Thursday, February 13, 2014

What is dreamed and math that should never be (part 3)

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="">arxiv:0901.2665</a>, but see later work by Polizzi."

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