  An integral equation based scheme is presented for the fast and accurate
computation of effective conductivities of two-component checkerboard-like
composites with complicated unit cells at very high contrast ratios. The scheme
extends recent work on multi-component checkerboards at medium contrast ratios.
General improvement include the simplification of a long-range preconditioner,
the use of a banded solver, and a more efficient placement of quadrature
points. This, together with a reduction in the number of unknowns, allows for a
substantial increase in achievable accuracy as well as in tractable system
size. Results, accurate to at least nine digits, are obtained for random
checkerboards with over a million squares in the unit cell at contrast ratio
10^6. Furthermore, the scheme is flexible enough to handle complex valued
conductivities and, using a homotopy method, purely negative contrast ratios.
Examples of the accurate computation of resonant spectra are given.

Computational Physics;Disordered Systems and Neural Networks;Materials Science;Numerical Analysis