论文标题
线性非矩阵铅笔的特征值问题的投影方法
Projection method for eigenvalue problems of linear nonsquare matrix pencils
论文作者
论文摘要
涉及复杂矩的特征者可以确定复杂平面中给定区域中的所有特征值以及常规线性基质铅笔的相应特征向量。复杂的力矩是从随机向量或矩阵中提取感兴趣的特征组件的过滤器。这项研究将常规本本特征问题的投影方法扩展到了奇异的非Quare情况，从而代替了与假符号分解的标准矩阵逆。扩展方法涉及与非矩阵相关的广义分解的轮廓积分给出的复杂矩。我们建立条件使该方法在复杂平面的规定区域中给出了所有有限特征值。在数值计算中，使用数值四元素近似轮廓积分。主要成本在于线性最小二乘问题的解决方案，这些问题是由正交点引起的，并且在实践中很容易平行。大型基质铅笔上的数值实验说明了此方法。该新方法比以前的方法更强大，更有效，并且基于实验结果，它在并行化的设置中被认为更有效。值得注意的是，所提出的方法在涉及成对非常接近特征值的情况下不会失败，并且克服了问题大小的问题。
Eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors of a regular linear matrix pencil. The complex moment acts as a filter for extracting eigencomponents of interest from random vectors or matrices. This study extends a projection method for regular eigenproblems to the singular nonsquare case, thus replacing the standard matrix inverse in the resolvent with the pseudoinverse. The extended method involves complex moments given by the contour integrals of generalized resolvents associated with nonsquare matrices. We establish conditions such that the method gives all finite eigenvalues in a prescribed region in the complex plane. In numerical computations, the contour integrals are approximated using numerical quadratures. The primary cost lies in the solutions of linear least squares problems that arise from quadrature points, and they can be readily parallelized in practice. Numerical experiments on large matrix pencils illustrate this method. The new method is more robust and efficient than previous methods, and based on experimental results, it is conjectured to be more efficient in parallelized settings. Notably, the proposed method does not fail in cases involving pairs of extremely close eigenvalues, and it overcomes the issue of problem size.
