Abstract
We generalize univariate multipoint evaluation of polynomials of degree n at sublinear amortized cost per point. More precisely, it is shown how to evaluate a bivariate polynomial p of maximum degree less than n, specified by its n 2 coefficients, simultaneously at n 2 given points using a total of \(\mathcal{O}(n^{2.667})\) arithmetic operations. In terms of the input size N being quadratic in n, this amounts to an amortized cost of \(\mathcal{O}(N^{0.334})\) per point.
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References
Bini, D., Pan, V.Y.: Polynomial and matrix computations. Progress in theoretical computer science, vol. 1. Birkhäuser, Boston (1994) ISBN 0-8176-3786-9, 3-7643-3786-9
Borodin, A., Munro, I.: The Computational Complexity of Algebraic and Numeric Problems. Theory of computation series, vol. 1. American Elsevier Publishing Company, New York (1975)
Brent, R.P., Kung, H.T.: Fast Algorithms for Manipulating Formal Power Series. Journal of the ACM 25(4), 581–595 (1978)
Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften, vol. 315. Springer, Heidelberg (1997)
Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. Journal of Symbolic Computation 9, 251–280 (1990)
Eppstein, D.: Re: Geometry problem: Optimal direction. Known results? Usenet news article (2004), http://mathforum.org/epigone/sci.math.research/slexyaxsle
Fiduccia, C.M.: Polynomial evaluation via the division algorithm: the fast Fourier transform revisited. In: Proceedings of the Fourth Annual ACM Symposium on the Theory of Computing, Denver CO, pp. 88–93. ACM Press, New York (1972)
von zur Gathen, J.: Functional Decomposition of Polynomials: the Tame Case. Journal of Symbolic Computation 9, 281–299 (1990)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003) ISBN 0-521-82646-2, http://www-math.upb.de/~aggathen/mca/ ; 1st edn. (1999)
Horowitz, E.: A fast method for interpolation using preconditioning. Information Processing Letters 1, 157–163 (1972)
Huang, X., Pan, V.Y.: Fast Rectangular Matrix Multiplication and Applications. Journal of Complexity 14, 257–299 (1998) ISSN 0885-064X
Lodha, S.K., Goldman, R.: A unified approach to evaluation algorithms for multivariate polynomials. Mathematics of Computation 66(220), 1521–1559 (1997) ISSN 0025-5718, http://www.ams.org/mcom/1997-66-220/S0025-5718-97-00862-4
Odlyzko, A.M., Schönhage, A.: Fast algorithms for multiple evaluations of the Riemann zeta function. Transactions of the American Mathematical Society 309(2), 797–809 (1988)
Pan, V.Y.: О способах вычиспения значени многочпенов. Успехи Математических Наук 21(1(127)), 103–134 (1966); Pan, V.Y.: Methods of computing values of polynomials. Russian Mathematical Surveys 21, 105–136 (1966)
Schönhage, A.: Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Informatica 7, 395–398 (1977)
Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)
Strassen, V.: Gaussian Elimination is not Optimal. Numerische Mathematik 13, 354–356 (1969)
Ziegler, M.: Fast Relative Approximation of Potential Fields, pp. 140–149, http://www.springerlink.com/openurl.asp?genre=article&issn=0302-9743&volume=2748&spage=140
Ziegler, M.: Quasi-optimal Arithmetic for Quaternion Polynomials, pp. 705–715, http://www.springerlink.com/openurl.asp?genre=article&issn=0302-9743&volume=2906&spage=705
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Nüsken, M., Ziegler, M. (2004). Fast Multipoint Evaluation of Bivariate Polynomials. In: Albers, S., Radzik, T. (eds) Algorithms – ESA 2004. ESA 2004. Lecture Notes in Computer Science, vol 3221. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30140-0_49
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DOI: https://doi.org/10.1007/978-3-540-30140-0_49
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