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Issue No.06 - June (2012 vol.61)
pp: 780-789
Pascal Giorgi , (LIRMM), CNRS, Université Montpellier 2, Montpellier
ABSTRACT
In a recent paper, Lima, Panario, and Wang have provided a new method to multiply polynomials expressed in Chebyshev basis which reduces the total number of multiplication for small degree polynomials. Although their method uses Karatsuba's multiplication, a quadratic number of operations are still needed. In this paper, we extend their result by providing a complete reduction to polynomial multiplication in monomial basis, which therefore offers many subquadratic methods. Our reduction scheme does not rely on basis conversions and we demonstrate that it is efficient in practice. Finally, we show a linear time equivalence between the polynomial multiplication problem under monomial basis and under Chebyshev basis.
INDEX TERMS
Theory of computation, computations on polynomials, arithmetic, polynomial multiplication, Chebyshev basis.
CITATION
Pascal Giorgi, "On Polynomial Multiplication in Chebyshev Basis", IEEE Transactions on Computers, vol.61, no. 6, pp. 780-789, June 2012, doi:10.1109/TC.2011.110
REFERENCES
[1] J.C. Mason and D.C. Handscomb, Chebyshev Polynomials. Chapman and Hall/CRC, 2002.
[2] J.P. Boyd, Chebyshev and Fourier Spectral Methods. Dover Publications, 2001.
[3] Z. Battles and L. Trefethen, “An Extension of Matlab to Continuous Fractions and Operators,” SIAM J. Scientific Computing, vol. 25, pp. 1743-1770, 2004.
[4] N. Brisebarre and M. Joldeş, “Chebyshev Interpolation Polynomial-Based Tools for Rigorous Computing,” Proc. Int'l Symp. Symbolic and Algebraic Computation, pp. 147-154, 2010.
[5] A. Karatsuba and Y. Ofman, “Multiplication of Multidigit Numbers on Automata,” Doklady Akademii Nauk SSSR, vol. 145, no. 2, pp. 293-294, 1962.
[6] A.L. Toom, “The Complexity of a Scheme of Functional Elements Realizing the Multiplication of Integers,” Soviet Math., vol. 3, pp. 714-716, 1963.
[7] S.A. Cook, “On the Minimum Computation Time of Functions,” master's thesis, Harvard Univ., May 1966.
[8] J. Cooley and J. Tukey, “An Algorithm for the Machine Calculation of Complex Fourier Series,” Math. of Computation, vol. 19, no. 90, pp. 297-301, 1965.
[9] J. von zur Gathen and J. Gerhard, Modern Computer Algebra. Cambridge Univ. Press, 2003.
[10] R. Brent and P. Zimmermann, Modern Computer Arithmetic. Cambridge Univ. Press, http://www.loria.fr/zimmerma/mcamca-cup-0.5.3.pdf , Aug. 2010.
[11] D.G. Cantor and E. Kaltofen, “On Fast Multiplication of Polynomials over Arbitrary Algebras,” Acta Informatica, vol. 28, no. 7, pp. 693-701, 1991.
[12] J.B. Lima, D. Panario, and Q. Wang, “A Karatsuba-Based Algorithm for Polynomial Multiplication in Chebyshev Form,” IEEE Trans. Computers, vol. 59, no. 6, pp. 835-841, June 2010.
[13] A. Bostan, B. Salvy, and E. Schost, “Power Series Composition and Change of Basis,” Proc. Int'l Symp. Symbolic and Algebraic Computation, pp. 269-276, 2008.
[14] A. Bostan, B. Salvy, and E. Schost, “Fast Conversion Algorithms for Orthogonal Polynomials,” Linear Algebra and Its Applications, vol. 432, no. 1, pp. 249-258, Jan. 2010.
[15] A. Schönhage and V. Strassen, “Schnelle Multiplikation Grosser Zahlen,” Computing, vol. 7, pp. 281-292, 1971.
[16] S. Johnson and M. Frigo, “A Modified Split-Radix FFT with Fewer Arithmetic Operations,” IEEE Trans. Signal Processing, vol. 55, no. 1, pp. 111-119, Jan. 2007.
[17] H. Sorensen, D. Jones, M. Heideman, and C. Burrus, “Real-Valued Fast Fourier Transform Algorithms,” IEEE Trans. Acoustics, Speech and Signal Processing, vol. ASSP-35, no. 6, pp. 849-863, June 1987.
[18] G. Baszenski and M. Tasche, “Fast Polynomial Multiplication and Convolution Related to the Discrete Cosine Transform,” Linear Algebra and Its Applications, vol. 252, nos. 1-3, pp. 1-25, 1997.
[19] S.C. Chan and K.L. Ho, “Direct Methods for Computing Discrete Sinusoidal Transforms,” Proc. IEE F Radar and Signal Processing, vol. 137, no. 6, pp. 433-442, Dec. 1990.
[20] P. Giorgi, “On Polynomial Multiplication in Chebyshev Basis,” technical report, arxiv.org, http://arxiv.org/abs1009.4597, 2010.
[21] Intel 64 and IA-32 Architectures Optimization Reference Manual, Intel Corp, http://www.intel.com/Assets/PDF/manual248966.pdf , 2011.
[22] D.R. Musser, G.J. Derge, and A. Saini, The STL Tutorial and Reference Guide: C++ Programming with the Standard Template Library. Addison Wesley Longman Publishing Co., Inc., 2002.
[23] M. Püschel, J.M.F. Moura, J. Johnson, D. Padua, M. Veloso, B. Singer, J. Xiong, F. Franchetti, A. Gacic, Y. Voronenko, K. Chen, R.W. Johnson, and N. Rizzolo, “SPIRAL: Code Generation for DSP Transforms,” Proc. IEEE, Special Issue on Program Generation, Optimization, and Platform Adaptation, vol. 93, no. 2, pp. 232-275, Feb. 2005.
[24] M. Frigo and S.G. Johnson, “The Design and Implementation of FFTW3,” Proc. IEEE, Special Issue on Program Generation, Optimization, and Platform Adaptation, vol. 93, no. 2, pp. 216-231, Feb. 2005.
[25] N.J. Higham, Accuracy and Stability of Numerical Algorithms. Soc. for Industrial and Applied Math., 2002.
[26] R. Lidl and H. Niederreiter, Finite Fields, second ed. Cambridge Univ. Press, 1997.
[27] A. Hasan and C. Negre, “Low Space Complexity Multiplication over Binary Fields with Dickson Polynomial Representation,” IEEE Trans. Computers, vol. 60, no. 4, pp. 602-607, Apr. 2010.
[28] J.B. Lima, D. Panario, and R. Campello de Souza, “Public-Key Encryption Based on Chebyshev Polynomials over gf(q),” Information Processing Letters, vol. 11, no. 2, pp. 51-56, 2010.
[29] D. Harvey and D.S. Roche, “An In-Place Truncated Fourier Transform and Applications to Polynomial Multiplication,” Proc. Int'l Symp. Symbolic and Algebraic Computation, pp. 325-329, 2010.
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