Hostname: page-component-6bf8c574d5-956mj Total loading time: 0 Render date: 2025-02-21T18:40:16.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Root distribution of complex coefficient polynomials

Published online by Cambridge University Press:  17 February 2025

T. N. Lucas
T. N. Lucas*
Affiliation:
7 Hoel Ty Nant, Llangeinor, Bridgend CF32 8HZ
Get access Rights & Permissions [Opens in a new window]

Extract

The Routh Array is known to be a very convenient way to determine the number of polynomial roots that lie in the right half of the complex plane using only the polynomial coefficients. Extensive use of this is made particularly in control system stability studies [1]. We now show that the same real-coefficient Routh algorithm can be used to obtain this information for polynomials with complex coefficients, which are prominent in areas of engineering such as electrical inductance theory [2]. The method is easy to apply and is seen to be more straightforward than existing methods for this task [e.g. 3,4]. If required, it also enables the use of appropriate root-finding methods for real-coefficient polynomials to obtain the roots of complex- coefficient polynomials.

Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 574 , March 2025 , pp. 69 - 74
Check for updates
Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dorf, R. C., Modern control systems (5th edn.) Addison-Wesley (1989) pp. 183–188.Google Scholar
Screekala, K., Extension of Routh’s stability criterion for the conditions of spontaneous excitation in induction generators, Int. J. Electrical and Electronics Engineering Research, 4(6) (2014) pp. 37–42.Google Scholar
Ohannessian, M. I., Kabalan, K. Y. and El-Hajj, A., Complex-coefficient polynomial roots by a stability criterion, Int. J. of Information and System Sciences, 1(1) (2005) pp. 89–104.Google Scholar
Barnett, S., A new formulation of the theorems of Routh, Hurwitz and Sturm, J. Inst. Mathematics and its Applications, 8 (1971) pp. 240–250.CrossRefGoogle Scholar
Lucas, T. N., Finding roots of polynomials by using the Routh Array, Electronics Letters, 32 (16) (1996) pp. 1519–1521.CrossRefGoogle Scholar
Lucas, T. N., Finding roots by deflated polynomial approximation, J Franklin Inst., 327 (5) (1990) pp. 819–830.CrossRefGoogle Scholar
Jenkins, M. A. and Traub, J. F., Zeros of a complex polynomial, Communications of the ACM, 15 (2) (1972) pp. 97–99.CrossRefGoogle Scholar