  Recently, Ogita and Aishima proposed an efficient eigendecomposition
refinement algorithm for the symmetric eigenproblem. Their basic algorithm
involves division by the difference of two approximate eigenvalues, and can
become unstable when there are multiple eigenvalues. To resolve this problem,
they proposed to replace those equations that casue instability with different
equations and gave a convergence proof of the resulting algorithm. However, it
is not straightforward to understand intuitively why the modified algorithm
works, because it removes some of the necessary and sufficient conditions for
obtaining the eigendecomposition. We give an answer to this question using
Banach's fixed-point theorem.
