We all know the famous Newton-Raphson method in order to solve $\Psi(\xi)=0$:
\begin{align} \xi^{(\ell+1)} &= \xi^{(\ell)} + \Delta\!\xi^{(\ell)}, \\ \frac{d}{d\xi}\Psi(\xi^{(\ell)})\cdot\Delta\!\xi^{(\ell)} &= - \Psi(\xi^{(\ell)}). \end{align}
Often this method is modified in a way that the Jacobian is only evaluated once:
\begin{align} \xi^{(\ell+1)} &= \xi^{(\ell)} + \Delta\!\xi^{(\ell)}, \\ \frac{d}{d\xi}\Psi(\xi^{(0)})\cdot\Delta\!\xi^{(\ell)} &= - \Psi(\xi^{(\ell)}). \end{align} How is this method commonly called?
What about the modification \begin{align} \xi^{(\ell+1)} &= \xi^{(\ell)} + \Delta\!\xi^{(\ell)}, \\ J\cdot\Delta\!\xi^{(\ell)} &= - \Psi(\xi^{(\ell)}), \end{align} where $J\approx\frac{d}{d\xi}\Psi(\xi^{(\ell)})$ is only a (numerical) approximation to the Jacobian? How would you call that?
I have the names “inexact”, “simplified” and “modified” Newton's method flying in my head, but I am not a native English speaker and I am not sure which name belongs to which method.
First variant should be "simplified" (as used in the proof of the implicit/inverse function theorem), second variant "inexact" or, depending on how $J$ is obtained, "quasi-Newton", like in Broydens method.