A Karatsuba-Based Algorithm for Polynomial Multiplication in Chebyshev Form

In this paper, we present a new method for multiplying polynomials in Chebyshev form. Our approach has two steps. First, the well-known Karatsuba's algorithm is applied to polynomials constructed by using Chebyshev coefficients. Then, from the obtained result, extra arithmetic operations are used to write the final result in Chebyshev form. The proposed algorithm has a quadratic computational complexity. We also compare our method to other approaches.

[1]  Michail Papamichail,et al.  Volterra Analysis Using Chebyshev Series , 2007, 2007 IEEE International Symposium on Circuits and Systems.

[2]  Joachim von zur Gathen,et al.  Modern Computer Algebra (3. ed.) , 2003 .

[3]  Manfred Tasche,et al.  Fast Polynomial Multiplication and Convolutions Related to the Discrete Cosine Transform , 1997 .

[4]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[5]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[6]  S. C. Chan,et al.  Direct methods for computing discrete sinusoidal transforms , 1990 .

[7]  Po-Jui Chiang,et al.  Robust Calculation of Chromatic Dispersion Coefficients of Optical Fibers From Numerically Determined Effective Indices Using Chebyshev–Lagrange Interpolation Polynomials , 2006, Journal of Lightwave Technology.

[8]  Peter L. Montgomery,et al.  Five, six, and seven-term Karatsuba-like formulae , 2005, IEEE Transactions on Computers.

[9]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[10]  Christof Paar,et al.  A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields , 1996, IEEE Trans. Computers.

[11]  Marc Moonen,et al.  Chebyshev interpolation for DMT modems , 2004, 2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577).

[12]  Reza R. Adhami,et al.  Arbitrary waveform DDFS utilizing Chebyshev polynomials interpolation , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.