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.