To determine the number of iterations for the bisection method, a formula involving logarithms is used. The necessary number of iterations is calculated based on the initial interval and the desired decimal place accuracy, rounding up to the nearest whole number. An example calculation shows you need at least 5 iterations for an accuracy of three decimal places.
To find the number of iterations needed for the bisection method to achieve a desired decimal place accuracy, we can use a specific formula derived from the premise that each iteration halves the interval containing the root.
Calculating the Number of Iterations
The number of iterations n can be found using the following formula:
n ≥ (log(b - a) - log(ε)) / log(2)
Where:
a and b are the initial interval endpoints
ε is the desired accuracy (for example, for 3 decimal places, ε = 0.001)
The calculation uses logarithms because the number of places that the decimal point moves is equivalent to the number of zeros in the corresponding power of ten. To ensure that the number of iterations is an integer, the resulting value from the formula must be rounded up to the next whole number.
Example
If you want to find a root to within 0.001 accuracy and you start with an interval of length 1 (ε = 0.001, a = 0, b = 1), then:
n ≥ (log(1 - 0) - log(0.001)) / log(2)
n ≥ (log(1) - log(1×10⁻³)) / log(2)
n ≥ (0 - (-3)) / 0.693 (approximating log(2) as 0.693)
n ≥ 4.33
Therefore, at least 5 iterations are required to achieve a three-decimal-place level of accuracy.