  We show the power of the algebra of subspace collections developed in Chapter
7 of the book "Extending the Theory of Composites to Other Areas of Science
(edited by Milton, 2016). Specifically we accelerate the Fast Fourier Transform
schemes of Moulinec and Suquet and Eyre and Milton (1994, 1998) for computing
the fields and effective tensor in a conducting periodic medium by substituting
a subspace collection with nonorthogonal subspaces inside one with orthogonal
subspaces. This can be done when the effective conductivity as a function of
the conductivity $\sigma_1$ of the inclusion phase (with the matrix phase
conductivity set to $1$) has its singularities confined to an interval
$[-\beta,-\alpha]$ of the negative real $\sigma_1$ axis. Numerical results of
Moulinec and Suquet show accelerated convergence for the model example of a
square array of squares at $25\%$ volume fraction. For other problems we show
how $Q^*_C$-convex functions can be used to restrict the region where
singularities of the effective tensor as a function of the component tensors
might be found.
