The Diamond Operator - Implementation of Exact Real Algebraic Numbers

The LEDA number type real is extended by the diamond operator, which allows to compute exactly with real algebraic numbers given as roots of polynomials. The coefficients of these polynomials can be arbitrary real algebraic numbers. The implementation is presented and experiments with two other existing implementations of real algebraic numbers (CORE, EXACUS) are done.

[1]  Leonidas J. Guibas,et al.  A Computational Framework for Handling Motion , 2004, ALENEX/ANALC.

[2]  Chee-Keng Yap,et al.  Recent progress in exact geometric computation , 2005, J. Log. Algebraic Methods Program..

[3]  Susanne Schmitt Improved separation bounds for the diamond operator , 2004 .

[4]  Kurt Mehlhorn,et al.  A Separation Bound for Real Algebraic Expressions , 2001, ESA.

[5]  Fabrice Rouillier,et al.  An environment for Symbolic and Numeric Computation , 2002 .

[6]  Chee-Keng Yap,et al.  Constructive root bound for k-ary rational input numbers , 2003, SCG '03.

[7]  Ioannis Z. Emiris,et al.  Comparing Real Algebraic Numbers of Small Degree , 2004, ESA.

[8]  Chee-Keng Yap,et al.  A new constructive root bound for algebraic expressions , 2001, SODA '01.

[9]  Jeremy R. Johnson,et al.  Polynomial real root isolation using approximate arithmetic , 1997, ISSAC.

[10]  Marie-Françoise Roy,et al.  Generic computation of the real closure of an ordered field , 1996 .

[11]  Renaud Rioboo,et al.  Towards faster real algebraic numbers , 2002, ISSAC '02.

[12]  Michael N. Vrahatis,et al.  On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree , 2002, J. Complex..

[13]  Kurt Mehlhorn,et al.  Effective Computational Geometry for Curves and Surfaces , 2005 .

[14]  Stefan Näher,et al.  LEDA: A Library of Efficient Data Types and Algorithms , 1989, STACS.

[15]  S. Basu,et al.  Algorithmic and Quantitative Real Algebraic Geometry: DIMACS Workshop, Algorithmic and Quantitative Aspects of Real Algebraic, Geometry in Mathematics and Computer Science, March 12-16, 2001, DIMACS Center , 2003 .

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

[17]  Kurt Mehlhorn,et al.  A Descartes Algorithm for Polynomials with Bit-Stream Coefficients , 2005, CASC.

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

[19]  Chee-Keng Yap,et al.  A core library for robust numeric and geometric computation , 1999, SCG '99.