Optimizing a Parallel Self-verified Method for Solving Linear Systems

Solvers for linear equation systems are commonly used in many different kinds of real applications, which deal with large matrices. Nevertheless, two key problems appear to limit the use of linear system solvers to a more extensive range of real applications: computing power and solution correctness. In a previous work, we proposed a method that employs high performance computing techniques together with verified computing techniques in order to eliminate the problems mentioned above. This paper presents an optimization of a previously proposed parallel self-verified method for solving dense linear systems of equations. Basically, improvements are related to the way communication primitives were employed and to the identification of the points in the algorithm in which mathematical accuracy is needed to achieve reliable results.

[1]  G. Bohlender.  What Do We Need Beyond IEEE Arithmetic? , 1990, Computer Arithmetic and Self-Validating Numerical Methods.

[2]  G. Corliss,et al.  C-Xsc: A C++ Class Library for Extended Scientific Computing , 1993 .

[3]  Siegfried M. Rump,et al.  Kleine Fehlerschranken bei Matrixproblemen , 1980 .

[4]  Siegfried M. Rump,et al.  INTLAB - INTerval LABoratory , 1998, SCAN.

[5]  Paulo Fernandes,et al.  Parallel Selfverified Method for Solving Linear Systems , 2006 .

[6]  Siegfried M. Rump,et al.  Accurate Sum and Dot Product , 2005, SIAM J. Sci. Comput..

[7]  Markus Grimmer,et al.  An MPI Extension for the Use of C-XSC in Parallel Environments , 2005 .

[8]  Jack Dongarra,et al.  MPI: The Complete Reference , 1996 .

[9]  Christian P. Ullrich,et al.  Computer Arithmetic and Self-Validating Numerical Methods , 1990, Notes and reports in mathematics in science and engineering.

[10]  T. Ogita,et al.  Accurate sum and dot product with applications , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[11]  Ulrich W. Kulisch,et al.  C++ Toolbox for Verified Scientific Computing I: Basic Numerical Problems , 1997 .