WOLFRAM|DEMONSTRATIONS PROJECT

The Eneström-Kakeya Bounds for Roots of a Polynomial with Positive Coefficients

​
a
0
1
a
1
1
a
2
1
a
3
3
a
4
1
5
z
+
4
z
+3
3
z
+
2
z
+z+1
This Demonstration shows some bounds on the roots of a polynomial with positive coefficients of the form
5
z
+
a
4
4
z
+
a
3
3
z
+
a
2
2
z
+
a
1
z+
a
0
. The green circle (centered at the origin) has radius
max(
a
i
)+1
. All roots (represented by the red points) always lie within this circle. This is a bound obtained by applying Rouché's theorem in complex analysis. The two blue circles (centered at the origin) represent the bounds given by the Eneström–Kakeya theorem. The roots always lie between these two circles. By varying the coefficients, we see that for certain polynomials the Eneström–Kakeya bounds are much tighter than the one given by Rouché's theorem, but for other polynomials it can be the other way around.