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

Markus A. Hitz, Erich Kaltofen, Y. N. Lakshman
1999 Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99  
Acknowledgements: Rob Corless directed us to 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  ... 
doi:10.1145/309831.309937 dblp:conf/issac/HitzKL99 fatcat:xixr6sxr7nbfxouqjinvdcmlhy

Preface

Dario Andrea Bini, Victor Y. Pan, Jan Verschelde
2008 Theoretical Computer Science  
An important subject of SNC is 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  ... 
doi:10.1016/j.tcs.2008.09.001 fatcat:uwwutxg5svdkholg7mtcu2pvzy

Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials

Erich Kaltofen, Zhengfeng Yang, Lihong Zhi
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
As an application of 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.  ... 
doi:10.1145/1145768.1145799 dblp:conf/issac/KaltofenYZ06 fatcat:qkttkeutvzb7hkpablkrdyy6ea

Challenges of Symbolic Computation: My Favorite Open Problems

Erich Kaltofen
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.  ... 
doi:10.1006/jsco.2000.0370 fatcat:46vy6vmihrdgpirlmam4xaqgxm

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

Sharon Hutton, Erich L. Kaltofen, Lihong Zhi
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.  ... 
doi:10.1145/1837934.1837979 dblp:conf/issac/HuttonKZ10 fatcat:l4apnronovc5vgwoq254i3rt2a

Foreword

Gert Vegter, Chee K. Yap
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.  ... 
doi:10.1007/s11786-011-0088-z fatcat:iwrfa7zfuvfahosel3ewbf5bgy

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]

Xiaolin Qin, Yong Feng, Jingwei Chen, Jingzhong Zhang
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.  ... 
arXiv:0902.0828v1 fatcat:zrv2jy5dkzd2hktcy2saepsg2e

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

Xiaolin Qin, Yong Feng, Jingwei Chen, Jingzhong Zhang
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.  ... 
arXiv:1001.0649v1 fatcat:kek2owfp3nhmfa67s7pfgtrrhq

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

Guillaume Chèze, André Galligo, Bernard Mourrain, Jean-Claude Yakoubsohn
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).  ... 
doi:10.1016/j.tcs.2011.04.018 fatcat:bha46garvrd5xhbr4wazhccwwi

An efficient algorithm for computing permanental polynomials of graphs

Yan Huo, Heng Liang, Fengshan Bai
2006 Computer Physics Communications  
It is applied to fullerene-type graphs, 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.  ... 
doi:10.1016/j.cpc.2006.03.002 fatcat:x2f6qbidzfcrhjqwddqtrz4eau

A HYBRID APPROACH FOR DETERMINANT SIGNS OF MODERATE-SIZED MATRICES

TIM CULVER, JOHN KEYSER, DINESH MANOCHA, SHANKAR KRISHNAN
2003 International journal of computational geometry and applications  
We describe 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.  ... 
doi:10.1142/s0218195903001256 fatcat:wnrbbcrvsfepfbygopl66eegpu
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