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.

翻译：矩阵的约旦剖腹产形式对扰动非常敏感,其数字计算仍是一项艰巨的挑战。本文件提出了一个正规化理论,它确立了一个在最大迭代算法中找到最接近的阶梯分解的最小方块的问题。为计算数字约旦剖腹产形式,开发了两级算法。在第一阶段,该方法计算了矩阵的约旦结构,并初步近似于多个二次数值。然后,在第二阶段,通过迭代算法构建了楼梯分解。结果,约旦剖腹产的数值与多个二次数值一起可以非常精确地计算,即使基本矩阵被缠绕。