research-article

The approximate irreducible factorization of a univariate polynomial: revisited

Author:

Zhonggang ZengAuthors Info & Claims

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

Pages 367 - 374

https://doi.org/10.1145/1576702.1576752

Published: 28 July 2009 Publication History

Abstract

When multiple roots are present, factoring a univariate polynomial is an ill-posed problem in the sense that it is highly sensitive to data perturbations and the round-off error. In ISSAC '03, the author introduced an algorithm that is capable of calculating the polynomial roots and multiplicities accurately using floating point arithmetic without extending the hardware precision even if the coefficients are perturbed. This paper revisits the problem by establishing a rigorous theoretical framework for eliminating the ill-posedness and by introducing a redesigned algorithm. The new algorithm avoids error accumulation in the GCD computation and substantially improves the accuracy and robustness.

References

[1]

B. Beckermann and G. Labahn. A fast and numerically stable Euclidean-like algorithm for detecting relatively prime numerical polynomials. J. Symb. Comp., 26:691--714, 1998.

Digital Library

[2]

R. Corless, P. Gianni, B. Trager, and S. Watt. The singular value decomposition for polynomial systems. Proc. ISSAC '95, pp 195--207, 1995.

Digital Library

[3]

R. M. Corless, S. M. Watt, and L. Zhi. QR factoring to compute the GCD of univariate approximate polynomials. IEEE Trans. Signal Processing, 52:3394--3402, 2003.

Digital Library

[4]

J. E. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1983.

Digital Library

[5]

I. Z. Emiris, A. Galligo, and H. Lombardi. Certified approximate univariate GCDs. J. Pure Appl. Algebra, 117/118:229--251, 1997.

[6]

S. Gao, E. Kaltofen, J. May, Z. Yang, and L. Zhi. Approximate factorization of multivariate polynomials via differential equations. Proc. ISSAC '04, ACM Press, pp 167-174, 2004.

Digital Library

[7]

E. Kaltofen, Z. Yang, and L. Zhi. Approximate greatest common divisor of several polynomials with linearly constrained coefficients and singular polynomials. Proc. ISSAC'06, ACM Press, pp 169--176, 2006.

Digital Library

[8]

N. K. Karmarkar and Y. N. Lakshman. On approximate polynomial greatest common divisors. J. Symb. Comput., 26:653--666, 1998.

Digital Library

[9]

J. M. McNamee. A bibliography on roots of polynomials. J. Comput. Appl. Math., 47:391--394, 1993.

Digital Library

[10]

J. M. McNamee. Numerical methods for roots of polynomials. Part I. Elsevier, Amsterdam, 2007.

[11]

V. Y. Pan. Numerical computation of a polynomial GCD and extensions. Information and Computation, 167:71--85, 2001.

Digital Library

[12]

T. Sasaki and M. Noda. Approximate square-free decomposition and root-finding of ill-conditioned algebraic equations. J. Inform. Proc, 12:159--168, 1989.

Digital Library

[13]

A. Sch&#246;nhage. Quasi-GCD computations. J. Complexity, 1:118--137, 1985.

[14]

D. Y. Y. Yun. On square-free decomposition algorithms. Proc. ISSAC '76, ACM Press, pp 26--35, 1976.

Digital Library

[15]

Z. Zeng. A method computing multiple roots of inexact polynomials. Proc. ISSAC '03, ACM Press, pp 266--272, 2003.

Digital Library

[16]

Z. Zeng. The approximate GCD of inexact polynomials. Part I: A univariate algorithm. Preprint.

[17]

Z. Zeng. Algorithm 835: Multroot - A Matlab package for computing polynomial roots and multiplicities. ACM Trans. Math. Software, 30:218--235, 2004.

Digital Library

[18]

Z. Zeng. Computing multiple roots of inexact polynomials. Math. Comp., 74:869--903, 2005.

[19]

Z. Zeng. ApaTools: A Maple and Matlab toolbox for approximate polynomial algebra. In M. Stillman, N. Takayama, and J. Verschelde, editors, Software for Algebraic Geometry, IMA Volume 148, pages 149--167. Springer, 2008.

[20]

Z. Zeng. Regularization and matrix computation in numerical polynomial algebra. To appear in Approximate Commutative Algebra, Springer, 2009.

Cited By

Zeng Z(2023)A Newton’s iteration converges quadratically to nonisolated solutions tooMathematics of Computation10.1090/mcom/365792:344(2795-2824)Online publication date: 15-Jun-2023
https://doi.org/10.1090/mcom/3657
Zeng Z(2009)Regularization and Matrix Computation in Numerical Polynomial AlgebraApproximate Commutative Algebra10.1007/978-3-211-99314-9_5(125-162)Online publication date: 21-Aug-2009
https://doi.org/10.1007/978-3-211-99314-9_5

Index Terms

The approximate irreducible factorization of a univariate polynomial: revisited
1. Mathematics of computing
  1. Mathematical analysis
    1. Nonlinear equations
  2. Mathematical software

Recommendations

Exact polynomial factorization by approximate high degree algebraic numbers
SNC '09: Proceedings of the 2009 conference on Symbolic numeric computation

For factoring polynomials in two variables with rational coefficients, an algorithm using transcendental evaluation was presented by Hulst and Lenstra. In their algorithm, transcendence measure was computed. However, a constant c is necessary to compute ...
Polynomial-Time Reductions from Multivariate to Bi- and Univariate Integral Polynomial Factorization

Consider a polynomial f with an arbitrary but fixed number of variables and with integral coefficients. We present an algorithm which reduces the problem of finding the irreducible factors of f in polynomial-time in the total degree of f and the ...
New Algorithms for Polynomial Square-Free Decomposition over the Integers

Previously-known algorithms for polynomial square-free decomposition rely on greatest common divisor (gcd) computations over the same coefficient domain where the decomposition is to be performed. In particular, gcd of the given polynomial and its first ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

July 2009

402 pages

ISBN:9781605586090

DOI:10.1145/1576702

General Chairs:
Jeremy Johnson
Drexel University, USA
,
Hyungju Park
Korea Institute for Advanced Study, Korea
,
Program Chair:
Erich Kaltofen
North Carolina State University, USA

Copyright © 2009 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 28 July 2009

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ISSAC '09

Sponsor:

ISSAC '09: International Symposium on Symbolic and Algebraic Computation

July 28 - 31, 2009

Seoul, Republic of Korea

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
112
Total Downloads

Downloads (Last 12 months)2
Downloads (Last 6 weeks)0

Reflects downloads up to 05 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Zeng Z(2023)A Newton’s iteration converges quadratically to nonisolated solutions tooMathematics of Computation10.1090/mcom/365792:344(2795-2824)Online publication date: 15-Jun-2023
https://doi.org/10.1090/mcom/3657
Zeng Z(2009)Regularization and Matrix Computation in Numerical Polynomial AlgebraApproximate Commutative Algebra10.1007/978-3-211-99314-9_5(125-162)Online publication date: 21-Aug-2009
https://doi.org/10.1007/978-3-211-99314-9_5

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten