The Newton-Raphson method is an iterative numerical method used to find the roots of a function. The method starts with an initial guess for the root and then iteratively refines this guess until the desired level of accuracy is achieved.

The general procedure for the Newton-Raphson method is as follows:

- Start with an initial guess for the root, denoted by x0.
- Compute the value of the function at x0, denoted by f(x0).
- Compute the derivative of the function at x0, denoted by f'(x0).
- Calculate the next approximation of the root using the formula:

x1 = x0 – f(x0) / f'(x0)

- Check if the difference between x1 and x0 is within the desired level of accuracy. If not, repeat steps 2-4 with x1 as the new guess for the root.
- Continue iterating until the desired level of accuracy is achieved.

The formula in step 4 is based on the idea of using the tangent line at x0 as an approximation of the curve. The intersection of the tangent line and the x-axis is then taken as the next guess for the root.

Note that the Newton-Raphson method may not always converge to a root or may converge to a wrong root if the initial guess is not close enough to the actual root or if the function has a singularity or a multiple root. Therefore, it is important to carefully choose the initial guess and check the convergence of the method.

## Example.

Example 1:

Let’s take the function f(x) = x^3 – 2x – 5 and find one of its roots using the Newton-Raphson method.

- Choose an initial guess for the root, let’s say x0 = 2.
- Compute the value of the function at x0: f(x0) = 2^3 – 2(2) – 5 = 1.
- Compute the derivative of the function at x0: f'(x0) = 3x^2 – 2 = 10.
- Calculate the next approximation of the root using the formula:

x1 = x0 – f(x0) / f'(x0) = 2 – 1 / 10 = 1.9

- Check the difference between x1 and x0: |x1 – x0| = |1.9 – 2| = 0.125. Since this is larger than our desired level of accuracy, we need to continue with the next iteration.
- Set x0 = x1 and repeat steps 2-4 until the desired level of accuracy is achieved.

Let’s do another iteration:

- Compute the value of the function at x1: f(x1) = (1.875)^3 – 2(1.875) – 5 = -0.327
- Compute the derivative of the function at x1: f'(x1) = 3(1.875)^2 – 2 = 8.953
- Calculate the next approximation of the root using the formula:

x2 = x1 – f(x1) / f'(x1) = 1.875 – (-0.327) / 8.953 = 2.095

- Check the difference between x2 and x1: |x2 – x1| = |2.095 – 1.875| = 0.22. Since this is larger than our desired level of accuracy, we need to continue with the next iteration.
- Set x1 = x2 and repeat steps 2-4 until the desired level of accuracy is achieved.

After repeating steps 2-4 a few more times, we eventually get an approximation of the root to be x = 2.0945514815.

Example 2:

Use the Newton-Raphson method to find a root of the function f(x) = x^3 – 2x^2 + 2.

Solution:

- Choose an initial guess for the root, let’s say x0 = 1.
- Compute the value of the function at x0: f(x0) = (1)^3 – 2(1)^2 + 2 = 1.
- Compute the derivative of the function at x0: f'(x0) = 3(1)^2 – 4(1) = -1.
- Calculate the next approximation of the root using the formula:

x1 = x0 – f(x0) / f'(x0) = 1 – 1 / (-1) = 2.

- Check the difference between x1 and x0: |x1 – x0| = |2 – 1| = 1. Since this is larger than our desired level of accuracy, we need to continue with the next iteration.
- Set x0 = x1 and repeat steps 2-4 until the desired level of accuracy is achieved.

Let’s do another iteration:

- Compute the value of the function at x1: f(x1) = (2)^3 – 2(2)^2 + 2 = -2.
- Compute the derivative of the function at x1: f'(x1) = 3(2)^2 – 4(2) = 4.
- Calculate the next approximation of the root using the formula:

x2 = x1 – f(x1) / f'(x1) = 2 – (-2) / 4 = 2.5.

- Check the difference between x2 and x1: |x2 – x1| = |2.5 – 2| = 0.5. Since this is larger than our desired level of accuracy, we need to continue with the next iteration.
- Set x1 = x2 and repeat steps 2-4 until the desired level of accuracy is achieved.

After a few more iterations, we get an approximation of the root to be x = 1.5321 (rounded to 4 decimal places).