Broyden's method for solving F(x)=0

To initialize this algorithm, a starting point x⁽⁰⁾ and an estimate A₀ of J(x⁽⁰⁾)must be given. The iteration is then

$$\begin{eqnarray*}x^{(k+1)}&=&x^{(k)}-A_k^{-1}F(x^{(k)}),\\ s^{(k)}&=&x^{(k+1)}-... ...left(y^{(k)}-A_ks^{(k)}\right)(s^{(k)})^T}{s^{(k)}\cdot s^{(k)}} \end{eqnarray*}$$

for $k=1,2,3,\ldots$. The initial matrix A₀ is usually taken to be either J(x⁽⁰⁾) or a finite-difference approximation to it. The reader should notice that nothing in the development of Broyden's method guarantees that A_k is always nonsingular. Therefore, some enhancements to the algorithm are still necessary.