Let be a root of $f ∈ [, ]$ and $f$ is differentiable and $_$ be a sequence of iterates obtained in bisection method. Then show that ...

Question

Then show that $ − _n = −f(_) /f'(_)$ where $ < _ < _n$

I know this statement represents the mean value theorem, I think It can be proved using proof of the mean value theorem but Are there any other methods to prove it using the bisection method?

Answer 1

The result comes directly from Lagrange's theorem in the interval $[\alpha, x_n]$:

$$ \underbrace{f(\alpha)}_{=0}-f(x_n)= f'(\xi_n)(\alpha -x_n) \Leftrightarrow \alpha -x_n = -\frac{f(x_n)}{f'(\xi_n)}, \quad \xi_n \in (\alpha, x_n) $$

This is not specific to the bisection method. If $x_n$ is a sequence that is supposed to converge to $\alpha$, this just tells you that $|f(x_n)|< \varepsilon$ is not a good stopping criteria when $|f'|$ is small.