An eigenmode solution algorithm based on high-order power iteration with fractally ordered shifts Original Research Article

This paper presents a technique which allows the high-order power method with shifts to solve reliably for mid-spectrum eigenmodes. Normally, the method fails at high orders because undesirable modes can never be reduced below round-off error levels, even though analytically those modes have been reduced to much lower levels. This paper describes a technique of reordering the shifts which drastically reduces the round-off error effect. The ordering technique is based on a fractal mapping of the eigenmode spectrum which is self-similar at all scales along the spectrum. The technique is rather stunning in its capability. For example, an eigenmode problem of 10 000 unknowns based on the finite difference electromagnetic wave equations can be solved for the first 50 modes in a few hours on a PC, using single precision arithmetic. Furthermore, any given interval of the spectrum can be searched for an eigenmode without the need to find lower modes. And because the fractal mapping technique treats all regions and all scales of the spectrum identically, mid-spectrum modes pose no more difficult a problem than do the lowest or highest modes