High performance verified computing using C-XSC
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High performance verified computing using C-XSC Walter Krämer
Received: 13 March 2012 / Accepted: 24 December 2012 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
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. Keywords Verified computing · Self-validating methods · High performance computing · Parallelization · Thread-safety · Sparse methods · C-XSC Mathematics Subject Classification (2010)
Primary 65G20; Secondary 65G30
Communicated by Renata Hax Sander Reiser. To see/download the latest file release please consult the C-XSC web page http://www.math.uni-wuppertal. de/wrswt/xsc/cxsc_new.html. W. Krämer (B) Scientific Computing/Software Engineering, Faculty of Mathematics and Natural Sciences, University of Wuppertal, 42119 Wuppertal, Germany e-mail: [email protected]
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W. Krämer
1 Preliminary remarks This paper is based on a plenary talk the author has given at the conference CNMAC 2010, XXXIII Congresso Nacional de Matemática Aplicada e Computacional, Águas de Lindóia/SP, Brazil in 2010. The paper particularly summarizes newer developments concerning the C-XSC library and high performance verified computing (HPVC). One of the driving forces of these developments was the bilateral Brasilien/German Probral project HPVC (see the Acknowledgments at the end of this paper) funded by CAPES and DFG. The need for numerical validation is shown by simple examples and exemplarily the need for HPVC is demonstrated when functional enclosures of the solution of an integral equation are to be computed. Typical (sub)tasks require self-verifyin
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