Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2015: 0.134
Source Normalized Impact per Paper (SNIP) 2015: 0.686
Impact per Publication (IPP) 2015: 0.296

Open Access
Online
ISSN
1898-9934
See all formats and pricing
In This Section
Volume 14, Issue 4 (Jan 2006)

Issues

Schur's Theorem on the Stability of Networks

Christoph Schwarzweller
  • Institute of Computer Science, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
/ Agnieszka Rowińska-Schwarzweller
  • Chair of Display Technology, University of Stuttgart, Allmandring 3b, 70569 Stuttgart, Germany
Published Online: 2008-06-13 | DOI: https://doi.org/10.2478/v10037-006-0017-9

Schur's Theorem on the Stability of Networks

A complex polynomial is called a Hurwitz polynomial if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical networks.

In this article we prove Schur's criterion [17] that allows to decide whether a polynomial p(x) is Hurwitz without explicitly computing its roots: Schur's recursive algorithm successively constructs polynomials pi(x) of lesser degree by division with x - c, ℜ {c} < 0, such that pi(x) is Hurwitz if and only if p(x) is.

  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

  • [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

  • [3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.

  • [4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

  • [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

  • [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

  • [7] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.

  • [8] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.

  • [9] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2):265-269, 2001.

  • [10] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.

  • [11] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.

  • [12] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.

  • [13] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.

  • [14] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.

  • [15] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.

  • [16] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.

  • [17] J. Schur. Über algebraische Gleichungen, die nur Wurzeln mit negativen Realteilen besitzen. Zeitschrift für angewandte Mathematik und Mechanik, 1:307-311, 1921.

  • [18] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

  • [19] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.

  • [20] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.

  • [21] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

  • [22] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.

  • [23] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.

  • [24] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

  • [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

  • [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

  • [27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

About the article


Published Online: 2008-06-13

Published in Print: 2006-01-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-006-0017-9. Export Citation

This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Hiroyuki Okazaki and Yasunari Shidama
Formalized Mathematics, 2008, Volume 16, Number 2

Comments (0)

Please log in or register to comment.
Log in