Convergence analysis of an algorithm for accurate inverse Cholesky factorization

This paper is concerned with factorization of symmetric and positive definite matrices which are extremely ill-conditioned. Following the results by Rump (1990), Oishi et al. (2007, 2009) and Ogita (2010), Ogita and Oishi (2012) derived an iterative algorithm for an accurate inverse matrix factorization based on Cholesky factorization for such ill-conditioned matrices. We analyze the behavior of the algorithm in detail and give reasons for convergency by the use of numerical error analysis. Main analysis is that each iteration reduces the condition number of a preconditioned matrix by a factor around the relative rounding error unit until convergence. This behavior is consistent with the numerical results.