When solving $f(x)=0$ we can use Newton's method iteration $x_{k+1} = x_k - \frac{ f(x_k) }{f'(x_k)} $. What if we replace $f'(x_k)$ by a constant say $\alpha$:
$$ x_{k+1} = x_k - \frac{ f(x_k) }{\alpha} $$
Will this scheme always be locally convergent? If so, what is convergence rate?
try
if maybe see this as fixed-point iteration by making $g(x) = x- \frac{f(x) }{\alpha}$, then $g'(x) = 1 - \frac{f'(x) }{\alpha} $. If $x^*$ is the root, then $|g'(x^*)| = |1 - \frac{ f'(x^*) }{\alpha} |<1$
So $\alpha < f'(x^*) $ gives locally convergence. how can we find convergence rate?