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.
@article{bwmeta1.element.doi-10_2478_v10037-006-0017-9,
author = {Christoph Schwarzweller and Agnieszka Rowi\'nska-Schwarzweller},
title = {Schur's Theorem on the Stability of Networks},
journal = {Formalized Mathematics},
volume = {14},
year = {2006},
pages = {135-142},
zbl = {1293.30016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-006-0017-9}
}
Christoph Schwarzweller; Agnieszka Rowińska-Schwarzweller. Schur's Theorem on the Stability of Networks. Formalized Mathematics, Tome 14 (2006) pp. 135-142. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-006-0017-9/
[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858
[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. | Zbl 48.0082.03
[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.