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# convergence watch

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1. An iteration of the form X n+1 =g(Xn) converges when the gradient of y=g(x) at the point of intersection with the line y=x satisfies the condition |g'(x)|<1, provided a suitable value for X1 is chosen.

can someone explain this to me with an example
2. (Original post by man111111)
An iteration of the form X n+1 =g(Xn) converges when the gradient of y=g(x) at the point of intersection with the line y=x satisfies the condition |g'(x)|<1, provided a suitable value for X1 is chosen.

can someone explain this to me with an example
The iteration scheme produces a staircase/cobweb diagram which you would've covered at A2 level.

If we wanted to approximate solutions to the equation then we can rearrange this into which graphically can be interpreted as the intersection point(s) between and . We proceed by the iteration scheme . For reference, it looks like this:

Spoiler:
Show

Now note that with . Testing the values of intersection, we get that and so we should expect and not towards for a suitably chosen .
This can be clearly observed:
Starting on produces a staircase to the root. Starting on produces a staircase to the expected root as well. But starting produces a staircase that doesn't stop (more or less). So, we never get convergence to as expected.

The reason and analysis for should be covered in your notes.

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Updated: February 14, 2018
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