计算若尔当典范形的一种数值方法
A numerical method for computing the Jordan Canonical Form
摘要：
矩阵的若尔当典范形对扰动高度敏感，其数值计算仍然是一个艰巨的挑战。本文提出了一个正则化理论，建立了一个适定的最小二乘问题，找到最近的楼梯分解的最高余维的矩阵束。提出了一种计算数值若尔当典范形的两阶段算法。在第一阶段，该方法计算矩阵的Jordan结构和多重特征值的初始近似。然后在第二阶段通过迭代算法构造阶梯分解。因此，即使底层矩阵受到扰动，也可以高精度地计算数值Jordan正则分解和多重特征值。
英文摘要：
The Jordan Canonical Form of a matrix is highly sensitive to perturbations, and its numerical computation remains a formidable challenge. This paper presents a regularization theory that establishes a well-posed least squares problem of finding the nearest staircase decomposition in the matrix bundle of the highest codimension. A two-staged algorithm is developed for computing the numerical Jordan Canonical Form. At the first stage, the method calculates the Jordan structure of the matrix and an initial approximation to the multiple eigenvalues. The staircase decomposition is then constructed by an iterative algorithm at the second stage. As a result, the numerical Jordan Canonical decomposition along with multiple eigenvalues can be computed with high accuracy even if the underlying matrix is perturbed.
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