# Convergents and Fixed Points

• Nov 6th 2006, 04:31 PM
Mr_Green
Convergents and Fixed Points
Hey all. Could someone check my work here and get back to me before 9 central time. THanks, GREEN!

*****Part A:

Let f(x) = 1 / (x^2 + 3x -2)

Find the fixed points for f.

I got x = 0.85106383, x = -0.3191489, and x = -3.510638.

Are these correct?

*****Part B:

Try different seeds to find out which, if any are convergent.

To me it seems that it converges at -0.343379569, so according to this none are convergent.

GREEN
• Nov 6th 2006, 04:49 PM
ThePerfectHacker
Quote:

Originally Posted by Mr_Green
[B]Hey all. Could someone check my work here and get back to me before 9 central time. THanks, GREEN!

*****Part A:

Let f(x) = 1 / (x^2 + 3x -2)

Find the fixed points for f.

I got x = 0.85106383, x = -0.3191489, and x = -3.510638
Are these correct?

A fixed point has to exist by the Brouwer Fixed Point theorem (somebody correct me if I am wrong).
If we solve,
$\displaystyle x=\frac{1}{x^2+3x-2}$
We get,
$\displaystyle x^3+3x^2-2x-1=0$
The solutions to this equation are roots.
• Nov 6th 2006, 05:11 PM
Mr_Green
ok i checked my work, and i believe these are my new and improved answers ...

part a:

x = 0.834
x = -0.343
x = -3.491

part b:

remains the same, except now one of the fixed points are convergent.

How's this look guys?

Here is part c now:

Zoom in on a convergent point until the graph of f looks like a straight line (this phenomenon is known as "local linerity.") Find the coordinates of two points on f - one to the left and one to the right of the convergent point. Then find the slope (rise over run) of the segment connecting these two points.

I would also like to check my slope with someone, so if you find it, please let me know.

Thanks again,
GREEN