A Survey of Condition Number Estimation for Triangular Matrices

We survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix, and make recommendations concerning the use of the estimates in applications. Each of the methods is shown to bound the condition number; the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms. For each bound we examine by how much, at worst, it can overestimate or underestimate the condition number. Numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators. Key words, matrix condition number, triangular matrix, LINPACK, QR decomposition, rank estimation AMS(MOS) subject classification. 65F35