A quasi chaotic polynomial.

While fooling around with a VB polynomial root finder application, I discovered a 20th order polynomial which was extremely sensitive to small changes in the value of the independent variable.

My application had determined a root, but when I evaluated the polynomial at that root, the value was about -4*10^5. I used Newton Raphson to improve the precision of the root. The new value of the root was about 1*10^ -12 larger. When I evaluated the polynomial at the improved root, the value was about +1.5*10^5

It looked like chaotic behavior in an ordinary polynomial. A change in the 12th digit to the right of the decimal point resulted in an incredible change in the value of the polynomial. From about minus 370,000 to plus 150,000

Then I evaluated the derivative near that root. It was about 5*10^17, which explains the behavior. The slope of the curve is almost vertical. Any slight change in the independent variable results in a huge change in the value of the polynomial.

The only word I understood was 'root', and probably for the wrong reasons. But I do think I found a picture of you.

http://www.vbforums.com/images/ieimages/2002/06/17.gif

:)

Very interesting Guv. It it's very obvious that your mathematical skills are light years ahead of mine. So that being said i an unable to comment on your findings. :( Still very interesting though. :D

Yeh, I understood. I was trying to fudge around and work out what kind of quadratic it was... Can you let me know?

In plain English - Guv found an 'equation'. He found that if you made a ridiculously small change in one of the 'input' values he got a huge change in the root (solution). He did not expect the result, so he first believed the polynomial was showing chaotic behavior. The polynomial looked completely vanilla, which is why he got interested.

Chaos (fractals, turbulence, clouds, non-linear dynamics) deals with objects (to use a programming term) that display this kind of unpredictable behavior.

He found that the slope of the curve where the behavior was unexpected was very steep, almost asymptotic. So, it wasn't really chaotic. Tiny changes made the root values change unexpectedly.

It would have been really interesting if it were in fact chaotic. Like possibly worthy of publication interesting.