## Introduction to Bisection Method Matlab

Bisection method is used to find the root of equations in mathematics and numerical problems. This method can be used to find the root of a polynomial equation; given that the roots must lie in the interval defined by [a, b] and the function must be continuous in this interval. The bisection method requires 2 guesses initially and so is referred to as ‘close bracket’ type. In comparison with other root-finding methods, this method is relatively slow as it converges in a linear, steady, and slow manner.

In MATLAB, we do not have a pre-defined bisection method, so we create one to get the roots using this method.

### How to create the Bisection method in MATLAB?

Let us now understand the syntax to create the bisection method in MATLAB:

#### Syntax

**1. Z = input** (‘Enter the input function and set right hand side equal to zero:’,’s’);

**2. f = inline (Z);**

`initialValue = input ('Enter the initial value for guess interval:');`

- [Asking user to enter ‘Initial value’ which is used as the first value for 1
^{st}iteration]

`endValue = input ('Enter the last value for guess interval:');`

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- [Asking user to enter ‘End value’ which is used as the last value for 1
^{st}iteration]

`allowedError = input ('Enter the error allowed:');`

- [Asking the user to enter tolerable error]

**Code:**

`if f (endValue) * f (initialValue) < 0`

else

fprintf('The guess entered is incorrect. Please enter the new guess\n');

initialValue = input ('Enter the initial value for guess interval:\n') ;

endValue = input ('Enter the last value for guess interval:\n');

end

for I = 2 : 500

iter1 = (endValue + initialValue) / 2;

if f (endValue) * f (iter1) < 0

initialValue = iter1;

else

endValue = iter1;

end

if f (initialValue) * f (iter1) < 0

endValue = iter1;

else

initialValue = iter1;

end

iterN (1) = 0;

iterN (i) = iter1;

if abs ((iterN (i) – iterN (i-1)) / iterN (i)) <allowedError, break, end

end

outStr = ['Required root for the input equation is: ', num2str(iter1), '']

**1.** Mid-point for the range [ initialValue, endValue ] is calculated. We have defined the mid point as iter1, which is defined as:

`iter1 = (initialValue + endValue) / 2`

**2.** The value of input function is calculated at ‘iter1’, giving following scenarios:

- If f (iter1) = 0; iter1 will be the required root
- If f (endValue) * f (iter1)> 0; i.e, the product f (endValue) * f (iter1) is positive, then the root of input function will lie between the range [firstValue, iter1].
- If f (endValue) * f (iter1)< 0; i.e, the product f (endValue) * f (iter1) is negative, then the root of input function will lie between the range [ endValue, iter1]]

### Example to Implement Bisection Method Matlab

Below are the examples mentioned:

#### Example #1

In this example, we will take a polynomial function of degree 2 and will find its roots using the bisection method. We will use the code above and will pass the inputs as asked. For our first example, we will input the following values:

- Pass the input function as 2*x.^2 + 3
- Pass the firstValue as 1
- Pass the endValue as 2
- Pass the allowed error as 0.001
- If the root does not lie between 1 & 2, then the code will again ask for firstValue and endValue
- Pass the firstValue as 2 and endValue as 3 this time

**Output:**

As we can see in the output, we have obtained the root of our input function as 2.5.

### Example #2

In this example, we will take a polynomial function of degree 3 and will find its roots using the bisection method. We will use the code above and will pass the inputs as asked. For this example, we will input the following values:

- Pass the input function as 3*x.^3 + 2*x.^2
- Pass the firstValue as 1
- Pass the endValue as 3
- Pass the allowed error as 0.001
- The code will again ask for firstValue and endValue
- Pass the firstValue as 2 and endValue as 3 this time

**Output:**

As we can see in the output, we have obtained the root of our input function as 2.5.

Next, let us see an example where we are not asked for the guess values the second time

#### Example #3

In this example, we will take a polynomial function of degree 2 and will find its roots using the bisection method. We will use the code above and will pass the inputs as asked. For this example, we will input the following values:

- Pass the input function as x.^2 – 3
- Pass the firstValue as 1
- Pass the endValue as 2
- Pass the allowed error as 0.001

**Output:**

As we can see in the output, we have obtained the root of our input function as 1.7344, after we input our guess values for the first time.

### Conclusion

The bisection method is used to find the root of any polynomial function. As there is no direct function for the bisection rule in MATLAB, we define the code or logic for it manually.

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