Understanding the Bisection Method
The Bisection Method is a robust numerical technique used to find the roots of a continuous function $f(x)$. The core idea is to repeatedly halve the interval $[a, b]$ containing a root, checking the midpoint $c = \frac{a+b}{2}$ until the desired level of accuracy is reached.
The Problem Statement
Given an initial interval $[0, 8]$ where $f(0) = -1$ and $f(8) = 1$, we want to find the number of iterations $n$ required to obtain an approximation with an accuracy of $0.25$.
The Mathematical Formula
The maximum error after $n$ iterations is given by the formula:
$$\text{Error} \leq \frac{b - a}{2^n}$$
Where:
- $b - a$ is the width of the initial interval.
- $n$ is the number of iterations.
Step-by-Step Solution
Define the variables:
- Initial interval $[a, b] = [0, 8]$
- Interval width $L = 8 - 0 = 8$
- Required accuracy $\epsilon = 0.25$
Set up the inequality: We need the error after $n$ steps to be less than or equal to $0.25$: $$\frac{8}{2^n} \leq 0.25$$
Solve for $n$: Multiply both sides by $2^n$ and divide by $0.25$: $$\frac{8}{0.25} \leq 2^n$$ $$32 \leq 2^n$$
Evaluate the power of 2: We know that $2^5 = 32$. Therefore, $n$ must be at least $5$.
Conclusion
To achieve an accuracy of $0.25$ starting from an interval of width $8$, 5 steps of the bisection method are required.
Why does this work?
At each step, we divide the current interval by 2.
- Start: width = 8
- Step 1: width = 4
- Step 2: width = 2
- Step 3: width = 1
- Step 4: width = 0.5
- Step 5: width = 0.25
After the 5th iteration, the interval width is exactly $0.25$, guaranteeing that our approximation is within the required tolerance.