Equation (9.7) gives the rank-two \(\mathrm{BFGS}\) update of the inverse Hessian \(\mathbf{C}_{t}\) to \(\mathbf{C}_{t+1}\). Instead of using a two-rank update, we can consider a one-rank update, in which \(\mathbf{C}_{t}\) is updated to \(\mathbf{C}_{t+1}\) by the general rank-one formula:

\[ \mathbf{C}_{t+1}=\mathbf{C}_{t}+v_{t} \boldsymbol{r}_{t} \boldsymbol{r}_{t}^{\top} \]

Find values for the scalar \(v_{t}\) and vector \(r_{t}\), such that \(\mathbf{C}_{t+1}\) satisfies the secant condition \(\mathbf{C}_{t+1} g_{t}=\boldsymbol{\delta}_{t}\).