Band-overlaps and dot-product

The (S-)orthonormalisation and subspace diagonalisation operations both require dot products between all pairs of bands. In the case of the (S-)orthonormalisation the overlaps are between all pairs of bands of a wavefunction, and so the result is Hermitian. For the subspace diagonalisation the dot-product is between each band of a wavefunction with each band of a different wavefunction - in fact since the second wavefunction is the result of a Hermitian operator applied to the first wavefunction the result should also be Hermitian, though at the moment we do not exploit this^4.1.

We distribute this by passing our data $n$ places to the right, and receiving data from $n$ places to the left, and dotting the received data with our local data. Then we increment $n$ by 1 and repeat the process until $n\geq N_b/2$.