# BFGS update

A way to update an approximation of the hessian, used for unconstrained optimization. The update is

$LaTeX: B' = B + \frac{(q - Bp)(q - Bp)^T}{p^T(q - Bp)},$

where $LaTeX: B'$ is the update from $LaTeX: B$ and $LaTeX: p$ and $LaTeX: q$ are defined in the BFGS method for the $LaTeX: k$-th iteration. Taking the inverse to compare with the DFP update,

$LaTeX: H' = H + \frac{1 + q^T H q}{(q^T p)^2} \left( pp^T \right) - \frac{1}{p^T q} \left( pq^T H + H qp^T \right)$

where $LaTeX: H'$ is the update from $LaTeX: H$.