Towards Verified Rounding Error Analysis for Stationary Iterative Methods

Abstract

Iterative methods for solving linear systems serve as a basic building block for computational science. The computational cost of these methods can be significantly influenced by the round-off errors that accumulate as a result of their implementation in finite precision. In the extreme case, round-off errors that occur in practice can completely prevent an implementation from satisfying the accuracy and convergence behavior prescribed by its underlying algorithm. In the exascale era where cost is paramount, a thorough and rigorous analysis of the delay of convergence due to round-off should not be ignored. In this paper, we use a small model problem and the Jacobi iterative method to demonstrate how the Coq proof assistant can be used to formally specify the floating-point behavior of iterative methods, and to rigorously prove the accuracy of these methods.