Fixed-point analysis of Ogita-Aishima's symmetric eigendecomposition refinement algorithm for multiple eigenvalues

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.