Condition Number Estimation in a Solution Process of a Large Sparse Symmetric Linear System

Abstract

Various scientific computations require solving a system of linear equations with a large and sparse coefficient matrix, for which Krylov subspace methods like the CG method are widely used. In such iterative methods, a computed approximate solution is typically evaluated using the (relative) residual norm; however, the (relative) error norm is also of interest. There is a relationship between these two values, with the condition number playing a crucial role in this context. Motivated by this, the presented study focuses on a linear system with a large, sparse, real, symmetric, and positive definite coefficient matrix, and considers methods for estimating the condition number of the coefficient matrix during the solution process of the linear system. Specifically, we consider approaches based on the Lanczos method and the ES (Error vector Sampling) method. Through numerical experiments using sufficiently large sparse matrices, we evaluate the two approaches in terms of accuracy and execution time.