Values of x1 for which an iterative formula converges or diverges?

[21joanna12]

I was wondering if there is a way of finding out the values of x₁ for which an iterative formula converges.
Specifically, I am looking at the formula x_n+1=0.25(x_n³+1) and I am thinking about it in the context of finding roots of a function, although it would be great if there was a general way of finding whether a sequence converges or diverges.

I haven't had any ideas yet, so thank you in advance for any help! :)

 

[21joanna12]

I found the answer: it converges if the absolute value of the derivative of the function f(x) where your formula is x_n+1=f(x_n) must be less 1. I get that, but now II have another question...

For the function x³-4x+1, if you want to find the roots using this iterative process you can use x_n+1=0.25(x_n³+1) or x_n+1=cube root(4x_n-1). I know that there are 3 roots. I have only considered the two in the interval 0-1 and 1-2, but it seems like for each of these roots only one of the formulae converges, the other diverges. Is it the case that there will always be one fomrula which converges, or are there some roots that can never be found by this process? Can this be shown mathematically?

Thank you :)

 

[Ray Vickson]

I was wondering if there is a way of finding out the values of x₁ for which an iterative formula converges.
Specifically, I am looking at the formula x_n+1=0.25(x_n³+1) and I am thinking about it in the context of finding roots of a function, although it would be great if there was a general way of finding whether a sequence converges or diverges.

I haven't had any ideas yet, so thank you in advance for any help! :)

You will gain a lot of insight into your problem if you look at it using a "cobweb diagram" methodology. See, eg., http://en.wikipedia.org/wiki/Cobweb_plot or
https://www.math.ubc.ca/~andrewr/620341/pdfs/ga_sum.pdf . This last one has examples similar to yours.