Accurate Numerical Algorithm by Iterative Refinement in High Precision

Abstract

This paper is concerned with an iterative algorithm for an accurate inverse matrix factorization which requires an algorithm for accurate dot product, which helps to treat ill-conditioned matrices. Following the results by Rump[15], Ogita[8], Ogita and Oishi[9] derived an such iterative algorithm. Firstly, we explicate Rumpʼs method[15] for inverting an ill-conditioned matrix. We then focus on the algorithm for an accurate inverse Cholesky factorization via the adaption of Rumpʼs framework directly to shifted Cholesky factorization of symmetric and positive definite matrices. Furthermore, we present some numerical results from a comparison of the algorithm with a standard Cholesky factorization using long precision arithmetic [5, 6], in terms of measured computing time for verifying the positive definiteness of an input matrix.