Bisection Method

Definition

The Bisection Method is a fundamental numerical technique used to find the real roots of a continuous function $f(x) = 0$. It is a "bracketing method" that relies on the Intermediate Value Theorem, which states that if a continuous function has values of opposite signs at the endpoints of an interval $[a, b]$, there must be at least one root within that interval.

Main Content

1. The Intermediate Value Theorem

The method requires an initial interval $[a, b]$ such that $f(a) \cdot f(b) < 0$. This indicates that one value is positive and the other is negative, ensuring the graph crosses the x-axis.
Since the function is continuous, it must cross the x-axis at least once between $a$ and $b$ to transition from positive to negative (or vice versa).

2. The Concept of Halving

The method repeatedly divides the interval in half by calculating the midpoint $c = \frac{a+b}{2}$.
By checking the sign of $f(c)$, we determine which half of the interval contains the root, allowing us to discard the other half.

3. Convergence and Precision

With every iteration, the interval size is halved, meaning the error is reduced by half.
The process continues until the interval $[a, b]$ becomes sufficiently small or the value $|f(c)|$ is close enough to zero, satisfying a predefined tolerance level.

Visualizing the search for a root:

      f(x) ^
           |      *
           |    *   *
-----------|---*------*------> x
           |  a  c  b
           |*

(The root lies between a and b; we test midpoint c)

Working / Process

1. Initialization

Choose two initial points $a$ and $b$ such that $f(a)$ and $f(b)$ have opposite signs.
Define the desired tolerance (error bound) $\epsilon$ to stop the calculation.

2. Iteration

Calculate the midpoint $c = \frac{a + b}{2}$.
Evaluate the function at the midpoint, $f(c)$.
If $f(c) = 0$, then $c$ is the exact root. Stop the process.

3. Updating the Interval

If $f(a) \cdot f(c) < 0$, the root lies between $a$ and $c$. Set $b = c$.
If $f(b) \cdot f(c) < 0$, the root lies between $c$ and $b$. Set $a = c$.
Repeat the process until the interval $(b - a) < \epsilon$.

Advantages / Applications

It is always convergent, provided the function is continuous and a sign change is found.
It is simple to implement and does not require calculating derivatives, unlike the Newton-Raphson method.
It is highly robust and used in engineering problems where we need to find approximate solutions to non-linear equations where algebraic solutions are impossible.

Summary

The Bisection Method is a reliable, iterative numerical approach that repeatedly halves an interval to isolate a root of a continuous function. By ensuring a sign change exists between two points, the method systematically narrows the search area until a solution is found within a desired level of accuracy. Important terms to remember include Bracketing, Midpoint, Convergence, and Tolerance.