# On a Theorem by Van Vleck Regarding Sturm Sequences

• Volume: 7, Issue: 4, page 389-422
• ISSN: 1312-6555

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## Abstract

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In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizing one of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” in Van Vleck’s theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck’s theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete.

## How to cite

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Akritas, Alkiviadis, Malaschonok, Gennadi, and Vigklas, Panagiotis. "On a Theorem by Van Vleck Regarding Sturm Sequences." Serdica Journal of Computing 7.4 (2013): 389-422. <http://eudml.org/doc/268656>.

@article{Akritas2013,
abstract = {In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizing one of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” in Van Vleck’s theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck’s theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete.},
journal = {Serdica Journal of Computing},
keywords = {Polynomials; Real Roots; Sturm Sequences; Sylvester’s Matrices; Matrix Triangularization},
language = {eng},
number = {4},
pages = {389-422},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On a Theorem by Van Vleck Regarding Sturm Sequences},
url = {http://eudml.org/doc/268656},
volume = {7},
year = {2013},
}

TY - JOUR
AU - Vigklas, Panagiotis
TI - On a Theorem by Van Vleck Regarding Sturm Sequences
JO - Serdica Journal of Computing
PY - 2013
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 7
IS - 4
SP - 389
EP - 422
AB - In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizing one of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” in Van Vleck’s theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck’s theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete.
LA - eng
KW - Polynomials; Real Roots; Sturm Sequences; Sylvester’s Matrices; Matrix Triangularization
UR - http://eudml.org/doc/268656
ER -

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