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Broyden’s Good Method

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).