论文标题
传达PSD矩阵上迭代特征值算法的动力学的可视化
A visualisation for conveying the dynamics of iterative eigenvalue algorithms over PSD matrices
论文作者
论文摘要
我们提出了一种可视化迭代特征值算法（例如QR算法）的动力学的新方法，该算法在PSD（阳性半明确）矩阵的重要特殊情况下。这种算法的许多微妙而重要的特性很容易找到这种方式。我们认为，这可能对数值线性代数的学生和研究人员都具有教学价值。迭代算法的固定点是视觉上获得的，并直观地分析其稳定性。很明显，迭代特征值算法“快速汇合”是一个模棱两可的问题，这取决于是否正在寻求特征值或特征向量。该演讲可能是一种新颖的，并且使用它，证明了有关一般迭代特征值算法的动力学的定理。目前在YouTube上托管的随附视频系列在充分利用可视化的互动性方面具有某些优势。
We propose a new way of visualising the dynamics of iterative eigenvalue algorithms such as the QR algorithm, over the important special case of PSD (positive semi-definite) matrices. Many subtle and important properties of such algorithms are easily found this way. We believe that this may have pedagogical value to both students and researchers of numerical linear algebra. The fixed points of iterative algorithms are obtained visually, and their stability is analysed intuitively. It becomes clear that what it means for an iterative eigenvalue algorithm to "converge quickly" is an ambiguous question, depending on whether eigenvalues or eigenvectors are being sought. The presentation is likely a novel one, and using it, a theorem about the dynamics of general iterative eigenvalue algorithms is proved. There is an accompanying video series, currently hosted on Youtube, that has certain advantages in terms of fully exploiting the interactivity of the visualisation.
