Bisection Method

The bisection method is a straightforward and reliable numerical technique used to find roots (solutions) of continuous functions. It works by repeatedly dividing an interval in half and selecting the subinterval where the function changes sign, thereby narrowing down the location of the root.

Given a function:

f(x) = 0

How the Bisection Method Works

Select Interval: Choose two points $a$ and $b$ such that $f(a)$ and $f(b)$ have opposite signs.
Midpoint Calculation: Compute the midpoint $C= \frac{a + b}{2}$ .
Evaluate: Check the sign of $f(c)$ .
Update Interval: If $f(c)$ is close enough to zero, $c$ is the root. Otherwise, update the interval.

Example

Given Function

x^2 = 4

Error Margin is 0.01

Step 1 : Rearranging the Equation:

f(x) = x^2 - 4 = 0

Step 2 : Choose Initial Points

To apply the Bisection Method, we first need to choose two initial points $a$ and $b$ such that:

f(a) \text{ and } f(b) \text{ have opposite signs.}

lets's choose $a = 0 \text{ and } b = 3$

f(0) = 0^2 - 4 = -4 \text{ (negative)}

f(3) = 3^2 - 4 = 5\text{ (positive)}

Since $f(0) < 0 \text{ and } f(3) > 0 ,$ we can proceed.

Step 3 : Calculate Midpoint

Next, we calculate the midpoint $C$ of the interval [ a , b ] :

C = \frac{ a + b }{2} = \frac{ 0 + 3 }{2} = 1.5

Step 4 : Update the Interval

Since $f(a) < 0 \text{ and } f(C) < 0 ,$ we know that the root must lie in the interval [ C , b ] :

\text{Set } a = C = 1.5

Step 5 : Repeat the Process

We repeat the steps:

Calculate new midpoint:
$C = \frac{1.5 + 3 }{2} = 2.25$
Evaluate function:
$f(2.25) = (2.25)^2 - 4 = 5.0625 - 4 = 1.0625$
Update interval:
Since $f(1.5) < 0 \text{ and } f(2.25) > 0$ set $b = C = 2.25$

Step 6 : Continuing the Process

Continuing this process, we narrow down the interval:

Calculate new midpoint:
$C = \frac{1.5 + 2.25 }{2} = 1.875$
Evaluate function:
$f(1.875) = (1.875)^2 - 4 = 3.515625 - 4 = −0.484375$
Update interval:
Since $f(1.875) < 0 \text{ and } f(2.25) > 0 ,$ set $a = 2.25$

Continue until Desired Accuracy

Continue this process until the difference between 𝑎 and 𝑏 is less than a desired tolerance (for example,0.01).

Eventually, you will converge on the root:

\sqrt{4} = 2

Conclusion

The Bisection Method is an efficient way to find the root of a function defined by a continuous equation. In this case, we demonstrated it for $x^2 = 4$ and found that the root is $x = 2$ The method guarantees convergence as long as you start with points that bracket the root, making it a reliable technique for root-finding problems.