RefinEig: a Program to Refine Eigensystems W. Kahan Mathematics, and E.E.&C.S. Depts. University of California Berkeley CA 94720-1776 Abstract: Software to compute eigenvalues and eigenvectors of matrices can hardly be deemed infallible. Results are often rather less accurate than deserved by the data, sometimes far less accurate than different software could have provided at the same cost. RefinEig attempts to tidy up those results at less cost than if the whole computation were repeated carrying higher precision throughout. RefinEig uses a novel iterative refinement algorithm designed to cope well with the most common cause of inaccuracy, namely isolated pairs of nearly coincident eigenvalues. ( No way can exist to cope economically with all possible pathologies.) Examples abound for which RefinEig improves accuracy spectacularly, but usually the improvement is modest. Curiously, RefinEig works far better with the most popular floating-point arithmetics than with the more expensive ones despite that all of them purport to conform to the same IEEE Standard 754 for floating-point arithmetic.