On Design and Implementation of a Generic Number Type for Real Algebraic Number Computations Based on Expression Dags
We report on the design and implementation of a number type called Real_algebraic. This number type allows us to compute the sign of arithmetic expressions involving the operations . The sign computation is always correct and, in this sense, not subject to rounding errors. We focus on modularity and...
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| Published in | Mathematics in computer science Vol. 4; no. 4; pp. 539 - 556 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Basel
SP Birkhäuser Verlag Basel
01.12.2010
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1661-8270 1661-8289 |
| DOI | 10.1007/s11786-011-0086-1 |
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| Abstract | We report on the design and implementation of a number type called Real_algebraic. This number type allows us to compute the sign of arithmetic expressions involving the operations
. The sign computation is always correct and, in this sense, not subject to rounding errors. We focus on modularity and use generic programming techniques to make key parts of the implementation exchangeable. Thus, our design allows for easily performing experiments with different implementations or thereby tailoring the number type for specific tasks. For many problems in computational geometry, instantiations of our number type Real_algebraic are a user-friendly alternative for implementing the exact geometric computation paradigm in order to abandon numerical robustness problems. |
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| AbstractList | We report on the design and implementation of a number type called Real_algebraic. This number type allows us to compute the sign of arithmetic expressions involving the operations
. The sign computation is always correct and, in this sense, not subject to rounding errors. We focus on modularity and use generic programming techniques to make key parts of the implementation exchangeable. Thus, our design allows for easily performing experiments with different implementations or thereby tailoring the number type for specific tasks. For many problems in computational geometry, instantiations of our number type Real_algebraic are a user-friendly alternative for implementing the exact geometric computation paradigm in order to abandon numerical robustness problems. |
| Author | Rössling, Ivo Schirra, Stefan Mörig, Marc |
| Author_xml | – sequence: 1 givenname: Marc surname: Mörig fullname: Mörig, Marc email: marc.moerig@ovgu.de organization: Department of Simulation and Graphics, Faculty of Computer Science, University of Magdeburg – sequence: 2 givenname: Ivo surname: Rössling fullname: Rössling, Ivo organization: Department of Simulation and Graphics, Faculty of Computer Science, University of Magdeburg – sequence: 3 givenname: Stefan surname: Schirra fullname: Schirra, Stefan organization: Department of Simulation and Graphics, Faculty of Computer Science, University of Magdeburg |
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| Cites_doi | 10.1145/363707.363723 10.1007/978-3-642-15582-6_23 10.1007/978-0-8176-4705-6 10.1016/0925-7721(95)00040-2 10.1016/j.comgeo.2004.12.007 10.1007/s00453-007-9132-4 10.1137/030601818 10.1007/978-3-642-15582-6_24 10.1145/777792.777831 10.1201/9781420035315.ch41 10.1007/978-3-642-03456-5_27 10.1007/BF01397083 10.1007/PL00009321 10.1145/304893.304989 10.1145/304893.304988 |
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| Keywords | Primary 68W30 Exact geometric computation Algorithm engineering Secondary 65D99 Symbolic numeric computation |
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| References_xml | – reference: Karamcheti, V., Li, C., Pechtchanski, I., Yap, C.: A core library for robust numeric and geometric computation. In: 15th ACM Symposium on Computational Geometry (SCG’99), pp. 351–359. ACM, New York (1999) – reference: GammaE.HelmR.JohnsonR.VlissidesJ.Design Patterns: Elements of Reusable Object-Oriented Software1995ReadingAddison-Wesley – reference: DekkerT.J.A floating-point technique for extending the available precisionNum. Math.19711822242422990070226.6503410.1007/BF01397083 – reference: KnuthD.E.Seminumerical algorithms. The Art of Computer Programming, vol. 219973ReadingAddison-Wesley – reference: YapC.Towards exact geometric computationComput. Geom. Theory Appl.199771-232314299050869.68104 – reference: KahanW.Further remarks on reducing truncation errorsCommun. ACM1965814010.1145/363707.363723 – reference: MPFR: A multiple precision floating-point library. http://www.mpfr.org/ – reference: OgitaT.RumpS.M.OishiS.Accurate sum and dot productSIAM J. Sci. Comput.20052661955198821965841084.6504110.1137/030601818 – reference: Shewchuk, J.R.: http://www.cs.cmu.edu/~quake/robust.html (1997) – reference: Schirra, S.: Much Ado about Zero. In: Efficient Algorithms. LNCS, vol. 5760, pp. 408–421, September 2009 – reference: Yap, C.-K.: Robust geometric computation. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, chap. 41, 2nd edn., pp. 927–952. Chapman & Hall/CRC (2004) – reference: GMP: The GNU multiple precision arithmetic library. http://www.gmplib.org/ – reference: Pion, S., Yap, C.: Constructive root bound for k-ary rational input numbers. In: Proceedings of the 19th ACM Symposium on Computational Geometry, pp. 256–263. ACM Press, San Diego, January 2003 – reference: Mörig, M., Schirra, S.: On the design and performance of reliable geometric predicates using error-free transformations and exact sign of sum algorithms. In: 19th Canadian Conference on Computational Geometry (CCCG’07), pp. 45–48, August 2007 – reference: AlexandrescuA.Modern C++ design: generic programming and design patterns applied2001BostonAddison-Wesley Longman Publishing Co. Inc. – reference: boost C++ Libraries. http://www.boost.org/ – reference: RealAlgebraic: A number type for exact geometric computation. http://www.isg.cs.uni-magdeburg.de/ag/RealAlgebraic/ – reference: LEDA: Library of Efficient Data Structures and Algorithms. http://www.algorithmic-solutions.com/ – reference: FunkeS.MehlhornK.NäherS.Structural filtering: a paradigm for efficient and exact geometric programsComput. Geometry20053131791941078.6501510.1016/j.comgeo.2004.12.007 – reference: Muller, J.-M., Brisebarre, N., de Dinechin, F., Jeannerod, C.-P., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.: Handbook of Floating-Point Arithmetic. Birkhäuser Boston 2010 – reference: Burnikel, C., Fleischer, R., Mehlhorn, K., Schirra, S.: Efficient exact geometric computation made easy. In: 15th ACM Symposium on Computational Geometry (SCG’99), pp. 341–350. ACM, New York (1999) – reference: Yu, J., Yap, C., Du, Z., Pion, S., Brönnimann, H.: The design of Core 2: a library for exact numeric computation in geometry and algebra. In: 3rd International Congress on Mathematical Software (ICMS 2010). LNCS, vol. 6327, September 2010 – reference: ShewchukJ.R.Adaptive precision floating-point arithmetic and fast robust geometric predicatesDiscrete Comput. Geom.199718330536314876470892.6809810.1007/PL00009321 – reference: Li, C., Yap, C.: A new constructive root bound for algebraic expressions. In: SODA ’01: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 496–505, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics (2001) – reference: Du, Z.: Guaranteed precision for transcendental and algebraic computation made easy. PhD thesis, Courant Institute of Mathematical Sciences, New York University, May 2006 – reference: Mörig, M.: Deferring dag construction by storing sums of floats speeds-up exact decision computations based on expression dags. In: 3rd International Congress on Mathematical Software (ICMS 2010). LNCS, vol. 6327, pp. 109–120, September 2010 – reference: CGAL: Computational Geometry Algorithms Library http://www.cgal.org/ – reference: BurnikelC.FunkeS.MehlhornK.SchirraS.SchmittS.A separation bound for real algebraic expressionsAlgorithmica2009551142825069251180.6830410.1007/s00453-007-9132-4 – volume: 8 start-page: 40 issue: 1 year: 1965 ident: 86_CR11 publication-title: Commun. 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| Title | On Design and Implementation of a Generic Number Type for Real Algebraic Number Computations Based on Expression Dags |
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