Abstract
Let \(f(x)=a_n x^n + a_{n-1} x^{n-1} + \cdots +a_1 x +a_0\) be a polynomial with real positive coefficients and \(p\in \mathbb {R}\). The pth Hadamard power of f is the polynomial \(f^{[p]}(x):=a_n^p x^n + a_{n-1}^p x^{n-1}+ \cdots + a_1^p x +a_0^p\). We give sufficient conditions for \(f^{[p]}\) to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all \(p>p_0\) or \(p<p_1\) with some positive \(p_0\) and negative \(p_1\) (without any assumption about stability of f). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378, 1998) asserts that if f is a stable polynomial with positive coefficients then \(f^{[p]}\) is stable for every \(p\ge 1\). We construct a counterexample to this statement.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
For a positive integer number n we consider
Let \(\mathbb {R}^+[n]\) be the family of all polynomials of the form (1). The polynomial
where \(p\in \mathbb {R}\) is called the pth Hadamard power of \(f\in \mathbb {R}^+[n]\). We say that the polynomial f with real coefficients is stable (f is a Hurwitz polynomial) if every zero of f has strictly negative real part. A necessary condition for a polynomial f with real coefficients to be stable is that f has all coefficients of the same sign. Let \(H_n\) be the family of all stable polynomials of degree n with positive coefficients.
In 1996 J.Garloff and D.G.Wagner proved in [1] that \(f\,\in \, H_n\) implies \(f^{[p]}\,\in \, H_n\) for all \(p\,\in \,\{1,2,3,\ldots \}\). The natural question arises of a set of real numbers p for which \(f^{[p]}\) is stable where \(f\in \mathbb {R}^+[n]\). We give some conditions on p and on f for \(f^{[p]}\) to belong to \(H_n\). Moreover, we show that \(f^{[p]}\) does not need to be stable for a stable polynomial f and an exponent \(p> 1\), contrary to Theorem 5 in [2].
Observe that if \(n=1\) or \(n=2\) then \(f^{[p]}\) is stable for every \(p\in \mathbb {R}\) and for all polynomials \(f\in \mathbb {R}^+[n]\). The case of \(n\ge 3\) is much more complicated, e.g., for \(f(x)=x^3+x^2+x+1\) we have \(f^{[p]}\not \in H_n\) for any \(p\in \mathbb {R}\). Therefore, we will consider only the case \(n\ge 3\).
1.1 Basic information
For relevant background material concerning Hurwitz polynomials and related topics see [5, Sec.11]. We list below selected theorems that will be useful in the paper.
The Hurwitz matrix H(f) associated to the polynomial \(f\in \mathbb {R}^+[n]\) is given as follows
Denote by \(D_i(p)\) for \(i=1,\ldots ,n\) the ith leading principal minor of the Hurwitz matrix \(H(f^{[p]})\), i.e.,
To simplify the writing, we put \(D_i:=D_i(1)\).
Theorem 1
Routh–Hurwitz criterion
If \(f\in \mathbb {R}^+[n]\) then \(f\in H_n\) if and only if \(D_i>0\) for all \(i=1,\ldots ,n\).
Theorem 2
(see [3, Th.2 and (1.10)])
If \(f\in H_n\) with \(n\ge 3\) then
Theorem 3
(see [4])
Let \(f\in \mathbb {R}^+[n]\) with \(n\ge 5\) and \(\gamma \) be the unique real root of the equation
If \(\gamma \, a_{n-i}\,a_{n-i-1}>a_{n-i+1}\,a_{n-i-2}\) for every \(i=1,\ldots ,n-2\) then \(f\in H_n\).
1.2 Counterexample
Theorem 5 in [2] asserts that if \(f\in H_n\) then \(f^{[p]}\in H_n\) for all \(p\ge 1\). We construct below a counterexample to this statement.
For a fixed polynomial \(f\in \mathbb {R}^+[n]\) with \(n\ge 3\) consider the following decomposition
It may be worth reminding the reader that g and h are called interlacing if
-
all zeros of g and h are real, negative and distinct,
-
between every two zeros of g there exists a zero of h and vice versa.
Among variants of Hermite–Biehler theorem we will apply the following one to construct a counterexample.
Theorem 4
(see [5, Chapter 6.3]) Every polynomial \(f\in \mathbb {R}^+[n]\) decomposed as in (3) is stable if and only if g and h are interlacing.
Counterexample 1
Let
Y. Wang and B. Zhang considered g in [6] and observed that for \(p=1.139\) the polynomial \(g^{[p]}\) has two nonreal zeros: \(-16.0617\pm 0.178468\,i\) (approximated value). Take now
and put
It is easy to verify that f is stable (e.g., by the Routh–Hurwitz criterion). We have \(f^{[p]}(x)=g^{[p]}(x^2)+x\,h^{[p]}(x^2)\) and thus, by Theorem 4 the polynomial \(f^{[p]}\) is not stable for \(p=1.139\). By means of Wolfram Mathematica 10.4 we found two zeros of \(f^{[1.139]}\) that have positive real part: \(0.00179025\pm 4.01279\,i\) (approximated value).
2 Main results
Now we will state and prove some sufficient conditions for \(f^{[p]}\) to be a Hurwitz polynomial for all \(p> p_0\) or \(p< p_1\) with some positive \(p_0\) and negative \(p_1\) depending only on coefficients of f. The polynomial f is assumed to be of the form (1) but need not to be stable. We will discuss separately three cases: \(n=3\), \(n=4\) and \(n\ge 5\). We start with a lemma and some necessary conditions for the Hurwitz stability.
2.1 Notations and preliminary results
For a fixed polynomial \(f\in \mathbb {R}^+[n]\) with \(n\ge 3\) and \(p\in \mathbb {R}\) we put
Moreover, for ease of notation, throughout the paper we write \(w_i\) for \(w_i(1)\). Let
It is worth noticing that
-
if \(w_i>0\) for all i then \(\overline{d}<1\),
-
if \(w_i<0\) for all i then \(\underline{d}>1\).
Lemma 1
Let \(\lambda \in (0,1)\) and \(f\in \mathbb {R}^+[n]\) with \(n\ge 3\). Put
-
1.
If \(w_i>0\) for all \(i=1,\ldots ,n-2\)
$$\begin{aligned}&then \ \ \ \ \lambda a_{n-i-1}^p \, a_{n-i}^p - a_{n-i-2}^p \, a_{n-i+1}^p> 0 \ \ \ \ {for \ all} \\ \qquad&i=1,\ldots ,n-2 \ \ \ and \ \ \ p>p_0>0.\ \ \end{aligned}$$ -
2.
If \(w_i<0\) for all \(i=1,\ldots ,n-2\)
$$\begin{aligned}&then \ \ \ \ \lambda a_{n-i-1}^p \, a_{n-i}^p - a_{n-i-2}^p \, a_{n-i+1}^p > 0 \ \ \ \ {for \ all} \\ \qquad&i=1,\ldots ,n-2 \ \ \ and \ \ \ p<p_1<0.\ \ \end{aligned}$$
Proof
Firstly we show statement 1. Since \(w_i>0\) for all i, it follows that \(\overline{d}<1\) and hence for a fixed \(p>p_0\) we have \(\overline{d}^p<\lambda \). From the definition of \(\overline{d}\) we can easily conclude that \(\overline{d}a_{n-i-1} a_{n-i} \ge a_{n-i-2} a_{n-i+1}\) for all i and so
In an analogous manner we can prove statement 2. Indeed, in this case we have \(\underline{d}>1\) and \(\underline{d}^p<\lambda \) for \(p<p_1\). From the definition of \(\underline{d}\) we get \(\underline{d}\, a_{n-i-1} \, a_{n-i} \le a_{n-i-2} \, a_{n-i+1}\) for all i. Hence
and the proof is completed.\(\square \)
We give below some sufficient conditions for \(f^{[p]}\) not to be stable. This is a direct consequence of Theorem 2.
Theorem 5
Let \(f\in \mathbb {R}^+[n]\) with \(n\ge 3\).
-
1.
If \(w_i\ge 0\) for some \(i\in \{1,\ldots ,n-2\}\) then \(f^{[p]}\not \in H_n\) for all \(p\le 0\).
-
2.
If \(w_i\le 0\) for some \(i\in \{1,\ldots ,n-2\}\) then \(f^{[p]}\not \in H_n\) for all \(p\ge 0\).
2.2 Case \(n=3\)
In this subsection we consider \(f(x)=a_3\ x^3 + a_2\ x^2 + a_1\ x + a_0\) with positive coefficients \(a_3,\, a_2,\, a_1, \, a_0\). For \(n=3\) the family of \(w_i\)’s [see (4)] is reduced to the unique element \(w_1=a_1\, a_2-a_0\, a_3\).
Theorem 6
For any polynomial \(f\in \mathbb {R}^+[n]\) with \(n=3\) we have
-
1.
If \(w_1>0\) then \(f^{[p]}\in H_3\) for all \(p> 0\).
-
2.
If \(w_1<0\) then \(f^{[p]}\in H_3\) for all \(p< 0\).
Proof
In order to prove statement 1, we observe that \(w_1>0\) implies \(w_1(p)=a_1^p\, a_2^p-a_0^p\, a_3^p>0\) for every \(p>0\). By the Routh–Hurwitz criterion we get the stability of \(f^{[p]}\) for \(p>0\), because
In an analogous manner we can prove statement 2. \(\square \)
2.3 Case \(n=4\)
We start this subsection with a simple characterization of stable polynomials of degree 4 with positive coefficients.
Proposition 7
Let \(f\in \mathbb {R}^+[n]\) with \(n=4\). The polynomial f is stable if and only if
Proof
It is easily computed that
By the Routh–Hurwitz criterion, \(f\in H_4\) implies \(D_3>0\), i.e.,
Dividing by \(a_1\, a_2\, a_3\) we obtain inequality (5).
For the reverse implication, we can conclude from (5) that
and hence \(D_2>0\). Moreover, an immediate consequence of (5) is \(D_3>0\), and so \(D_4>0\). Once again we use the Routh–Hurwitz criterion and get the stability of f. \(\square \)
Note that for \(n=4\) and any function \(f(x)=a_4 \ x^4 + a_3\ x^3 + a_2\ x^2 + a_1\ x + a_0\) we have only two \(w_i\)’s defined by (4):
and
It is worth recalling from the beginning of Sect. 2.1 that for f with positive coefficients we have \(\overline{d}< 1\) if all \(w_i\)’s are positive and \(\underline{d}> 1\) whenever all \(w_i\)’s are negative.
Theorem 8
Let \(f\in \mathbb {R}^+[n]\) with \(n=4\) and
-
1.
If \(w_1, \, w_2>0\) then \(f^{[p]}\in H_4\) for all \(p> p_0> 0\).
-
2.
If \(w_1, \, w_2<0\) then \(f^{[p]}\in H_4\) for all \(p< p_1<0\).
Moreover, the constants \(p_0\) and \(p_1\) are the best possible, i.e., for \(p_0\) it means that there exists a polynomial f of degree 4 with positive coefficients and \(w_1,w_2>0\) such that \(f^{[p]}\) is not stable for every \(p\le p_0\).
Proof
For the proof of statement 1, we use Lemma 1. For \(\lambda =1/2\) and \(p>p_0\) we have
Consequently,
and therefore,
By Proposition 7 we get the stability of \(f^{[p]}\) for \(p>p_0\). Statement 2 can be proved in an analogous fashion.
By Example 2 given below we show that the constants \(p_0\) and \(p_1\) cannot be improved. \(\square \)
Example 2
Consider the polynomial
In this case we have
and
Fix \(p\le p_0\). By Proposition 7, \(f^{[p]}\in H_4\) if and only if inequality (6) holds. We calculate
We see that inequality (6) does not hold and consequently \(f^{[p]}\) is not stable. Additionally, we can easily verify by Proposition 7 that polynomial f is stable.
Corollary 9
If \(f\in H_4\) then \(f^{[p]}\in H_4\) for all \(p\ge 1\).
Proof
Since \((t^p+s^p)^{1/p} \le t+s\) for all \(s,t\ge 0\) and \(p\ge 1\), we have
the last estimate being a consequence of the stability of f and Proposition 7. Once again we use Proposition 7 and we get the stability of \(f^{[p]}\). \(\square \)
2.4 Case \(n\ge 5\)
The main result of this subsection will be based on Theorem 3 that deals with \(n\ge 5\). We remind the reader that \(\gamma \) denotes the unique real root of the equation \(\gamma (\gamma +1)^2=1\). One can verify that \(\gamma \in (0.4655,0.466)\). Quantities \(w_1,\ldots ,w_{n-2}\) and \(\overline{d}, \underline{d}\) have been defined in the beginning of Sect. 2.1.
Theorem 10
Let \(f\in \mathbb {R}^+[n]\) with \(n\ge 5\) and
-
1.
If \(w_1,\ldots , w_{n-2}>0\) then \(f^{[p]}\in H_n\) for all \(p> p_0> 0\).
-
2.
If \(w_1,\ldots ,w_{n-2}<0\) then \(f^{[p]}\in H_n\) for all \(p< p_1<0\).
Proof
Take \(p>p_0\) in the case of \(w_1,\ldots , w_{n-2}>0\) or \(p<p_1\) in the case \(w_1,\ldots ,w_{n-2}<0\). In both cases, by Lemma 1 used for \(\lambda =\gamma \), we have \(\gamma a_{n-i-1}^p \, a_{n-i}^p - a_{n-i-2}^p \, a_{n-i+1}^p > 0\) for all \(i=1,\ldots ,n-2\). Thanks to Theorem 3 we obtain the stability of \(f^{[p]}\) and the proof is completed. \(\square \)
Let us observe that \(p_0\) and \(p_1\) in Theorem 10 are not far from being optimal as evidenced in the next example.
Example 3
Consider the polynomial
We have
and
The Hurwitz matrix H(f) associated to f is
The leading principal minors are
and therefore, by the Routh–Hurwitz criterion, f is not stable.
Now take \(p\in \mathbb {R}\) and compute the 4th leading principal minors of \(H(f^{[p]})\):
If we take p close to 1 then \(f^{[p]}\) is not stable because of the continuity of exponential functions and since \(D_4(1)<0\).
On the other hand, by Theorem 10, \(f^{[p]}\) is stable for all \(p\ge 1.1032\) as
We conclude that the quantity \(p_0\) given in Theorem 10 is close to the value, where the stability of \(f^{[p]}\) changes.
The above example shows also that Theorem 8 proved for \(n=4\) cannot be applied for polynomials of degree 5, because by Theorem 8 we get \(f^{[p]}\in H_n\) for all \(p>\frac{\log 0.5}{\log \overline{d}}\). However, for the polynomial f considered in Example 3 we have \(\frac{\log 0.5}{\log \overline{d}}=1\) and we see that \(f^{[p]}\) is not stable for p close to 1.
We can show by the next example that the constant \(\gamma \) in Theorem 3 is close to the optimal one.
Example 4
Let
Observe that f has all positive coefficients and for
that is close to \(\gamma \in (0.4655,0.466)\), we have
By Theorem 3 analogous inequalities satisfied for \(\gamma \) (instead of \(\lambda \)) imply the Hurwitz stability of f. However, in the considered case we get
and therefore, by the Routh–Hurwitz criterion f is not stable.
References
Garloff J, Wagner DG (1996) Hadamard products of stable polynomials are stable. J Math Anal Appl 202:797–809
Gregor J, Tišer J (1998) On Hadamard powers of polynomials. Math Control Signals Syst 11:372–378
Kemperman JHB (1982) A Hurwitz matrix is totally positive. SIAM J Math Anal 13:331–341
Lipatov AV,Sokolov NI (1978) On some sufficient conditions for stability and instability of linear continuous stationary systems. Avtomatika i Telemekhanika 9:30–37 (translated in: Automat Remote Control (1979) 39:1285–1291)
Rahman QI, Schmeisser G (2002) Analytic theory of polynomials, london mathematical society monographs, vol 26. Oxford University Press, Oxford
Wang Y, Zhang B (2013) Hadamard powers of polynomials with only real zeros. Linear Algebra Appl 439:3173–3176
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of Leokadia Białas-Cież was partially supported by the NCN Grant No. 2013/11/B/ST1/03693.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Białas, S., Białas-Cież, L. Comments on ‘On Hadamard powers of polynomials’. Math. Control Signals Syst. 29, 16 (2017). https://doi.org/10.1007/s00498-017-0202-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00498-017-0202-0