Given the equation: f(x) = x3 – x – 2
and the interval (1,2)
I was asked to calculate the iterations that are going to be needed
in order to get the root at 4 decimal places
I know how to apply the Bisection method, what I don't know is how to calculate the iterations for a given value of decimal places on the end result root.
Any help is greatly appreciated.
Best Answer
You need to iterate until the length of the bracketing interval is smaller than $10^{-5}$, so that rounding to 4 decimal places from both sides of the interval gives the same result.
The earliest that you can be cautiously confident about the first 4 decimal places is when the interval length is smaller $5·10^{-5}$, however there is then still a substantial chance that the last digit varies by $1$. Despite that the bound on the interval length is equivalent to a bound on the number of steps, here $15$ bisections, as $2^{-15}<3.1·10^{-5}$.
You can replace that condition by rounding the interval ends to 4 decimals and test for equality. While that gives absolute certainty, it can take a rather large number of steps to reach that point.
If you take $0.2$ and $0.7$ which are $0.5$ apart, and round both to integer, you get a different result. The same behavior under rounding to integer could happen with $0.497$ and $0.502$, so that interval length is not always a good measure to ensure stationarity in the decimal digits. Another case would be if the root were $0.4999…993$. Under bisection starting with the interval $[0,1]$ the upper interval bound will move in the first steps to $0.5$. Certainty that the first 4 decimal digits are indeed $0.4999$ requires reduction of the interval length to far below $5·10^{-5}$.