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Author, Editor(s)

Author(s):

Eigenwillig, Arno

dblp



BibTeX cite key*:

Eigenwillig2007a

Title

Title*:

On Multiple Roots in Descartes' Rule and Their Distance to Roots of Higher Derivatives

Journal

Journal Title*:

Journal of Computational and Applied Mathematics

Journal's URL:

http://www.elsevier.com/locate/cam

Download URL
for the article:

http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6TYH-4J5C7MG-3-1&_cdi=5619&_user=43521&_orig=browse&_coverDate=03%2F01%2F2007&_sk=997999998&view=c&wchp=dGLbVtz-zSkWA&md5=425b5e336f29c39a4894b0d0ba88ee05&ie=/sdarticle.pdf

Language:

English

Publisher

Publisher's
Name:

Elsevier

Publisher's URL:

http://www.elsevier.com

Publisher's
Address:

Amsterdam, The Netherlands

ISSN:

0377-0427

Vol, No, pp, Date

Volume*:

200

Number:

1

Publishing Date:

March 2007

Pages*:

226-230

Number of
VG Pages:


Page Start:

226

Page End:

230

Sequence Number:


DOI:

10.1016/j.cam.2005.12.016

Note, Abstract, ©

Note:


(LaTeX) Abstract:

If an open interval $I$ contains a $k$-fold root $\alpha$
of a real polynomial~$f$, then, after transforming $I$ to
$(0,\infty)$, Descartes' Rule of Signs counts
exactly $k$ roots of $f$ in~$I$, provided $I$ is such that
Descartes' Rule counts no roots of the $k$-th derivative of~$f$.
We give a simple proof using the Bernstein basis.

The above condition on $I$ holds if its width does not exceed the
minimum distance $\sigma$ from $\alpha$ to any complex root of the
$k$-th derivative. We relate $\sigma$ to the minimum distance $s$
from $\alpha$ to any other complex root of $f$ using Szeg{\H o}'s
composition theorem. For integer polynomials, $\log(1/\sigma)$
obeys the same asymptotic worst-case bound as $\log(1/s)$.

URL for the Abstract:

http://dx.doi.org/10.1016/j.cam.2005.12.016

Categories,
Keywords:

Descartes' Rule of Signs, Descartes-Jacobi Rule, Bernstein basis, root isolation, root separation

HyperLinks / References / URLs:


Copyright Message:

Copyright © 2006 Published by Elsevier B.V.
This article has been published in Journal of Computational and Applied Mathematics 200(1), March 2007, Pages 226-230.

Personal Comments:


Download
Access Level:

Internal

Correlation

MPG Unit:

Max-Planck-Institut für Informatik



MPG Subunit:

Algorithms and Complexity Group

Appearance:

MPII WWW Server, MPII FTP Server, MPG publications list, university publications list, working group publication list, Fachbeirat, VG Wort


BibTeX Entry:

@ARTICLE{Eigenwillig2007a,
AUTHOR = {Eigenwillig, Arno},
TITLE = {On Multiple Roots in {Descartes'} Rule and Their Distance to Roots of Higher Derivatives},
JOURNAL = {Journal of Computational and Applied Mathematics},
PUBLISHER = {Elsevier},
YEAR = {2007},
NUMBER = {1},
VOLUME = {200},
PAGES = {226--230},
ADDRESS = {Amsterdam, The Netherlands},
MONTH = {March},
ISBN = {0377-0427},
DOI = {10.1016/j.cam.2005.12.016},
}


Entry last modified by Anja Becker, 02/28/2008
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Editor(s)
Arno Eigenwillig
Created
01/09/2007 10:59:52 AM
Revisions
3.
2.
1.
0.
Editor(s)
Anja Becker
Uwe Brahm
Christine Kiesel
Arno Eigenwillig
Edit Dates
08.02.2008 14:20:49
07/07/2007 00:45:19
26.06.2007 10:42:28
01/09/2007 10:59:52 AM