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# FP2 Iteration Question

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1. I was revising this topic here https://goo.gl/qyE8Cn when I read that iterative methods will only cobweb or staircase to a root of f(x) when F'(x) is greater than -1 or smaller than 1, this seemed a bit difficult to believe, so I tested it out with a simple example.

f(x) = x^5 -x

F(x) = x^0.2

I used an initial value of 0.1, which obviously converges to 1 and does indeed staircase if y=x and x^0.2 are plotted. However:

F'(x) = 0.2x^(-0.8)

F'(0.1) = 1.262 > 1

What am I getting wrong, or is the video incorrect?
2. bump
3. (Original post by 16characterlimit)
I was revising this topic here https://goo.gl/qyE8Cn when I read that iterative methods will only cobweb or staircase to a root of f(x) when F'(x) is greater than -1 or smaller than 1, this seemed a bit difficult to believe, so I tested it out with a simple example.

f(x) = x^5 -x

F(x) = x^0.2

I used an initial value of 0.1, which obviously converges to 1 and does indeed staircase if y=x and x^0.2 are plotted. However:

F'(x) = 0.2x^(-0.8)

F'(0.1) = 1.262 > 1

What am I getting wrong, or is the video incorrect?

If we are using that iteration then it will converge (assuming we start at a sensible start point) provided |F'(x)| < 1 AT THE ROOT. Not at the start point. The root is x = 1 so considering the derivative here we see that F'(1) = 0.2 < 1 in magnitude so the iteration does converge.

In practice you might not know the root exactly when testing for convergence so might have to use a nearby point. This is usually OK provided the point you use is close enough. In this case however, 0.1 is too far away from 1 so F'(0.1) and F'(1) were notably different.
4. can anyone share the paper of 2016 jan ?
5. (Original post by 16Characters....)
If we are using that iteration then it will converge (assuming we start at a sensible start point) provided |F'(x)| < 1 AT THE ROOT. Not at the start point. The root is x = 1 so considering the derivative here we see that F'(1) = 0.2 < 1 in magnitude so the iteration does converge.

In practice you might not know the root exactly when testing for convergence so might have to use a nearby point. This is usually OK provided the point you use is close enough. In this case however, 0.1 is too far away from 1 so F'(0.1) and F'(1) were notably different.
Thanks that makes sense.

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