Solving x3-x-2 for the X-Axis: Newton's Method

[alpha01]

Homework Statement

using Newtons method with an initial estimate of x₀=2, find the point where the graph f(x)=x³-x-2 crosses the x-axis

Homework Equations

x_i+1 = x_i - f(x_i)/f'(x_i)

The Attempt at a Solution

Using a function plotter, I know the answer should be around 1.52138... But how am i supposed to know how many repetitions of Newton's method is required to get x where y=0 (i.e the x-axis).

 

[fluidistic]

Using a function plotter, I know the answer should be around 1.52138... But how am i supposed to know how many repetitions of Newton's method is required to get x where y=0 (i.e the x-axis).

Generally it will never reach the x value when y=0, so the number of repetition is often infinite. However it's different if you want say 5 correct digits or so. I don't remember very well how to find the number of repetitions but I remember that the method converges quadratically to the solution. Reading : http://en.wikipedia.org/wiki/Newton's_method will certainly help you.

 

[alpha01]

I had a look but still couldn't figure it out ...it wants the solution to 3 decimal places..

 

[HallsofIvy]

Homework Statement

using Newtons method with an initial estimate of x₀=2, find the point where the graph f(x)=x³-x-2 crosses the x-axis

Homework Equations

x_i+1 = x_i - f(x_i)/f'(x_i)

The Attempt at a Solution

Using a function plotter, I know the answer should be around 1.52138... But how am i supposed to know how many repetitions of Newton's method is required to get x where y=0 (i.e the x-axis).

You aren't- no number of repetitions will give the exact value. However that is not relevant to your problem. Because you can't get an exact value, you need to think about how accurate you want the answer to be. Generally speaking, a solution is as close to the correct value as it is to the previous iteration.

 

[alpha01]

it says give your answer "accurate to three decimal places" that is what I am unsure about... that sounds like i can just do one repetition and write the answer to 3 decimal places.. but that sounds too easy (its for my finals, it should be harder i think)

 

[fluidistic]

Then I think that 3 iterations are more than enough.
To be sure, do one iteration and keep the number you get. Do another one iteration and if the first 3 decimal places are the same, then it's done. If only the 2 first are equal, then do another iteration and you're done.