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