Let \mathbf{H}_k be the approximation of the inverse Jacobian \mathbf{J}_k^{-1} at iteration k . The update rule for \mathbf{H}_k is:
\mathbf{H}_{k+1} = \mathbf{H}_k + \frac{(\mathbf{s}_k - \mathbf{H}_k \mathbf{y}_k) \mathbf{s}_k^\top \mathbf{H}_k}{\mathbf{s}_k^\top \mathbf{H}_k \mathbf{y}_k},where \mathbf{y}_k = \mathbf{F}(\mathbf{x}_{k+1}) - \mathbf{F}(\mathbf{x}_k) . The new approximation \mathbf{H}_{k+1} satisfies the **inverse secant equation**:
\mathbf{H}_{k+1} \mathbf{y}_k = \mathbf{s}_k.The solution is updated as:
\mathbf{x}_{k+1} = \mathbf{x}_k - \mathbf{H}_k \mathbf{F}(\mathbf{x}_k).