  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 hardware such as analog devices and graphics
processing units. IR generally converges when the error associated with the
solution method is small, but 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.
Numerical experiments verify our theoretical results and illustrate the
effectiveness of our proposed scheme.
