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.

In this paper, we show how “lattices” have been used in cryptography. In particular, we explain the definition, the notion of the (approximate) shortest vector problem, and lattice reduction algorithms. Then, we recall one of the applications, i.e., lattice-based Coppersmithʼs methods to solve integer/modular equations with applications to RSA cryptanalyses.

Ranking systems are often based on a weighted sum of several quantitative variables. We discuss how to determine the weight. Anatural weighting method is standardization. Other possibilities are principal component analysis and factor analysis. The author recently proposed an objective general index as a different way of weighting. In this paper, we study a relationship between the objective general index and the existing methods.