From Lucas!

"You can also diagonalize large matrices without explicit
diagonalization, and it's also related to the inverse. The key is to
use the Green's function G(z) = (z S - H)^{-1}, defined for the
generalized eigenvalue problem H psi = S psi E. Taking a contour
integral C in the complex plane which encloses the eigenvalues you want,
you can get the density matrix rho = -1/(2 pi i) INT[z on C] G(z),
which is essentially psi psi^T. Go complex analysis and residue
theorems!

There's a neat method which avoids the full inverse
and approximates the contour integral nicely, and it's super
parallelizable. I think Zhenfei or Zenghui gave it at a previous group
talk. Or maybe John. You can find it at <a href="http://arxiv.org/abs/0901.2665 ">arxiv:0901.2665</a>, but see later work by Polizzi."

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