Error¶
- The error is always bounded by $\epsilon_n\le|b-a|$
- $\epsilon_{n+1} = \epsilon_n/2$ $\rightarrow$ linear convergence.
- $\epsilon_0$, $\epsilon_1=\epsilon_0/2$, $\epsilon_2 = \epsilon_1/2 = \epsilon_0/4 = \epsilon_0/2^2$
- $\rightarrow \epsilon_n = \epsilon_0/2^n$.
- Hence:
$$n = \log_2\left(\frac{\epsilon_0}{\epsilon_n}\right)$$
- That is, to reduce the error from $\epsilon_0 \le |a-b|$ to some desired $\epsilon_n$ requires $n=\log_2(\epsilon_0/\epsilon_n)$ iterations.