Convergence in Newtons Method when derivative is replaced by some constant.

Question

Suppose we replace the derivative by $d$ in Newtons method, i.e.

$$ x_{k+1} = x_k - \frac{f(x_k)}{d}.$$

For what conditions on $d$ will this be locally convergent?

Answer 1

Let $g(x)=x-f(x)/d$. If $x^*$ is a solution of $f(x)=0$, one condition would be that $$ |g'(x^*)|=\Bigl|1-\frac{f'(x^*)}{d}\Bigr|<1. $$ Since in general you do not know $^*$ a priori, a more practical condition is:

1. $[a,b]$ is an interval in which there is a root of the equation $f(x)=0$.
2. $$ |g'(x)|=\Bigl|1-\frac{f'(x^*)}{d}\Bigr|<1\quad\forall x\in[a,b]. $$