Bisection Method

In Mathematics, the Bisection Method is a straightforward method used to find numerical solutions of an equation with one unknown variable.

Definition

This method is a root-finding method that applies to any continuous functions with two known values of opposite signs. It is a very simple but cumbersome method.

The interval defined by these two values is bisected and a sub-interval in which the function changes sign is selected. This sub-interval must contain the root. These two steps are repeatedly executed until the root is in the form of the required precision level.

This method is also called as interval halving method, the binary method, or the dichotomy method. 

The Method: Explained

Let \(f\) be a continuous function defined on an interval \([a, b]\) where \(f(a)\) and \(f(b)\) have opposite signs. Bisection method is applicable for solving the equation \(f(x) = 0\) for a real variable \(x\).

At each step, the interval is divided into two parts/halves by computing the midpoint, \(c = \frac{(a+b)}{2}\), and the value of \(f(c)\) at that point. Unless the root is \(c\), there are two possibilities:

• \(f(a)\) and \(f(c)\) have opposite signs and bracket a root,
• \(f(c)\) and \(f(b)\) have opposite signs and bracket a root.

One of the sub-intervals is chosen as the new interval to be used in the next step. This process is carried out again and again until the interval is sufficiently small.

If \(f(a)\) and \(f(c)\) have opposite signs, then the value of \(b\) is replaced by \(c\).
If \(f(b)\) and \(f(c)\) have opposite signs, then the value of \(a\) is replaced by \(c\).
In the case that \(f(c) = 0, c\) will be taken as the solution and the process stops.

Bisection Method Algorithm

The algorithm for the bisection method is as below:

INPUT: Function \(f\), endpoint values \(a, b\), tolerance \(TOL\), maximum iterations \(NMAX\).

OUTPUT: value that differs from the root of \(f(x) = 0\) by less than \(TOL\).

N ← 1
while N <= NMAX do
    C ← (a + b)/2
    if f(c) = 0 or (b - a)/2 < TOL then
        Output(c)
        stop
    end if
    N ← N + 1
    if sign(f(c)) = sign(f(a)) then
        a ← c
    else
        b ← c
end while
Output(“Method failed.”)

Advantages & Disadvantages of Bisection Method

Solved Examples

Here, we have bisection method example problems with solution.

Question. Determine the root of the equation, \(f(x) = x^3 – x – 2\) for \(x ∈ [1, 2]\).

Solution. Given, 

\(f(x) = x^3 – x – 2\) for \(x ∈ [1, 2]\)

Take, \(a = 1, b = 2\)
\(f(1) = 1^3 – 1 – 2 = -2\)
\(f(2) = 2^3 – 2 – 2 = +4\)

As the function is continuous, a root must lie within [1, 2].

1^st Iteration:
\(a_1 = 1, b_1= 2\), 
\(c_1= \frac{2 + 1}{2} = 1.5\)

Hence, the function value at midpoint is,
\(f(c_1) = (1.5)^3 – (1.5) – 2 = -0.125\)

As \(f(c)\) gives a negative value, the value of \(a\) is replaced by \(c\). 

The iterations are concisely summarized into a table below:

From the above table, it can be pointed out that, after 13 iterations, it becomes apparent that the function converges to 1.521, which is concluded as the root of the polynomial.

Related Articles
1. Newton Raphson Method
2. Secant Method

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