## Introduction to Bisection Method Calculator:

Bisection method Calculator is an online tool that helps you to **evaluate the bisection** problem of a given function f(x). It is used to determine the root value of a continuous function f(x) within the given interval.

When you solve the bisection method problem manually you become exhausted in finding a solution because of its long process of calculation iteration. To avoid this tedious hassle you need a tool like our bisect calculator that gives you solutions quickly and easily.

## What is Bisection Method?

Bisection method is a numerical analysis process in which you find the **root of continuous function** over a specific interval. It is a straightforward but lengthy method for solving the function which cannot give a solution when using other methods like differentiation.

It always convergences with its initial root in the given interval. The bisection method does not provide as much accurate results as other numerical methods like the regular false that the Euler method gives which means the error of percentage is high in solution.

## Formula of Bisection Method:

The **formula** of the bisection method consists of two roots on the interval [a,b] that are bisected using the Meidan rule of a function f(x). The value of a and b varies because when you find the subsequent roots under the condition:

$$ f(a) . f(b) < 0 $$

$$ c \;=\; \frac{a + b}{2} $$

## How to Calculate the Bisection Method?

For the **calculation** of the bisection method, the bisection calculator bisects the root that gives solution in opposite sign of a function f(x) within a specified interval [a, b].

**Step 1**:

Determine the given function f(x) and select two root value a and b such that [a,b]

**Step 2**:

Put the roots value in the given function to check whether it gives opposite sign solution or not to satisfy the condition f(a) ⋅ f(b) < 0 for bisection method.

**Step 3**:

If the roots satisfy the given condition than put these roots value in the formula of bisection method. $$ c \;=\; \frac{a + b}{2} $$

**Step 4**:

The x0 value you get after finding the median value put it into the given function.

**Step 5**:

If the function has a negative value then select the positive root from the previous root value or if the function value is positive then select the negative root value from the previous root value.

**Step 6**:

Then again add the opposite sign root value in the bisection formula and then again repeat the above process add the value of x1 in the function and check the sign of function value is positive or negative. According to the value sign, choose the root value and put it into formula again.

**Step 7**:

Repeat this process, unti you do get the same root value when you bisect them in formula.

## Solved Example of Bisection Method:

An **example of bisection** method is given below to let you know how to do bisection method and the calculation process of bisection method calculator.

### Example: Find a root of an equation f(x) = 2x^{3} - 2x - 5 using bisection method

**Solution**:

Determine the given function,

$$ f(x) \;=\; 2x^3 - 2x - 5 $$

Lets suppose the initial roots a = 1 and b = 2 such that [1,2]. Add these roots in the given function one by one to check whether it satisfy the given condition or not.

$$ f(a) . f(b) < 0 $$

As both the value has opposite sign, that means roots lie between them.

**1st Iteration**:

Here f(1) = -5 < 0 and f(2) = 7 > 0.

To find the bisection point put a and b value in it,

$$ x_0 \;=\; \frac{1 + 2}{2} \;=\; 1.5 $$

Put the x0 value in the above function as f(x0).

$$ f(x_0) \;=\; f(1.5) = 2 . 1.5^3 - 2 . 1.5 - 5 \;=\; -1.25 < 0 $$

For the second interaction again choose a root that gives a positive value but f(x_0) has a negative value.

**2nd iteration**:

Here f(1.5) = -1.25 < 0 and f(2) = 7 > 0.

Put these roots in the bisection formula,

$$ x_1 \;=\; \frac{1.5 + 2}{2} \;=\; 1.75 $$

Again put the root value in the given function as f(x1).

$$ f(x_1) \;=\; f(1.75) \;=\; 2 . 1.75^3 - 2 . 1.75 - 5 \;=\; 2.21875 > 0 $$

For the third interaction again choose a root that gives a negative value but f(x_1) has a positive value.

**3rd iteration**:

Here f(1.5) \;=\; -1.25 < 0 and f(1.75) = 2.21875 > 0. Root lies between 1.5 and 1.75.

Put these roots in the bisection formula,

$$ x_2 \;=\; \frac{1.5 + 1.75}{2} \;=\; 1.625 $$

Again put the root value in the given function as f(x2),

$$ f(x_2) \;=\; f(1.625) \;=\; 2 . 1.625^3 - 2 . 1.625 - 5 \;=\; 0.33203 > 0 $$

For the fourth interaction again choose a root that gives a negative value but f(x_3) has a positive value.

**4rd iteration**:

Here f(1.5) = -1.25 < 0 and f(1.625) = 0.33203 > 0. Now root lies between 1.5 and 1.625.

Put these roots in the bisection formula,

$$ x_3 \;=\; \frac{1.5 + 1.625}{2} \;=\; 1.5625 $$

Again put the root value in the given function as f(x3).

$$ f(x_3) \;=\; f(1.5625) \;=\; 2 . 1.5625^3 - 2 . 1.5625 - 5 \;=\; -0.49561 < 0 $$

For the fifth interaction again choose a root that gives a positive value but f(x3) has a negative value. Here f(1.5625) = -0.49561 < 0 and f(1.625) = 0.33203 > 0. Now root lies between 1.5625 and 1.625.

Put these roots in the bisection formula,

$$ x_4 \;=\; \frac{1.5625 + 1.625}{2} \;=\; 1.59375 $$

Again put the root value in the given function as f(x4):

$$ f(x_4) \;=\; f(1.59375) \;=\; 2 . 1.59375^3 - 2 . 1.59375 - 5 \;=\; -0.09113 < 0 $$

For the sixth interaction again choose a root that gives a positive value but f(x_4) has a negative value.

**6th iteration**: here f(1.59375) = -0.09113 < 0 and f(1.625) = 0.33203 > 0. Now root lies between 1.59375 and 1.625.

Put these roots in the bisection formula,

$$ x_5 \;=\; \frac{1.59375 + 1.625}{2} \;=\; 1.60938 $$

Again put the root value in the given function as f(x5),

$$ f(x_5) \;=\; f(1.60938) \;=\; 2 . 1.60938^3 - 2 . 1.60938 - 5 \;=\; 0.1181 > 0 $$

For the seventh interaction again choose a root that gives a negative value but f(x_5) has a postive value.

**7th iteration**: here f(1.59375) = -0.09113 < 0 and f(1.60938) = 0.1181 > 0. Now root lies between 1.59375 and 1.60938.

Put these roots in the bisection formula,

$$ x_6 \;=\; \frac{1.59375 + 1.60938}{2} \;=\; 1.60165 $$

Again put the root value in the given function as f(x6).

$$ f(x_6) \;=\; f(1.60165) \;=\; 2 . 1.60156^3 - 2 . 1.60165 - 5 \;=\; 0.0129 > 0 $$

For the eighth interaction again choose a root that gives a negative value but f(x6) has a positive value. Here f(1.59375) = -0.09113 < 0 and f(1.60156) = 0.0129 > 0. Now root lies between 1.59375 and 1.60156.

Put these roots in the bisection formula,

$$ x_7 \;=\; \frac{1.59375 + 1.60156}{2} \;=\; 1.59766 $$

Again put the root value in the given function as f(x7).

$$ f(x_7) \;=\; f(1.59766) \;=\; 2 . 1.59766^3 - 2 . 1.59766 - 5 \;=\; -0.03926 < 0 $$

For the ninth interaction again choose a root that gives a positive value but f(x_7) has a negative value. Here f(1.59766) = -0.03926 < 0 and f(1.60156) = 0.0129 > 0. Now root lies between 1.59766 and 1.60156.

Put these roots in the bisection formula,

$$ x_8 \;=\; \frac{1.59766 + 1.60156}{2} \;=\; 1.59961 $$

The approximate value of the bisection root value method is 1.59901.

## How to Use Bisection Method Calculator?

The bisect calculator has a simple layout, so everyone can use it to calculate the given function using bisection method. Before adding the input value, you must follow some instructions. These instructions are:

- Enter the value of the given bisection method function f(x) that you want to find in the input box.
- Enter the root value of the closed interval in the form of a and b for bisection method in the next input box.
- Review your input function value before hitting the calculate button of bisection calculator to start the calculation process to find the solution to the bisection method problem.
- Click on the “
**Calculate**” button to get the required result for the bisection method problem. - If you are trying our bisection method online calculator for the first time, then you can use the load example to check accuracy in solution.
- Click on the “Recalculate” button to get a new page for solving more bisection method problem over an interval [a, b].

## Final Result of Bisect Calculator:

Bisection method Calculator gives you the **solution** to a function when you input the value into it. It may be included as:

**Result Option**:

You can click on the result option, and it will provide you with a solution for the bisection method for the given function.

**Possible Step**:

The Possible Steps option provides you with a solution in which evaluation steps are given for an bisection method over closed interval.

## Advantages of Bisection Calculator:

The Bisection Method Online Calculator has many **valuable advantages** that help you get the solution of the bisection method problem whenever you use it. These advantages are:

- It saves your time and effort from doing lengthy and complex calculations of the bisection method question in less than a minute.
- Our calculator is a free tool that you can use to find the bisection method for the question without paying any charge.
- The bisect calculator is an easy-to-use tool, so you do not need any technical expertise.
- It is a reliable tool that provides you with accurate solutions every time whenever you use it to calculate the bisection method problems easily.
- Our Calculator provides you with a solution procedure in a step-by-step method for more clarity.
- Bisection method calculator is an educational tool so you can solve as many problem of bisection method as you can.