RESEARCH ARTICLE

An extended version of Schur-Cohn-Fujiwara theorem in stability theory

  • Yongjian HU ,
  • Xuzhou ZHAN ,
  • Gongning CHEN
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  • School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received date: 09 May 2014

Accepted date: 28 Jan 2015

Published date: 24 Jun 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.

Cite this article

Yongjian HU , Xuzhou ZHAN , Gongning CHEN . An extended version of Schur-Cohn-Fujiwara theorem in stability theory[J]. Frontiers of Mathematics in China, 2015 , 10(5) : 1113 -1122 . DOI: 10.1007/s11464-015-0453-3

1
Barnett S. Polynomials and Linear Control Systems. New York: Marcel Dekker, 1983

2
Barnett S, Storey C. Matrix Methods in Stability Theory. London: Nelson, 1970

3
Chen G, Zhang H. Note on product of Bezoutians and Hankel matrices. Linear Algebra Appl, 1995, 225: 23-35

DOI

4
Fielder M, Pták V. Loewner and Bezout matrices. Linear Algebra Appl, 1988, 101: 187-220

DOI

5
Fujiwara M. Über die Wurzelanzahl algebraischer Gleichungen innerhalb und auf dem Einheitskreis. Math Z, 1924, 19: 161-169

6
Heinig G, Rost K. Algebraic Methods for Toeplitz-like Matrices and Operators. Operator Theory, Vol 13, Basel: Birkhäuser, 1984

DOI

7
Holtz O, Tyaglov M. Structured matrices, continued fractions, and root localization of polynomial. SIAM Review, 2012, 54: 421-509

DOI

8
Krein M G. To the theory of symmetric polynomials. Mat Sb, 1933, 40(3): 271-283

9
Krein M G, Naimark M A. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear Multilinear Algebra, 1981, 10: 265-308

DOI

10
Lancaster P, Tismenetsky M. The Theory of Matrices with Applications. 2nd ed. New York: Academic Press, 1985

11
Rogers J W. Location of roots of polynomials. SIAM Rev, 1983, 25: 327-342

DOI

12
Uspensky J V. Theory of Equations. New York: McGraw-Hill, 1948

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