Jury stability criterion

Method

If the characteristic polynomial of the system is given by

$f(z)=a_{0}z^{n}+a_{1}z^{n-1}+a_{2}z^{n-2}+\cdots +a_{n-1}z+a_{n}$

then the table is constructed as follows:

row	z0	z1	z2	z....	zn-1	zn
1	a0	a1	a2	...	an-1	an
2	an	an-1	an-2	...	a1	a0
3	b0	b1	...	bn-2	bn-1
4	bn-1	bn-2	...	b1	b0
5	c0	c1	...	cn-2
6	cn-2	cn-3	...	c0
...	...	...	...	...	...	...
2n-5	p3	p2	p1	p0
2n-4	p0	p1	p2	p3
2n-3	q2	q1	q0

That is, the first row is constructed of the polynomial coefficients in order, and the second row is the first row in reverse order and conjugated.

The third row of the table is calculated by subtracting ${\frac {a_{n}}{a_{0}}}$ times the second row from the first row, and the fourth row is the third row with the first n elements reversed (as the final element is zero).

${\begin{aligned}a_{0}\;\;&a_{1}\;\;&\dots \;\;&a_{n-1}\;\;&a_{n}\\a_{n}\;\;&a_{n-1}\;\;&\dots \;\;&a_{1}\;\;&a_{0}\\\left(a_{0}-a_{n}{\frac {a_{n}}{a_{0}}}\right)\;\;&\left(a_{1}-a_{n-1}{\frac {a_{n}}{a_{0}}}\right)\;\;&\dots \;\;&\left(a_{n-1}-a_{1}{\frac {a_{n}}{a_{0}}}\right)\;\;&0\\\left(a_{n-1}-a_{1}{\frac {a_{n}}{a_{0}}}\right)\;\;&\dots \;\;&\left(a_{1}-a_{n-1}{\frac {a_{n}}{a_{0}}}\right)\;\;&\left(a_{0}-a_{n}{\frac {a_{n}}{a_{0}}}\right)\;\;&0\\\end{aligned}}$

The expansion of the table is continued in this manner until a row containing only one non zero element is reached.

Note the ${\frac {a_{n}}{a_{0}}}$ is for the 1st two rows. Then for 3rd and 4th row the coefficient changes (i.e. ${\frac {b_{n-1}}{b_{0}}}$) . This can be viewed as the new polynomial which has one less degree and then continuing.