Liénard-Chipart Stability Criterion

Question

Consider the real polynomial $f(z) = a_0z^4 + a_1z^3 + a_2z^2 + a_3z + a_4$ with $a_0 > 0$. According to "The Theory of Matrices" by Gantmacher, the Liénard-Chipart Stability Criterion states that the roots of $f(z)$ will have negative real parts if any of the following conditions are met:

1. $a_n > 0, a_{n-2} > 0, \dots; \Delta_1 > 0, \Delta_3 > 0, \dots$
2. $a_n > 0, a_{n-2} > 0, \dots; \Delta_2 > 0, \Delta_4 > 0, \dots$
3. $a_n > 0, a_{n-1} > 0, a_{n-3} > 0, \dots; \Delta_1 > 0, \Delta_3 > 0, \dots$
4. $a_n > 0, a_{n-1} > 0, a_{n-3} > 0, \dots; \Delta_2 > 0, \Delta_4 > 0, \dots$

Where $\Delta_i$ is the $i$-th principal minor of the corresponding Hurwitz matrix.

While looking for more information about the Liénard-Chipart stability criterion, I ran across what seems to be lecture notes of a course taught in University of Colorado. Following is an excerpt from the notes to chapter 4:

On p. 221 of [4.27] it is stated that the Liénard-Chipart criterion can be expressed in four equivalent forms, labeled 1 through 4. But form 1 is only valid for even n and form 4 for odd n, as can be seen by comparing those statements with Table 4.1. Forms 2 and 3 are always wrong. Like most Russian books, Gantmacher shies away from numerical examples. Any such computation would have immediately displayed the mistake.

(4.27 is "The Theory of Matrices" by Gantmacher)

Ignoring the slightly offensive tone of the notes, I believe the authors are correct and that the conditions given by Gantmacher are false or at least incomplete. Two simple counterexamples:

1. Counterexample for form 2: $z^4 + z^3 + z^2 + 1$. The inequalities for this form all hold, but the polynomial does have roots with positive real parts.
2. Similar counterexample for form 4: $z^4 + z^3 + z^2 + 0.1z + 1$. The inequalities for this form all hold, but again the polynomial does have roots with positive real parts

Are my counterexamples somehow incorrect, or is the theorem as stated by Gantmacher false? Do you know of any decent source of information about this criterion which does not simply cite Gantmacher?

Answer 1

The cited counter-example 1 with coefficients $(a_0,...,a_4)=(1,1,1,0,1)$ has Hurwitz determinants

$$\Delta_2=\begin{vmatrix}a_1&a_3\\a_0&a_2\end{vmatrix}=\begin{vmatrix}1&0\\1&1\end{vmatrix}=1$$

$$\Delta_3=\begin{vmatrix}a_1&a_3&0\\a_0&a_2&a_4\\0&a_1&a_3\end{vmatrix} =\begin{vmatrix}1&0&0\\1&1&1\\0&1&0\end{vmatrix}=-1$$

$$\Delta_4 =\begin{vmatrix} a_1&a_3&0&0\\ a_0&a_2&a_4&0\\ 0&a_1&a_3&0\\ 0&a_0&a_2&a_4 \end{vmatrix} =\begin{vmatrix}1&0&0&0\\1&1&1&0\\0&1&0&0\\0&1&1&1\end{vmatrix} =\begin{vmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&1&1\end{vmatrix} =-1$$

so that it violates all forms of the Liénard-Chipart stability criterion.

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Example 2 with coefficients $(a_0,...,a_4)=(1,1,1,0.1,1)$ has Hurwitz determinants

$$\Delta_2=\begin{vmatrix}a_1&a_3\\a_0&a_2\end{vmatrix}=\begin{vmatrix}1&0.1\\1&1\end{vmatrix}=0.9$$

$$ \Delta_3=\begin{vmatrix}a_1&a_3&0\\a_0&a_2&a_4\\0&a_1&a_3\end{vmatrix} =\begin{vmatrix}1&0.1&0\\1&1&1\\0&1&0.1\end{vmatrix} =\begin{vmatrix}1&0.1&0\\0&0.9&1\\0&1&0.1\end{vmatrix} =-0.91$$

$$\Delta_4 =\begin{vmatrix} a_1&a_3&0&0\\ a_0&a_2&a_4&0\\ 0&a_1&a_3&0\\ 0&a_0&a_2&a_4 \end{vmatrix} =\begin{vmatrix}1&0.1&0&0\\1&1&1&0\\0&1&0.1&0\\0&1&1&1\end{vmatrix} =\begin{vmatrix}1&0.1&0\\0&0.9&1\\0&1&0.1\end{vmatrix} =-0.91$$

so that again all forms of the Liénard-Chipart stability criterion are violated.

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The cited lecture notes are no longer available, so I can not comment on other counter-examples that might have been provided there.