The roots of a polynomial
This is a Java applet which calculates the roots of the polynomial
xn + an-1xn-1 + ... + a1x
+ a0 = 0.
On the left hand side there are control panels where you can change the
order of the polynomial and its coefficients. Just manipulate the scrollbar or
enter a number into the number field. The result is show graphically in the
complex number plane (x-axis is the real part, y-axis is the
imaginary part). On the right-hand side the numerical result is shown. Note
that the sign of the real part decides the color code.
Below the roots the Hurwitz determinants are
shown. Their color code is determined by their sign.
They are needed for the Routh-Hurwitz theorem.
| Fun with the applet |
Here are nice games you can play:
- Try to get all roots onto the left side of the imaginary axis (i.e.,
the vertical line).
- Try to get all roots on the horizontal line.
The difficulty of these games increases with the order of the polynomial.
|
The Routh-Hurwitz theorem
Calculating analytically the roots of a polynomial of order higher than two
is very cumbersome and in general impossible if the order is
larger than five. But often you do not need the roots of a polynomial explicitely.
For example, when you do stability analysis for a fixed point in a (nonlinear)
dynamical system you get a characteristic polynomial for
the eigenvalues of the Jacobian. To answer the question on stability you
don't need the eigenvalues. You only want to know whether all eigenvalues have
negative real parts or not. And here comes the neat
theorem of Routh and Hurwitz
which helps you a lot. It says:
The real part of all roots of the polynomial
xn + an-1xn-1 + ... + a1x
+ a0 = 0
are negative if and only if all
upper-left determinants of the matrix
with am=0 for m<0, are positive.
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© 1998 Franz-Josef Elmer, 
elmer@ubaclu.unibas.ch last modified Thursday, July 16, 1998.