  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.
