# Schur's Theorem on the Stability of Networks

Formalized Mathematics (2006)

• Volume: 14, Issue: 4, page 135-142
• ISSN: 1426-2630

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## Abstract

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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.

## How to cite

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Christoph Schwarzweller, and Agnieszka Rowińska-Schwarzweller. "Schur's Theorem on the Stability of Networks." Formalized Mathematics 14.4 (2006): 135-142. <http://eudml.org/doc/267400>.

@article{ChristophSchwarzweller2006,
abstract = {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.},
author = {Christoph Schwarzweller, Agnieszka Rowińska-Schwarzweller},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {135-142},
title = {Schur's Theorem on the Stability of Networks},
url = {http://eudml.org/doc/267400},
volume = {14},
year = {2006},
}

TY - JOUR
AU - Christoph Schwarzweller
AU - Agnieszka Rowińska-Schwarzweller
TI - Schur's Theorem on the Stability of Networks
JO - Formalized Mathematics
PY - 2006
VL - 14
IS - 4
SP - 135
EP - 142
AB - 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.
LA - eng
UR - http://eudml.org/doc/267400
ER -

## References

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