Correct BFGS update

[timothy-nunn]

I believe that the BFGS update of the Lagrangian function Hessian in PyVMCON is incorrect. If the revision of the B matrix were incorrect, this could explain the "Workspace allocation error" being observed in ukaea/PROCESS#3619 (which was happening because B was not positive definite and hence the quadratic programming problem failed because it was non-convex, the error would now have a different message because of #21).

Crane report

The Crane report states the update of the B matrix occurs as follows:

It should be noted there is a typo in the second term that was identified during PyVMCON's development and it should read $\frac{\gamma\gamma^T}{\xi^T\gamma}$. (In some other places the denominator is $\gamma^T\xi$ but that does not change the result.)

Powell

Powell's paper agrees with the Crane report (as the Crane report is a summary of Powell's paper) except for the second term is correct.

BFGS Update

The BFGS wiki states that the update occurs as follows:

Note that this disagrees with both Powell's paper and the Crane report in that the second terms denominator (the first term in the other two papers) reads $B\xi\xi^TB^T$: the final B is transposed.

When I update PyVMCON to reflect this, none of the unit tests fail but it does allow the skipped PROCESS file to converge (albeit after 177 iterations).

[timothy-nunn]

As @mkovari has pointed out, the B matrix should be symmetric so a transpose should not change anything?

Having looked through the mathematics of the BFGS update equation, its possible PyVMCON implements it incorrectly and is not calculating the update matrices correctly. We should verify the shapes of the intermediate update matrices to ensure parity with NEA implementation VMCON.