Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real numbers. We improve the previously known bound by a factor of dτ, where d is the polynomial degree and τ bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is ο~ B (d 4 τ 2 ) using a standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library SYNAPS, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.

[1]  George E. Collins,et al.  Cylindrical Algebraic Decomposition I: The Basic Algorithm , 1984, SIAM J. Comput..

[2]  Maurice Mignotte,et al.  Mathematics for computer algebra , 1991 .

[3]  David G. Cantor,et al.  A continued fraction algorithm for real algebraic numbers , 1972 .

[4]  Alkiviadis G. Akritas,et al.  Polynomial real root isolation using Descarte's rule of signs , 1976, SYMSAC '76.

[5]  Kurt Mehlhorn,et al.  New bounds for the Descartes method , 2005, SIGS.

[6]  G. E. Collins,et al.  Real Zeros of Polynomials , 1983 .

[7]  T. A. Brown,et al.  Theory of Equations. , 1950, The Mathematical Gazette.

[8]  Chee-Keng Yap,et al.  Almost tight recursion tree bounds for the Descartes method , 2006, ISSAC '06.

[9]  Bernard Mourrain,et al.  SYNAPS: A library for symbolic-numeric computation , 2005 .

[10]  Jeffrey Shallit,et al.  A Continued Fraction Algorithm for Approximating All Real Polynomial Roots , 1978 .

[11]  Charles G. Moore,et al.  Introduction to Continued Fractions , 1964 .

[12]  Alkiviadis G. Akritas,et al.  A Comparative Study of Two Real Root Isolation Methods , 2005 .

[13]  W. Burnside,et al.  Theory of equations , 1886 .

[14]  P. Zimmermann,et al.  Efficient isolation of polynomial's real roots , 2004 .

[15]  Richard P. Brent,et al.  A Comparative Study of Algorithms for Computing Continued Fractions of Algebraic Numbers , 1996, ANTS.

[16]  Ioannis Z. Emiris,et al.  The predicates for the Voronoi diagram of ellipses , 2006, SCG '06.

[17]  Alfred J. van der Poorten,et al.  Continued Fractions of Algebraic Numbers , 1995 .

[18]  Doru Stefanescu,et al.  New Bounds for Positive Roots of Polynomials , 2005, J. Univers. Comput. Sci..

[19]  Alkiviadis G. Akritas,et al.  Elements of Computer Algebra with Applications , 1989 .

[20]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[21]  S. Rump.  Ten methods to bound multiple roots of polynomials , 2003 .

[22]  Alkiviadis G. Akritas,et al.  An implementation of Vincent's theorem , 1980 .

[23]  A. Sluis.  Upperbounds for roots of polynomials , 1970 .

[24]  Joachim von zur Gathen,et al.  Fast algorithms for Taylor shifts and certain difference equations , 1997, ISSAC.

[25]  J B Kiostelikis,et al.  Bounds for positive roots of polynomials , 1986 .

[26]  D. Hensley,et al.  The largest digit in the continued fraction expansion of a rational number. , 1991 .

[27]  Gradimir V. Milovanović,et al.  Inequalities for Polynomial Zeros , 2000 .

[28]  A. J. van der Poorten.  Diophantine Analysis: An introduction to continued fractions , 1986 .

[29]  Giuseppe Fiorentino,et al.  Design, analysis, and implementation of a multiprecision polynomial rootfinder , 2000, Numerical Algorithms.

[30]  下山 武司.  Cylindrical Algebraic Decomposition と実代数制約(数式処理における理論とその応用の研究) , 1995 .

[31]  Jonathan M. Borwein,et al.  On the Khintchine constant , 1997, Math. Comput..

[32]  Mukarram Ahmad,et al.  Continued fractions , 2019, Quadratic Number Theory.

[33]  Dario Bini,et al.  Numerical computation of polynomial zeros by means of Aberth's method , 1996, Numerical Algorithms.

[34]  L. Zoretti.  Sur la résolution des équations numériques , 1909 .

[35]  C. Yap,et al.  Amortized Bound for Root Isolation via Sturm Sequences , 2007 .

[36]  Bernard Mourrain,et al.  Real Algebraic Numbers: Complexity Analysis and Experimentation , 2008, Reliable Implementation of Real Number Algorithms.

[37]  Maurice Mignotte,et al.  On the distance between the roots of a polynomial , 1995, Applicable Algebra in Engineering, Communication and Computing.

[38]  Alkiviadis G. Akritas,et al.  There is no “Uspensky's method.” , 1986, SYMSAC '86.

[39]  Victor Y. Pan,et al.  Solving a Polynomial Equation: Some History and Recent Progress , 1997, SIAM Rev..

[40]  Alkiviadis G. Akritas,et al.  Implementation of real root isolation algorithms in Mathematica , 1994 .

[41]  Helmut Groh,et al.  Zur Berechnung der Eigenwerte der Mathieuschen Differentialgleichung , 1962 .

[42]  Dirk van Dalen,et al.  How the Mathematical Objects Determine the Mathematical Principles , 2005, J. Univers. Comput. Sci..

[43]  Ioannis Z. Emiris,et al.  Computations with one and two real algebraic numbers , 2005, ArXiv.