AI 中文总结
研究一般复矩阵特征值问题的雅可比型方法，给出渐近二次收敛新证明及收敛区域界，引入基于理论结果的两种预处理器，通过混合精度计算降低成本，数值实验表明算法稳健且特征值准确。
AI 中文摘要
本文研究了一种用于具有简单特征值的一般复矩阵特征值问题的雅可比型方法。该方法应用基本三角相似变换来消除选定的非对角元素，收敛时能产生高精度特征值。我们给出了其渐近二次收敛的新证明，并推导了一个明确、可验证的界来描述保证收敛的区域。为使该方法适用于远未接近对角形式的矩阵，引入了预处理策略。使用了两种基于理论收敛结果的预处理器，预处理器以较低精度计算以降低计算成本，相关相似变换以工作精度或更高精度应用以保留谱信息，主算法以工作精度执行。数值实验表明所得算法稳健且能产生非常准确的特征值。
英文摘要
The paper studies a Jacobi-type method for the eigenvalue problem of general complex matrices with simple eigenvalues. The method applies elementary triangular similarity transformations in order to annihilate selected off-diagonal elements and, when convergent, produces highly accurate eigenvalues. We give a new proof of its asymptotic quadratic convergence and derive an explicit, verifiable bound that describes the region in which this convergence is guaranteed. To make the method applicable well beyond matrices already close to the diagonal form, we introduce a preconditioning strategy. We use two types of preconditioners, both based on theoretical convergence results. The preconditioner is computed at lower precision to reduce computational cost, the associated similarity transformation is applied either at working or at higher precision, to preserve spectral information, while the main algorithm performs at working precision. Numerical experiments demonstrate that the resulting algorithm is robust and produces very accurate eigenvalues.
Comments29 pages, 12 figures, 5 tables
