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).