Bisection Method

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Description

In numerical, the bisection method is a method used to find the root that can be applied to any continuous functions for which two values of opposite signs are known. It is the simplest iterative method which is also known as the half-interval method or bolzano method.

In this method first, a sufficiently small interval  is found out containing the root by the method of tabulation.

Now, if

PROCEDURE

We start the sequence of iteration by setting,

Next, we have to set

and we have to repeat the above process till we obtain

with the desired accuracy.

GRAPHICAL REPRESENTATION OF BISECTION METHOD:

Algorithm

INPUT:

OUTPUT:

PROCESS:

Step 1: [Defining a function to find the root]

function f(x)

return the value of f(x)
[End of function f(x)]

Step 2: [Method to perform bisection method]

Step 2.1: Set a <- -1.0
Set b <- 0.0
Set x <- 0.0
Set y <- 0.0

Step 2.2: While(f(a) × f(b) > 0) repeat
Set a <- b
Set b <- b + 1.0
[end of ‘while’ loop]

Step 2.3: Take the error (the accuracy) from the user

Step 2.4: if f(a) > 0 then
Set a <- a + b
Set b <- a - b
Set a <- a - b
[End of ‘if’]

Step 2.5: Set y <- x
Set x <- (a + b)/2
If f(x) < 0 then
Set a <- x
else
Set b <- x
[End of ‘if’]
Print a, b, x, f(x)
While fabs(x - y) > err repeat Step 2.5
[End of bisection method]

Code

1. It is a simple method.

2. The method is unconditionally and surely convergent.

3. The approximate value of the desired root of the equation does not depend on the values of $\mathrm{f}\left({\mathrm{x}}_{\mathrm{n}}\right)$ where ${\mathrm{x}}_{\mathrm{n}}$ is the ${\mathrm{n}}^{\mathrm{th}}$ iteration of x, but it depends on their signs only.

1. It is very slow to converge to its root.

2. To obtain moderately accurate results, a large number of iterations is required.

3. This method is laborious for the complex transcendental equations.

APPLICATION

1. It is used to find the roots of continuous functions.

2. It can be used in thermistors.

3. It is used to solve mathematical problems numerically.