## Definition

The Bisection Method, also known as the dichotomy method or interval halving method, is a widely used root-finding algorithm employed to locate the root of a continuous function within a defined interval. This method works by repeatedly dividing the interval in half and selecting the subinterval that contains the root. This process is repeated until the root is found within a specified degree of accuracy.

### What is Bisection Method in C?

The Bisection Method is a numerical technique used to find a root (or zero) of a continuous function within an interval. The method involves repeatedly bisecting the interval into smaller subintervals and determining which subinterval contains the root. In C, the Bisection Method can be implemented by defining a function that takes the left and right endpoints of the interval as inputs calculate the midpoint of the interval, and checks if the value of the function at the midpoint is close enough to zero. If not, the method updates the interval based on the sign of the function value and continues the process until a satisfactory approximation of the root is found.

### Explanation

Its explained with an example:

**1.** Initially. we need to define the function f(x) which we want to find the root for:

You need to write a function in C that takes in a value x and returns the value of the function you want to find the root of. For example, the function f(x) is defined as x^2 – 4.

**2.** Define the interval [a, b] that contains the root:

You need to choose an interval [a, b] that contains the root of the function you defined in step 1. For example, let’s say the interval is [1, 5].

**3.** Set a tolerance value for the error in the approximation of the root:

Choose a tolerance value that represents the error tolerance for your approximation of the root. For example, let’s say the tolerance value is 10^-6.

**4.** Initialize variables:

Initialize two variables left and right with values a and b, respectively. Also, initialize a variable mid to store the midpoint of the interval, and a variable error to store the error in the approximation of the root.

**5.** Evaluate the function at the midpoint:

Calculate the midpoint of the interval by taking the average of left and right: mid = (left + right) / 2. Then evaluate the function f(x) at the midpoint and store the result in a variable f_mid.

**6.** Determine if the deviation from the true value falls within the specified tolerance level:

If f_mid is close enough to 0 (i.e., the error |f_mid| is less than the tolerance value), return the midpoint as the approximate root.

**7.** Update the interval:

If f_mid is positive, update the interval by setting right = mid. If f_mid is negative, update the interval by setting left = mid.

**8.** Repeat steps 5 to 7 until the deviation from the true value falls within the specified tolerance level:

Repeat steps 5 to 7 until the error |f_mid| is less than the tolerance value.

**9.** Return the approximate root:

Once the error is within the tolerance, return the value of mid as the approximate root of the function.

### Algorithm for Bisection Method in C

The algorithm for the Bisection Method in C can be described as follows:

- Input the function func whose root is to be found, the left and right endpoints of the interval l and r, and the desired tolerance EPS.
- Calculate the midpoint mid of the interval (l + r)/2.
- Evaluate the function at the midpoint, f(mid).
- If |f(mid)| < EPS, return mid as the approximation of the root.
- Determine if the root lies in the left or right half of the interval by checking the sign of f(mid) * f(l).
- If f(mid) * f(l) < 0, update r to be mid. If f(mid) * f(l) > 0, update l to be mid.
- Go back to step 2.

This would be an example execution of the Bisection Method in C:

**Code:**

```
#include <math.h>
double bisection(double l, double r) {
double mid = (l + r) / 2;
while (fabs(func(mid)) >= EPS) {
if (func(mid) * func(l) < 0)
r = mid;
else
l = mid;
mid = (l + r) / 2;
}
return mid;
}
```

### Examples of Bisection Method in C

Below are the different examples of bisection method in C:

#### Example #1

**Code:**

```
#include <stdio.h>
#include <math.h>
double func(double x) {
return pow(x, 3) - x - 2;
}
double bisection(double l, double r) {
double mid = (l + r) / 2;
while (fabs(func(mid)) >= 1e-6) {
if (func(mid) * func(l) < 0)
r = mid;
else
l = mid;
mid = (l + r) / 2;
}
return mid;
}
int main() {
double root = bisection(1, 2);
printf("The root of the function is approximately: %.6f\n", root);
return 0;
}
```

**Output:**

**Explanation:**

1. The function func is defined to be x^3 – x – 2.

2. The bisection function takes in two arguments, l, and r, which are the left and right endpoints of the interval where the root is to be found.

3. The function calculates the midpoint of the interval and stores it in mid.

4. The function uses a while loop to repeat the following steps until the absolute value of the function evaluated at the midpoint is less than 1e-6:

- If the product of the function values at the midpoint and one of the endpoints is negative, the interval is updated to be the left half.
- If the product is positive, the interval is updated to be the right half.
- The midpoint is recalculated based on the updated interval.

5. The midpoint is returned as the approximation of the root.

6. In the main function, the bisection function is called with the interval (1, 2) and the result is stored in root.

7. The outcome is printed to the console which portrays the approximate root of the function.

#### Example #2

**Code:**

```
#include <stdio.h>
#include <math.h>
double func(double x) {
return pow(x, 2) - 4;
}
double bisection(double l, double r) {
double mid = (l + r) / 2;
while (fabs(func(mid)) >= 1e-6) {
if (func(mid) * func(l) < 0)
r = mid;
else
l = mid;
mid = (l + r) / 2;
}
return mid;
}
int main() {
double root = bisection(1, 4);
printf("The root of the function is approximately: %.6f\n", root);
return 0;
}
```

**Output:**

**Explanation:**

1. The function func is defined to be x^2 – 4.

2. The bisection function takes in two arguments, l, and r, which are the left and right endpoints of the interval where the root is to be found.

3. The function calculates the midpoint of the interval and stores it in mid.

4. The function uses a while loop to repeat the following steps until the absolute value of the function evaluated at the midpoint is less than 1e-6:

- If the product of the function values at the midpoint and one of the endpoints is negative, the interval is updated to be the left half.
- If the product is positive, the interval is updated to be the right half.
- The midpoint is recalculated based on the updated interval.
- The midpoint is returned as the approximation of the root.

5. In the main function, the bisection function is called with the interval (1, 4) and the result is stored in root.

6. The outcome is printed to the console, displaying the approximate root of the function x^2 – 4.

### Features of Bisection Method

**Simple and easy to understand:**The Bisection Method is one of the simplest methods for finding the roots of a function and is easy to understand, even for those with limited mathematical knowledge.**Convergence:**The Bisection Method is a convergent method, meaning that it approaches the root as the number of iterations increases.**Guaranteed to Converge:**The Bisection Method is ensured to meet to a root as long as the function being settled is continuous on the interval being considered.**Robustness:**The Bisection Method is a robust method, meaning that it is not easily affected by changes in the function or the interval being considered.**Bracketing the Root:**The Bisection Method is a bracketing method, meaning that it always brackets the root, which makes it useful in cases where the root may be difficult to determine.**Slow Convergence:**One of the main disadvantages of the Bisection Method is that it has slow convergence. It requires a large number of iterations to reach a solution that is accurate to a high degree of precision.**Requires a Starting Interval:**The Bisection Method requires that a starting interval be specified, which contains the root being sought. This can present a challenge as the root may not be known in advance.

### Advantages and Disadvantages of Bisection Method

Advantages |
Disadvantages |

The Bisection Method is simple to comprehend and execute, making it available to all individuals. | The method only provides approximate solutions, and the degree of accuracy depends on the number of iterations. |

As long as the function is continuous within the interval, the method will always approach a root. | The method takes a longer time to find a solution that is accurate to a high degree of precision, resulting in a slower convergence rate. |

The method is resilient to modifications in the function or interval. | The method is not suitable for functions with multiple roots and may converge to a different root than desired. |

The method always brackets the root, making it useful in cases where the root is difficult to determine. | The method requires a starting interval that contains the root being sought, which can be challenging to determine. In some cases, the method may converge to a value that is not the actual root, as it may become trapped in local minima. |

### Conclusion

The Bisection Method is a simple and robust method for finding the roots of a function. It is guaranteed to converge to a root as long as the function is continuous on the interval being considered. The method is easy to understand and implement, making it accessible to people with limited mathematical knowledge. However, it has slow convergence and only provides approximate solutions, and it may become trapped in local minima in some cases. The Bisection Method is also limited to single root functions and requires a starting interval to be specified, which can be challenging to determine in some cases.

### Recommended Article

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