Stable iterative refinement for solving linear systems with inaccurate computation

Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision or inaccurate hardware such as graphics processing units and analog devices. IR generally converges when the error associated with the inaccurate method is small, but it is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Computational experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme on two important types of inaccurate computing architectures, namely stochastic analog computing and low-precision digital arithmetic.