Abstract
Semi-algebraic objects are subsets or functions that can be described by finite boolean combinations of polynomials with real coefficients. In this paper we provide sharp estimates for the the precision and the number of trials needed in the MonteCarlo integration method to achieve a given error with a fixed confidence when approximating the mean value of semi-algebraic functions. Our study extends to the functional case the results of P. Koiran ([7]) for approximating the volume of semi-algebraic sets.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Castro, D., Montaña, J.L., Pardo, L.M., San Martin, J.: The Distribution of Condition Numbers of Rational Data of Bounded Bit Length. Found. of Comput. Math. 1(1), 1–52 (2002)
Castro, D., Pardo, L.M., San Martín, J.: Systems of rational polynomial equations have polynomial size approximate zeros on the average. J. Complexity 19(2), 161–209 (2003)
Cucker, F., Smale, S.: On the Mathematical foundations of learning. Bulletin (New Series) Of the AMS 39(1), 1–4 (2001)
Davenport, H.: On a Principle of Lipschitz. J. London Math. Soc. 26, 179–183 (1951)
Erdos, P., Turan, P.: On a Problem in the Theory of Uniform Distribution. I, Indagationes Math. 10, 370–378, 57 (1948)
Erdos, P., Turan, P.: On a Problem in the Theory of Uniform Distribution. II, Indagationes Math. 10, 406–413 (1948)
Koiran, P.: Approximating the volume of definable sets. In: Proc. 36th IEEE Symposyum on Foundations of Computer Science FOCS 1995, pp. 134–141 (1995)
Milnor, J.: On the Betti Numbers of Real Varieties. Proc. Amer. Math. Soc. 15, 275–280 (1964)
Montaña, J.L., Pardo, L.M.: Lower bounds for Arithmetic Networks. Applicable Algebra in Engineering Communications and Computing 4, 1–24 (1993)
Mordell, L.J.: On some Arithmetical Results in the Geometry of Numbers. Compositio Mathematica 1, 248–253 (1934)
Oleinik, O.A.: Estimates of the Betti Numbers of Real Algebraic Hypersurfaces. Mat. Sbornik 70, 635–640 (1951)
Oleinik, O.A., Petrovsky, I.B.: On the topology of Real Algebraic Surfaces. Izv. Akad. Nauk SSSR (in Trans. of the Amer. Math. Soc.) 1, 399–417 (1962)
Pardo, L.M.: How Lower and Upper Complexity Bounds Meet in Elimination Theory. In: Cohen, G., Giusti, M., Mora, T. (eds.) AAECC 1995. LNCS, vol. 948, pp. 33–69. Springer, Heidelberg (1995)
Pollard, P.: Convergence of Stocastic Processes. Springer, Heidelberg (1984)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, RAND Corporation, Santa Monica, Calif. (1948)
Thom, R.: Sur l’Homologie des Varietes Alg’ebriques Réeelles. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 255–265. Princeton Univ. Press, Princeton (1965)
Valiant, L.G.: A theory of the learneable. Communications of the ACM 27, 1134–1142 (1984)
Vapnik, V.: Statistical learning theory. John Wiley & Sons, Chichester (1998)
Warren, H.E.: Lower Bounds for Approximation by non Linear Manifolds. Trans. A.M.S. 133, 167–178 (1968)
Weinzierl, S.: Introduction to MonteCarlo methods. Technical Report NIKHEF-00-012 Theory Group (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alonso, C.L., Montaña, J.L., Pardo, L.M. (2005). On the Number of Random Digits Required in MonteCarlo Integration of Definable Functions. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_9
Download citation
DOI: https://doi.org/10.1007/11549345_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28702-5
Online ISBN: 978-3-540-31867-5
eBook Packages: Computer ScienceComputer Science (R0)