Algorithm Steps
The steps of Broyden’s algorithm (Good Method) are as follows:
- Initialize \( \mathbf{x}_0 \) and \( \mathbf{B}_0 \) (e.g., \( \mathbf{B}_0 = \mathbf{I} \), the identity matrix).
- For \( k = 0, 1, 2, \dots \):
- Compute the step: \( \mathbf{s}_k = -\mathbf{B}_k^{-1} \mathbf{F}(\mathbf{x}_k) \).
- Update the solution: \( \mathbf{x}_{k+1} = \mathbf{x}_k + \mathbf{s}_k \).
- Compute \( \mathbf{y}_k = \mathbf{F}(\mathbf{x}_{k+1}) - \mathbf{F}(\mathbf{x}_k) \).
- Update the Jacobian approximation:
$$
\mathbf{B}_{k+1} = \mathbf{B}_k + \frac{(\mathbf{y}_k - \mathbf{B}_k \mathbf{s}_k) \mathbf{s}_k^\top}{\mathbf{s}_k^\top \mathbf{s}_k}.
$$
- Check for convergence: If \( \|\mathbf{F}(\mathbf{x}_{k+1})\| < \epsilon \), stop.