Newton Raphson Method Analysis

Find the solution of the following equation using the Newton Raphson Method:

newton raphson method python

Write a Python 3 code for this method.

Solution

Hand Solution

We must choose an initial value of x:

formula

Let’s plug the above value into the equation:

newton raphson method

The next value of x will be found according to the following formula:

newton raphson method formula
The derivative of the function is:

newton raphson method sample
The derivative of the function is:

newton raphson method assignment

We will continue finding the next x values until the difference between the adjacent values is small enough:

newton raphson method assignment example

newton raphson method formula

newton raphson method formula

newton raphson method formula

newton raphson method formula example

newton raphson method formula sample

As the

    \[x_{13}\]

and

    \[x_{14}\]

values are equal, the root of the equation is 0.73518.

Python 3 Code

Firstly, we will define a function for the equation:

newton raphson method

Also, we should write a function for numerical calculation of the equation’s derivative. We will use the central difference method with the step of 

    \[10^{-6}\]

for each side:

newton raphson method

And before writing the main function, we will define a function for the next x value calculation:

newton raphson method

Lastly, let’s define the main function:

newton raphson method

To simplify reading the variables and find the next x value:

newton raphson method

Let’s repeat the above procedure until the difference between the adjacent x values is greater than

    \[10^{-6}\]

:

At last, the x value is returned:

The entire code:

newton raphson method python

newton raphson method python

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