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

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.

Keywords

Inertia of polynomial / inertia of matrix / Bezout matrix / Schur-Cohn-Fujiwara theorem / Schur-Cohn matrix

References

[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

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

[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

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

[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

[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

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

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg