An Accurate and Efficient Selfverifying Solver for Systems with Banded Coefficient Matrix

摘要

In this paper we discuss a selfverifying solver for systems of linear equations Aχ = b with banded matrices A and the future adaptation of the algorithms to cluster computers. We present an implementation of an algorithm to compute efficiently componentwise good enclosures for the solution of a sparse linear system on typical cluster computers. Our implementation works with point as well as interval data (data afflicted with tolerances). The algorithm is implemented using C-XSC library (a C~(++) class library for extended scientific computing). Our goal is to provide software for validated numerics in high performance environments using C-XSC in connection with the MPICH library. Actually, our solver for linear system with banded matrices runs on two different clusters: ALiCE at the University of Wuppertal and LabTeC at UFRGS. Our preliminary tests with matrix-matrix multiplication show that the C-XSC library needs to be optimized in several ways to be efficient in a high performance environment (up to now the main goal of C-XSC was functionality and portability, not speed). This research is based on a joint research project between German and Brazilian universities (BUGH, UKA, UFRGS and PUCRS).