I have been working on a problem involving the bisection method and the estimation of the number of iterations required to approximate the root of a given function to three decimal places. I have made some progress and wanted to seek your insights and expertise on my approach.
The problem statement is as follows: I am tasked with finding the total number of iterations needed to approximate the root of the equation $f(x) = x^4 - x^3 -2=0$ within three decimal places, in the interval $[1,2]$.
The error estimate is given by the relation : $\left| x_n - \hat{x}\right| \leq 0.5 \times 10^{-3}$ and it suffices to show that the upper bound of the error is : $\frac{2-1}{2^n}\leq 0.5 \times 10^{-3}$
So in order to estimate the number of iterations required I used the following formula: $n \geq \frac{{\log(2 - 1) - \log(\epsilon)}}{{\log(2)}}$, where :
- n is the number of iterations.
- $\epsilon$ is the desired accuracy, in this case, $\epsilon = 0.5 \times 10^{-3}$ for three decimal places and rounding.
My calculations resulted in $n \geq 10.96578$, which I rounded up to $n=11$ indicating that at least 11 iterations are needed to approximate the root within three decimal places.
My question and concern: In the 11th iteration, the estimated root $x_{11} = 1.54345703$ rounds to $1.543$, whereas the real root is $\hat{x} = 1.54368901$ (which rounds to $1.544$). As you can see, we have not approximated the root in 3 decimal places as expected to do from the minimum number of iterations we found above. I am interested in discussing this small discrepancy between the estimated and real root. (If we do 12 iterations we obtain the correct approximation to $1.544$).
Is this level of discrepancy expected, even though we required the accuracy to be less than $0.5 \times 10^{-3}$? If so, why does it occur?
Thank you for your time and assistance.