High performance verified computing using C-XSC

Abstract

So called self-validating or self-verifying numerical methods allow to prove mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying mathematical theorems only fast finite precision machine arithmetic is used. The results are absolutely rigorous. We report on the accuracy as well as on the efficiency of the C++ class library C-XSC, our well known open source software tool designed to facilitate self-verifying numerical calculations. We focus mainly on solvers for dense and sparse interval linear systems. In recent years, these solvers have been improved significantly with respect to high performance computing within our bilateral Probral project HPVC (see Acknowledgments). As a motivating nontrivial example, where we need in an intermediate step an efficient solver for large dense interval linear systems, the computation of a verified functional enclosure for the solution of an integral equation is briefly discussed. The newest version C-XSC 2.5.1 released on June 9, 2011 allows using C-XSC in multi-threaded environments. The library as well as some further packages not mentioned in this paper are open source and freely available from the web site of the author’s research group Scientific Computing/Software Engineering at the University of Wuppertal: http://www2.math.uni-wuppertal.de/org/WRST/index_de.html.