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.