Accurate solution of dense linear systems, Part II: Algorithms using directed rounding

In Part I and this Part II of our paper we investigate how extra-precise evaluation of dot products can be used to solve ill-conditioned linear systems rigorously and accurately. In Part I only rounding to nearest is used. In this Part II we improve the results significantly by permitting directed rounding. Linear systems with tolerances in the data are treated, and a comfortable way is described to compute error bounds for extremely illconditioned linear systems with condition numbers up to about u−2/n, where u denotes the relative rounding error unit in a given working precision. We improve a method by Hansen/Bliek/Rohn/Ning/Kearfott/Neumaier. Of the known methods by Krawczyk, Rump, Hansen et al., Ogita and Nguyen we show that our presented Algorithm LssErrBnd seems the best compromise between accuracy and speed. Moreover, for input data with tolerances, a new method to compute componentwise inner bounds is presented. For not too wide input data they demonstrate that the computed inclusions are often almost optimal. All algorithms are given in executable Matlab code and are available from my homepage. © 2012 Elsevier B.V. All rights reserved.