合混合精度预处理器，在原始矩阵有聚类特征值时，该策略显著优于经典贪心方法。
AI 中文摘要
对称矩阵的扰动理论表明，谱隙小的特征值对非对角扰动更敏感，这意味着不同元素对特征值的影响不均衡。基于此，我们将谱隙信息纳入对称特征值问题的雅可比方法，提出一种新的选主元策略，该策略与仅由非对角元素大小决定的经典策略完全不同。结合将矩阵对角化到低精度的混合精度预处理器，数值实验表明，当原始矩阵具有聚类特征值时，所得策略能显著优于经典贪心方法。
英文摘要
Perturbation theory for symmetric matrices shows that eigenvalues with small spectral gaps are more sensitive to off-diagonal perturbation, implying that different entries affect the eigenvalues unevenly. Building on this insight, we incorporate spectral gap information into the Jacobi method for symmetric eigenvalue problems and propose a new pivoting strategy, which is completely different from classical ones governed solely by the magnitude of the off-diagonal entries. When combined with a mixed-precision preconditioner that diagonalizes the matrix to low precision, numerical experiments demonstrate that the resulting strategy can significantly outperform the classical greedy approach when the original matrix has clustered eigenvalues.
