Question

How do you use Newton's method to find the approximate solution to the equation `x^3-10x+4=0, x>1`?

Answer 1

Compute the first derivative. Write the equation. Start with a seed value `>1`. Recursively perform the computation (I recommend using a spreadsheet), until it converges.

Explanation:

This reference on Newton's Method gives us this equation:

`x_(n+1) = x_n - f(x_n)/(f'(x_n))`

Compute the first derivative:

`f'(x) = 3x^2 - 10`

The equation is:

`x_(n+1) = x_n - (x_n^3 - 10x_n + 4)/(3x_n^2 - 10)`

I am going tell you how to use an Excel spreadsheet to perform the recursion:

Enter the number 2 into cell A1.

Into cell A2, enter the following Excel formula:

=A1 - (A1^3 - 10A1 + 4)/(3A1^2 - 10)

Use the block paste feature of Excel to copy and paste the formula into cells A3 through A20.

The computation converges quickly onto the root 2.939235

You can find the other 2 roots by playing with the seed value in cell A1