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:
$$ \mathbf{x}_{k+1} = \mathbf{x}_k - \mathbf{H}_k \mathbf{F}(\mathbf{x}_k). $$