  We introduce a new Projected Rayleigh Quotient Iteration aimed at improving
the convergence behaviour of classic Rayleigh Quotient iteration (RQI) by
incorporating approximate information about the target eigenvector at each
step. While classic RQI exhibits local cubic convergence for Hermitian
matrices, its global behaviour can be unpredictable, whereby it may converge to
an eigenvalue far away from the target, even when started with accurate initial
conditions. This problem is exacerbated when the eigenvalues are closely
spaced. The key idea of the new algorithm is at each step to add a
complex-valued projection to the original matrix (that depends on the current
eigenvector approximation), such that the unwanted eigenvalues are lifted into
the complex plane while the target stays close to the real line, thereby
increasing the spacing between the target eigenvalue and the rest of the
spectrum. Making better use of the eigenvector approximation leads to more
robust convergence behaviour and the new method converges reliably to the
correct target eigenpair for a significantly wider range of initial vectors
than does classic RQI. We prove that the method converges locally cubically and
we present several numerical examples demonstrating the improved global
convergence behaviour. In particular, we apply it to compute eigenvalues in a
band-gap spectrum of a Sturm-Liouville operator used to model photonic crystal
fibres, where the target and unwanted eigenvalues are closely spaced. The
examples show that the new method converges to the desired eigenpair even when
the eigenvalue spacing is very small, often succeeding when classic RQI fails.
