Discussion on: “On the generation of random stable polynomials”. (English) Zbl 1227.65007
Summary: There are several very interesting open problems mentioned in the paper by P. Shcherbakov and F. Dabbene [ibid. 17, No. 2, 145–159 (2011; Zbl 1227.65008)] and a number of appealing directions can be considered for further investigations. For example, it is very interesting to explore the root distribution issue of generated stable polynomials.
This discussion focuses on two issues.
First, the introduction of a new name (LD parameters) for the parameters obtained via Schur-Cohn stability test f is a bit questionable. The Schur-Cohn stability test and Levinson recursion are well known. However, the parameters obtained by this recursive procedure have already too many different names in different research fields.
Second, uniform distribution for LD parameters leads to the uniform distribution of the polynomial coefficients. Unfortunately, all the roots of ULD-polynomials cluster close to the stability boundary for high order polynomials. The question is how to modify the proposed methods so that the root location will be better?
[This note concludes with final comments by P. Shcherbakov and F. Dabbene on p. 161.]
This discussion focuses on two issues.
First, the introduction of a new name (LD parameters) for the parameters obtained via Schur-Cohn stability test f is a bit questionable. The Schur-Cohn stability test and Levinson recursion are well known. However, the parameters obtained by this recursive procedure have already too many different names in different research fields.
Second, uniform distribution for LD parameters leads to the uniform distribution of the polynomial coefficients. Unfortunately, all the roots of ULD-polynomials cluster close to the stability boundary for high order polynomials. The question is how to modify the proposed methods so that the root location will be better?
[This note concludes with final comments by P. Shcherbakov and F. Dabbene on p. 161.]
MSC:
| 65C05 | Monte Carlo methods |
| 65D20 | Computation of special functions and constants, construction of tables |
| 33F05 | Numerical approximation and evaluation of special functions |
| 26C05 | Real polynomials: analytic properties, etc. |
Keywords:
random generation; stable polynomials; polynomial roots; Levinson-Durbin parameters; stability conditions; Schur-Cohn stability; Levinson recursionCitations:
Zbl 1227.65008Software:
RACTReferences:
| [1] | Nurges, Ü., New stability conditions via reflection coefficients of polynomials, IEEE TAC, 50, 9, 1354-1360 (2005) · Zbl 1365.93352 |
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