Abstract
Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda_real.
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References
C. Burnikel, R. Fleischer, K. Mehlhorn, and S. Schirra. Exact efficient computational geometry made easy. In Proceedings of the 15th Annual Symposium on Computational Geometry (SCG’99), pages 341–350, 1999. http://www.mpi-sb.mpg.de/~mehlhorn/ftp/egcme.ps.
C. Burnikel, R. Fleischer, K. Mehlhorn, S. Schirra. A strong and easily computable separation bound for arithmetic expressions involving radicals. Algorithmica, 27:87–99, 2000.
C. Burnikel, K. Mehlhorn, and S. Schirra. How to compute the Voronoi diagram of line segments: Theoretical and experimental results. In Springer, editor, Proceedings of the 2nd Annual European Symposium on Algorithms-ESA’ 94, volume 855 of Lecture Notes in Computer Science, pages 227–239, 1994.
C. Burnikel, K. Mehlhorn, and S. Schirra. The LEDA class real number. Technical Report MPI-I-96-1-001, Max-Planck-Institut für Informatik, Saarbrücken, 1996.
J.F. Canny. The Complexity of Robot Motion Planning. The MIT Press, 1987.
E. Hecke. Vorlesungen über die Theorie der algebraischen Zahlen. Chelsea, New York, 1970.
V. Karamcheti, C. Li, I. Pechtchanski, and Chee Yap. A core library for robust numeric and geometric computation. In Proceedings of the 15th Annual ACM Symposium on Computational Geometry, pages 351–359, Miami, Florida, 1999.
R. Loos. Computing in algebraic extensions. In B. Buchberger, G. E. Collins, and R. Loos, editors, Computer Algebra. Symbolic and Algebraic Computation, volume 4 of Computing Supplementum, pages 173–188. Springer-Verlag, 1982.
C. Li and C. Yap. A new constructive root bound for algebraic expressions. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 496–505, 2001.
M. Mignotte. Identification of Algebraic Numbers. Journal of Algorithms, 3(3):197–204, September 1982.
M. Mignotte. Mathematics for Computer Algebra. Springer, 1992.
K. Mehlhorn and S. Näher. The LEDA Platformfor Combinatorial and Geometric Computing. Cambridge University Press, 1999. http://www.mpi-sb.mpg.de/LEDA/leda.html.
K. Mehlhorn and St. Schirra. Exact computation with ledareal-theory and geometric applications. In G. Alefeld, J. Rohn, S. Rumpf, and T. Yamamoto, editors, Symbolic Algebraic Methods and Verification Methods. Springer Verlag, Vienna, 2001.
J. Neukirch. Algebraische Zahlentheorie. Springer-Verlag, 1990.
E. R. Scheinerman. When close enough is close enough. American Mathematical Monthly, 107:489–499, 2000.
C.K. Yap. Towards exact geometric computation. CGTA: Computational Geometry: Theory and Applications, 7, 1997.
C.K. Yap. Fundamental Problems in Algorithmic Algebra. Oxford University Press, 1999.
C.K. Yap and T. Dube. The exact computation paradigm. In Computing in Euclidean Geometry II. World Scientific Press, 1995.
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Burnikel, C., Funke, S., Mehlhorn, K., Schirra, S., Schmitt, S. (2001). A Separation Bound for Real Algebraic Expressions. In: auf der Heide, F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44676-1_21
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DOI: https://doi.org/10.1007/3-540-44676-1_21
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