Numerically accurate code synthesis for Gauss pivoting method to solve linear systems coming from mechanics

Abstract

In numerical analysis of mechanical problems, we usually have to solve huge linear systems, which may be non-symmetric or ill-conditioned. For these reasons, it is necessary to develop original and domain specific approaches to treat these family of systems. In this work, we introduce a new methodology to synthesize numerically accurate programs for the Gauss pivoting method. The synthesis is based on program transformation techniques and it is guided in its estimation of accuracy by interval arithmetic that computes the propagation of roundoff errors. We apply our code synthesis to the resolution of systems coming from finite element method arising from problems of Mechanics. We test our synthesizer on two problems concerning the flexion of a beam and the sliding contact of a viscoelastic body on a rigid foundation. Our experimental results show that the specialized synthesized code to solve the families of systems given in input is far more accurate and faster than the standard implementation of the Gauss method.