Understanding the Bisection Method
The Bisection Method is a simple and robust numerical technique used to find the roots of a continuous function $f(x) = 0$. The core idea is based on the Intermediate Value Theorem: if a continuous function $f(x)$ changes sign over an interval $[a, b]$, there must be at least one root within that interval.
The Algorithm
- Identify an interval $[a, b]$ such that $f(a) \cdot f(b) < 0$.
- Calculate the midpoint $c = \frac{a + b}{2}$.
- Evaluate $f(c)$:
- If $f(c) = 0$, then $c$ is the root.
- If $f(a) \cdot f(c) < 0$, the root lies in $[a, c]$. Set $b = c$.
- If $f(c) \cdot f(b) < 0$, the root lies in $[c, b]$. Set $a = c$.
- Repeat until the desired precision is reached.
Solving $x^3 - 2x - 5 = 0$
Let $f(x) = x^3 - 2x - 5$. We are looking for the root in the interval $(2, 3)$.
- $f(2) = 2^3 - 2(2) - 5 = 8 - 4 - 5 = -1$ (Negative)
- $f(3) = 3^3 - 2(3) - 5 = 27 - 6 - 5 = 16$ (Positive)
Since $f(2) < 0$ and $f(3) > 0$, the root lies in $(2, 3)$.
Iteration Table
| Iteration | $a$ | $b$ | $c = (a+b)/2$ | $f(c)$ | New Interval |
|---|---|---|---|---|---|
| 1 | 2.0 | 3.0 | 2.5 | 5.625 | $[2.0, 2.5]$ |
| 2 | 2.0 | 2.5 | 2.25 | 1.8906 | $[2.0, 2.25]$ |
| 3 | 2.0 | 2.25 | 2.125 | 0.3457 | $[2.0, 2.125]$ |
| 4 | 2.0 | 2.125 | 2.0625 | -0.3513 | $[2.0625, 2.125]$ |
| 5 | 2.0625 | 2.125 | 2.09375 | -0.0089 | $[2.09375, 2.125]$ |
| 6 | 2.09375 | 2.125 | 2.109375 | 0.1667 | $[2.09375, 2.109375]$ |
| 7 | 2.09375 | 2.109375 | 2.1015625 | 0.0785 | $[2.09375, 2.1015625]$ |
| 8 | 2.09375 | 2.1015625 | 2.09765625 | 0.0347 | $[2.09375, 2.09765625]$ |
| 9 | 2.09375 | 2.09765625 | 2.095703125 | 0.0129 | $[2.09375, 2.095703125]$ |
Continuing this process until the change in $c$ is smaller than 0.001, we converge to the root.
Final Result
After several iterations, the value of $x$ converges to approximately 2.095.