Newton's Method: Roots

Question

Use Newton's method to estimate the positive and negative fourth roots of 2. Start at x₀ = 1 and x₀ = -1 and find x₂ in both cases.

Answer 1

Step 1: Create your function.

In this case since you want the fourth root of 2 you can say x⁴ = 2 or x⁴ - 2 = 0 so your function is f(x) = x⁴ - 2

Step 2: Create the equation for finding the next iteration:

Use the "Newton's Method" equation x_n+1 = x_n - f(x_n)/f '(x_n)

In this case f '(x) = 4x³ so the equation becomes x_n+1 = x_n - (x_n⁴ - 2)/(4x_n³)

Step 3: Pick a starting point (x₀)

In this case, we are told to start at x₀ = 1 for the positive root and x₀ = -1 for the negative root.. Working first to find the positive root:

x_n+1 = x_n - (x_n⁴ - 2)/(4x_n³)

x₁ = x₀ - (x₀⁴ - 2)/(4x₀³)

x₁ = 1 - (1⁴ - 2)/(4•(1)³)

x₁ = 1.25

Then repeat with x_n = 1.25

x₂ = 1.25 - (1.25⁴ - 2)/(4•(1.25)³)

x₂ = 1.1935

If you were going to find the actual answer, you would keep going until there is no change in the number when you do the iteration. In this case, we are just asked to find x₂.

Repeat this process with x₀ = -1 to find the negative root:

x₁ = (-1) - ((-1)⁴ - 2)/(4•(-1)³)

x₁ = -1.25

x₂ = (-1.25) - ((-1.25)⁴ - 2)/(4•(-1.25)³)

x₂ = -1.1935