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Efficient algorithms for computing the nearest polynomial with a real root and related problems

1999
*
Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99
*

Acknowledgements: Rob Corless directed us to

doi:10.1145/309831.309937
dblp:conf/issac/HitzKL99
fatcat:xixr6sxr7nbfxouqjinvdcmlhy
*the*results about*the**nearest**polynomial**with**a*given*root*in [1, 15] . Gilles Villard brought [14] to our attention. ... We also thank*the*reviewers*for*their cogent comments. ... We can solve our specific*problem**efficiently*, that is in*polynomial*-time in*the*degree*and*input length, because an explicit expression*for**the*distance to*the**nearest**polynomial**with**a**real**root*can ...##
###
Preface

2008
*
Theoretical Computer Science
*

An important subject of SNC is

doi:10.1016/j.tcs.2008.09.001
fatcat:uwwutxg5svdkholg7mtcu2pvzy
*the*solution of*the*classical ill conditioned*problems*of numerical approximation of complex or*real**roots*of*a*univariate*polynomial**and**a*system of multivariate*polynomials*... These*problems*are fundamental*for*symbolic*computations*, but numerical techniques (combined*with*some symbolic methods) are*the*basis*for**the*most successful packages*for**the*univariate*polynomial**root*-finding ... Emiris*and*Tsigaridas*compute*rational isolating points*for*all*real**roots*of an integer*polynomial*of degree up to five, show*the*extension to isolating*the**real**roots*of*a*pair of bivariate*polynomials*...##
###
Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials

2006
*
Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06
*

As an application of

doi:10.1145/1145768.1145799
dblp:conf/issac/KaltofenYZ06
fatcat:qkttkeutvzb7hkpablkrdyy6ea
*the*linearly constrained approximate GCD*problem*we present an STLN-based method that*computes**a**real*or complex*polynomial**the**nearest**real*or complex*polynomial*that has*a**root*of ... We present an*algorithm*based on*a*version of*the*structured total least norm (STLN) method*and*demonstrate on*a*diverse set of benchmark*polynomials*that*the**algorithm*in practice*computes*globally minimal ... We thank Rong Xiao*and*Bican Xia*for*helping us*compute**the*non-monic global minimum of Example 4.2. ...##
###
Challenges of Symbolic Computation: My Favorite Open Problems

2000
*
Journal of symbolic computation
*

*The*mathematics

*and*

*computer*science in

*the*design

*and*implementation of our

*algorithms*are sophisticated. ...

*The*research challenges in symbolic

*computation*at

*the*close of

*the*20th century are formidable. I state my favorite eight open

*problems*in symbolic

*computation*. They range ...

*A*

*problem*in

*the*same spirit as above is to

*efficiently*computef ;ĝ "

*nearest*to" f ; g that have

*a*common

*root*. ...

##
###
Page 5683 of Mathematical Reviews Vol. , Issue 93j
[page]

1993
*
Mathematical Reviews
*

Summary: “We present tight bounds

*for*distances from differences of*roots*of*the**polynomial*f(x) €*A*[x] over*a*discrete normed 68Q Theory of*computing*93j:68076 commutative ring # without zero divisors ...*for**the*class of*polynomial*time*algorithms*. ...##
###
Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg's method

2010
*
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10
*

*The*

*computation*of

*the*

*nearest*

*polynomial*

*with*

*a*

*real*

*root*can be interpreted as

*a*dual of Seidenberg's method that decides if

*a*

*real*hypersurface contains

*a*

*real*point. ...

*The*radius of positive (or negative) semidefiniteness is

*the*distance to

*the*

*nearest*

*polynomial*

*with*

*a*

*real*

*root*, which has been thoroughly studied before. ... We thank Mohab Safey El Din

*for*his comments on Seidenberg's

*problem*,

*and*

*the*reviewers

*for*their remarks. ...

##
###
Foreword

2010
*
Mathematics in Computer Science
*

*For*instance, how can we faithfully

*and*

*efficiently*discretize

*a*continuous geometric object, or

*a*continuous

*problem*?

*Efficient*new

*algorithmic*techniques must be developed

*and*analyzed. ... Sagraloff's paper falls under

*the*first framework, addressing

*the*

*root*isolation

*problem*

*for*square-free

*real*

*polynomials*. ...

##
###
Page 3369 of Mathematical Reviews Vol. , Issue 2001E
[page]

2001
*
Mathematical Reviews
*

(F-NANC-LRI; Vandoeuvre-les-Nancy)
Algebraic geometry

*and**computer*vision:*polynomial*systems,*real**and*complex*roots*. (English summary) J. Math. Imaging Vision 10 (1999), no. 3, 191-220. ... We illustrate this on approximation*algorithms**for**the*following*problems*: vertex cover, set cover, feedback vertex set, generalized Steiner forest,*and**related**problems*. ...##
###
Finding Exact Minimal Polynomial by Approximations
[article]

2009
*
arXiv
*
pre-print

*The*

*algorithm*is applicable

*for*finding exact minimal

*polynomial*by its approximate

*root*. ... This also enables us to provide an

*efficient*method of converting

*the*rational approximation representation to

*the*minimal

*polynomial*representation,

*and*devise

*a*simple

*algorithm*to factor multivariate ... In this paper, we present

*a*new

*algorithm*

*for*finding exact minimal

*polynomial*

*and*reconstructing

*the*exact

*root*by approximate value. ...

##
###
A complete algorithm to find exact minimal polynomial by approximations
[article]

2010
*
arXiv
*
pre-print

*The*

*algorithm*is applicable

*for*finding exact minimal

*polynomial*of an algebraic number by its approximate

*root*. ... We present

*a*complete

*algorithm*

*for*finding an exact minimal

*polynomial*from its approximate value by using an improved parameterized integer

*relation*construction method. ... In this paper, we present

*a*new

*algorithm*

*for*finding exact minimal

*polynomial*

*and*reconstructing

*the*exact

*root*by approximate value. ...

##
###
Page 5807 of Mathematical Reviews Vol. , Issue 2001H
[page]

2001
*
Mathematical Reviews
*

*the*actual automatic

*computations*of separation bounds,

*the*authors

*compute*

*root*separation bounds sep(£)

*for*known

*problems*from

*computer*algebra

*and*

*computational*geometry [see, e.g., M. ...

*The*

*problem*of placing

*a*maximum number of hiding people is almost as hard to approximate as

*the*maximum clique

*problem*, i.e., it cannot be approximated by any

*polynomial*- time

*algorithm*

*with*an approximation ...

##
###
Page 6797 of Mathematical Reviews Vol. , Issue 2002I
[page]

2002
*
Mathematical Reviews
*

*Comput*. Sci.

*Comput*. Sci. Ser. 12 (2001), no. 1, 55-66. Summary: “This paper deals

*with*

*the*Boolean routing

*problem*,

*a*well-known strategy to code distributed routing

*algorithms*in

*a*compact way. ... However, there are quite

*a*few

*polynomially*solvable

*problems*as well. ...

##
###
A subdivision method for computing nearest gcd with certification

2011
*
Theoretical Computer Science
*

*A*new subdivision method

*for*

*computing*

*the*

*nearest*univariate gcd is described

*and*analyzed. It is based on an exclusion test

*and*an inclusion test. ... Under

*the*condition of simple

*roots*

*for*

*the*distance minimization

*problem*, we analyze

*the*complexity of

*the*

*algorithm*in terms of

*a*condition number, which is

*the*inverse of

*the*distance to

*the*set of ... Acknowledgements

*The*authors thank Prof. Lihong Zhi

*for*her comments during

*the*presentation at SNC'09 about cases

*with*

*roots*at infinity (Example 2). ...

##
###
An efficient algorithm for computing permanental polynomials of graphs

2006
*
Computer Physics Communications
*

It is applied to fullerene-type graphs,

doi:10.1016/j.cpc.2006.03.002
fatcat:x2f6qbidzfcrhjqwddqtrz4eau
*and*works*for*C 56 , while*the*largest fullerene*computed*before is C 40 . Extensive numerical*computations*show that*the**algorithm*is fast*and*stable. ... An*efficient*numerical method*for**computing*permanental*polynomials*of graphs is proposed. It adapts multi-entry expansion of FFT,*and*is parallel in nature. ... Acknowledgements*The*authors thank referees*and*editors*for*their careful reading*and*constructive suggestions*and*comments. ...##
###
A HYBRID APPROACH FOR DETERMINANT SIGNS OF MODERATE-SIZED MATRICES

2003
*
International journal of computational geometry and applications
*

We describe

doi:10.1142/s0218195903001256
fatcat:wnrbbcrvsfepfbygopl66eegpu
*a*hybrid method*for**computing**the*sign of*the*determinant. ... In this paper, we outline*the*earlier methods*for**computing*exact determinant signs,*and*evaluate their effectiveness on moderate-sized matrices. ...*For*systems of*polynomial*equations, our approach is to construct*a*univariate*polynomial*whose*roots*are*related*to*the**roots*of*the*system. ...
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