{"id":10718,"date":"2019-07-18T12:03:44","date_gmt":"2019-07-18T12:03:44","guid":{"rendered":"https:\/\/assignment.essayshark.com\/blog\/?p=10718"},"modified":"2021-12-30T09:39:29","modified_gmt":"2021-12-30T09:39:29","slug":"newton-raphson-method-analysis","status":"publish","type":"post","link":"https:\/\/assignmentshark.com\/blog\/newton-raphson-method-analysis\/","title":{"rendered":"Newton Raphson Method Analysis"},"content":{"rendered":"

Find the solution of the following equation using the Newton Raphson Method:<\/p>\n

\"newton<\/a><\/p>\n

Write a Python 3 code for this method.<\/p>\n

Solution<\/strong><\/p>\n

Hand Solution<\/p>\n

We must choose an initial value of x:<\/p>\n

\"formula\"<\/a><\/p>\n

Let\u2019s plug the above value into the equation:<\/p>\n

\"newton<\/a><\/p>\n

The next value of x will be found according to the following formula:<\/p>\n

\"newton<\/a>
\nThe derivative of the function is:<\/p>\n

\"newton<\/a>
\nThe derivative of the function is:<\/p>\n

\"newton<\/a><\/p>\n

We will continue finding the next x values until the difference between the adjacent values is small enough:<\/p>\n

\"newton<\/a><\/p>\n

\"newton<\/a><\/p>\n

\"newton<\/a><\/p>\n

\"newton<\/a><\/p>\n

\"newton<\/a><\/p>\n

\"newton<\/a><\/p>\n

As the <\/p>\n

  <\/span>   <\/span>\"\[x_{13}\]\"<\/p>\n

and <\/p>\n

  <\/span>   <\/span>\"\[x_{14}\]\"<\/p>\n

values are equal, the root of the equation is 0.73518.<\/strong><\/p>\n

Python 3 Code<\/p>\n

Firstly, we will define a function for the equation:<\/p>\n

\"newton<\/a><\/p>\n

Also, we should write a function for numerical calculation of the equation\u2019s derivative. We will use the central difference method with the step of\u00a0 <\/p>\n

  <\/span>   <\/span>\"\[10^{-6}\]\"<\/p>\n

for each side:<\/p>\n

\"newton<\/a><\/p>\n

And before writing the main function, we will define a function for the next x value calculation:<\/p>\n

\"newton<\/a><\/p>\n

Lastly, let\u2019s define the main function:<\/p>\n

\"newton<\/a><\/p>\n

To simplify reading the variables and find the next x value:<\/p>\n

\"newton<\/a><\/p>\n

Let\u2019s repeat the above procedure until the difference between the adjacent x values is greater than <\/p>\n

  <\/span>   <\/span>\"\[10^{-6}\]\"<\/p>\n

:<\/p>\n

\"\"<\/a><\/p>\n

At last, the x value is returned:<\/p>\n

\"\"<\/a><\/p>\n

The entire code:<\/p>\n

\"newton<\/a><\/p>\n

\"newton<\/a><\/p>\n

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