Concept:

Bisection method:

Used to find the root for a function. Root of a function f(x) = a such that f(a)= 0

Property: if a function f(x) is continuous on the interval [a…b] and sign of f(a) ≠  sign of f(b). There is a value c belongs to [a…b] such that f(c) = 0, means c is a root in between [a….b]

Note:

Bisection method cut the interval into 2 halves and check which half contains a root of the equation.

1) Suppose interval [a…b] .

2) Cut interval in the middle to find m : \(m =\frac{{a+b}}{{2}}\)

3) sign of f(m) not matches with f(a) proceed the search in the new interval.

Calculation:

The bisection method is applied to a given problem with [1, 9]

After 1 iteration

\({x_1} = \frac{{1\; + \;9}}{2} = 5\)

Now since f(x₁) > 0, x₂ replaces x₁

Now, x₀ = 1 and x₁ = 5

And after 2^nd iteration 

\({x_2} = \frac{{1\; + \;5}}{2} = 3\)

Now since f(x₁) f(x₂) > 0, x₂ replaces x₁ and x₀ = 1 and x₁ = 3 and after 3^rd iteration

\({x_2} = \frac{{1\; + \;3}}{2} = 2\)

Now, f(x₂) = f(2) = 2⁴ – 2³ – 2² – 4 = 0

So the method converges exactly to the root in 3 iterations.