Are you curious to know what is bisection method? You have come to the right place as I am going to tell you everything about bisection method in a very simple explanation. Without further discussion let’s begin to know what is bisection method?
If you’re a student of mathematics or computer science, you may have heard of the bisection method. The bisection method is a numerical method that is used to find the roots of a given equation. In this blog post, we’ll explore what the bisection method is and how it works.
What Is Bisection Method?
The bisection method is a numerical method that is used to find the roots of a given equation. A root of an equation is a value of the variable that makes the equation equal to zero. For example, the root of equation x^2 – 4 = 0 is x = 2.
The bisection method works by dividing the interval that contains the root in half and then repeatedly narrowing the interval until the root is found. The method requires that the equation be continuous and that there is only one root in the interval.
How Does The Bisection Method Work?
To understand how the bisection method works, let’s consider an example. Suppose we want to find the root of the equation x^2 – 4 = 0 in the interval [1, 3]. We begin by evaluating the equation at the midpoint of the interval:
f((1 + 3) / 2) = f(2) = 2^2 – 4 = 0
Since the function evaluates to zero at the midpoint of the interval, we know that the root must be either in the interval [1, 2] or [2, 3]. We repeat the process by evaluating the function at the midpoint of each interval:
f((1 + 2) / 2) = f(1.5) = (1.5)^2 – 4 = -0.75
f((2 + 3) / 2) = f(2.5) = (2.5)^2 – 4 = 1.25
Since the function evaluates to a negative value at the midpoint of the interval [1, 2] and a positive value at the midpoint of the interval [2, 3], we know that the root must be in the interval [2, 3]. We repeat the process until the interval becomes small enough to approximate the root.
Why Use The Bisection Method?
The bisection method is a simple and reliable method for finding the roots of an equation. It does not require any initial guesses, as it systematically narrows down the interval until the root is found. The method is also guaranteed to converge to a root, as long as the equation is continuous and there is only one root in the interval.
Conclusion
In conclusion, the bisection method is a numerical method that is used to find the roots of a given equation. The method works by dividing the interval that contains the root in half and repeatedly narrowing the interval until the root is found. The bisection method is a simple and reliable method for finding roots, and is guaranteed to converge to a root as long as the equation is continuous and there is only one root in the interval.
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FAQ
What Do You Mean By The Bisection Method?
The bisection method is an approximation method to find the roots of the given equation by repeatedly dividing the interval. This method will divide the interval until the resulting interval is found, which is extremely small.
What Are The Types Of Bisection?
The most often considered types of bisectors are the segment bisector (a line that passes through the midpoint of a given segment) and the angle bisector (a line that passes through the apex of an angle, that divides it into two equal angles).
What Are The Advantages Of The Bisection Method?
Advantages of the Bisection Method
- Guaranteed convergence.
- Errors can be managed.
- Doesn’t demand complicated calculations.
- Error bound is guaranteed.
- The bisection method is simple and straightforward to program on a computer.
- In the case of several roots, the bisection procedure is quick.
Why Newton Raphson’s Method Is Used?
The Newton Raphson Method is referred to as one of the most commonly used techniques for finding the roots of given equations. It can be efficiently generalised to find solutions to a system of equations. Moreover, we can show that when we approach the root, the method is quadratically convergent.
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