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Iteration of a quadratic

Given

f(x)=x2+32x−518f(x)=x^2+\frac{3}{2}x-\frac{5}{18}

iterate the function from x0=0x_0=0 and predict its long term behaviour.

So, the first few terms are:

x0=0x_0=0

x1=−518x_1=-\frac{5}{18}

x2=−5081x_2=-\frac{50}{81}

x3=−1079513122x_3=-\frac{10795}{13122}

I can see they're getting smaller, but how do I predict what (if anything) they'll converge to? I can see that the x2x^2 would tend to zero, but what happens to the middle term?
(edited 13 years ago)

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Original post by Plato's Trousers
Given

f(x)=x2+32x−518f(x)=x^2+\frac{3}{2}x-\frac{5}{18}

iterate the function from x0=0x_0=0 and predict its long term behaviour.

So, the first few terms are:

x0=0x_0=0

x1=−518x_1=-\frac{5}{18}

x2=−5081x_2=-\frac{50}{81}

x3=−1079513122x_3=-\frac{10795}{13122}

I can see they're getting smaller, but how do I predict what (if anything) they'll converge to? I can see that the x2x^2 would tend to zero, but what happens to the middle term?

I'm not sure if this will help you predict anything (I'm really inexperienced with numerical methods) but maybe complete the square on f(x) and notice that x is heading closer and closer to maybe -0.9 or so (that's a guess).
Original post by Plato's Trousers
Given

f(x)=x2+32x−518f(x)=x^2+\frac{3}{2}x-\frac{5}{18}

iterate the function from x0=0x_0=0 and predict its long term behaviour.

So, the first few terms are:

x0=0x_0=0

x1=−518x_1=-\frac{5}{18}

x2=−5081x_2=-\frac{50}{81}

x3=−1079513122x_3=-\frac{10795}{13122}

I can see they're getting smaller, but how do I predict what (if anything) they'll converge to? I can see that the x2x^2 would tend to zero, but what happens to the middle term?


Using decimals and finding a few more terms would help you. You have misled yourself if you think it will continue to get smaller.
You are really solving the equation:

x=x2+3x2−518x = x^2 + \frac{3x}{2} -\frac{5}{18}

Can you rearrange and factorise this quadratic?
Original post by Farhan.Hanif93
I'm not sure if this will help you predict anything (I'm really inexperienced with numerical methods) but maybe complete the square on f(x) and notice that x is heading closer and closer to maybe -0.9 or so (that's a guess).


hmm.. I shall try that. Apparently it converges to -0.83333

Original post by Mr M
Using decimals and finding a few more terms would help you. You have misled yourself if you think it will continue to get smaller.


Thanks
hmm... completing the square gives me this

(x+34)2−122144(x+\frac{3}{4})^2-\frac{122}{144}

But not sure if that helps..

:confused:
Original post by Plato's Trousers
hmm.. I shall try that. Apparently it converges to -0.83333

Thanks


Otherwise known as -5/6.

Do you know the conditions for when an iterative formula will cobweb or staircase?
Just computed it on MATLAB, and I got the following plot.
https://docs.google.com/leaf?id=0B59VcSaKK77cOTI5OWFkMjQtOTBiOC00NjdkLWFkYmEtMjNiYjk5MGY3NmIx&hl=en_GB&authkey=CP7x2MYO
Hope it helps. Also if anyone can direct me to an image hoster that doesn't require signing up, it'll be appreciated :wink:
(edited 13 years ago)
Why would you complete the square??!
Reply 9
Avatar for roar558
roar558
11
Original post by Plato's Trousers
hmm... completing the square gives me this

(x+34)2−122144(x+\frac{3}{4})^2-\frac{122}{144}

But not sure if that helps..

:confused:


If you work it out, you get 1/18 (6x-1)(3x+5) if you're trying to factorise the quadratic, the limit for iterating that function seems to be -5/6 though.
(edited 13 years ago)
Original post by roar558
If you work it out, you get 1/18 (6x-1)(3x+5) if you're trying to factorise the quadratic


Nah.
Reply 11
Avatar for roar558
roar558
11
Original post by Mr M
Nah.


ah sorry didn't see the x= part
well I got 0=1/18 (3x-1)(6x+5)
Original post by roar558
ah sorry didn't see the x= part
well I got 0=1/18 (3x-1)(6x+5)


So it is not surprising that x = -5/6 is a solution then.
Original post by roar558
ah sorry didn't see the x= part
well I got 0=1/18 (3x-1)(6x+5)


I see a -5/6.
Original post by gunmetalpanda
Why would you complete the square??!


Well I'm not sure FH knew what he was doing here but it actually is useful as it tells us the minimum point of the parabola and the change in the sign of the gradient expains why the iterative sequence stops approaching the root from one side and starts to approach alternately from either side.
Original post by Mr M
Otherwise known as -5/6.

Do you know the conditions for when an iterative formula will cobweb or staircase?


Yes, I know that -5/6 is a fixed point (from solving the fixed point equation) and I know it's attracting (by looking at the condition |f'(x)|<0). But there's also a fixed point at 1/3

How do I decide which it will converge to?


Original post by Mr M
Otherwise known as -5/6.

Do you know the conditions for when an iterative formula will cobweb or staircase?



Original post by hxecute
Just computed it on MATLAB, and I got the following plot.
https://docs.google.com/leaf?id=0B59VcSaKK77cOTI5OWFkMjQtOTBiOC00NjdkLWFkYmEtMjNiYjk5MGY3NmIx&amp;hl=en_GB&amp;authkey=CP7x2MYO
Hope it helps. Also if anyone can direct me to an image hoster that doesn't require signing up, it'll be appreciated :wink:


You can post an image as an attachment to your post right here on TSR ("Go advanced" then "manage attachments")
(edited 13 years ago)
Original post by Plato's Trousers
You can post an image as an attachment to your post right here on TSR ("Go advanced" then "manage attachments")


Yes, but I use the quick reply feature alot :redface:
Original post by hxecute
Yes, but I use the quick reply feature alot :redface:


but surely it's faster to used advanced edit on replies and attach your image right there and then, rather than go and find and image host?
Original post by Plato's Trousers
How do I decide which it will converge to?


I don't know which way you are supposed to be doing it but a sketch would tell you. You could differentiate as well and check the gradient if you liked. When x = 1/3 (the other root), you might note that the gradient is rather large.
Plato pm me with an email address and I will send you a Powerpoint and a .pdf of some worksheets that will help your understanding.

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