Computable Function Representations Using Effective Chebyshev Polynomial
We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.
[1] Howard Anton, Calculus With Analytic Geometry, Fourth edition. Anton
Textbooks, Inc., 1992.
[2] Nicolas Brisebarre, Jean-Michel Muller, and Arnaud Tisserand, Computing
Machine-Efficient Polynomial Approximation. ACM Transactions on
Mathematical Software, Number 2, Volume 32,pp. 236-256 ,2006.
[3] B. D-Aguanno, A. Nobile, and E. Roman, CHPACK: A Package For
The Manipulation of Chebyshev Approximations. Computer Physics
Communications, Number 29, pp. 361-374 ,1983.
[4] David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of
Scientific Computing. Brooks/Cole, 2002.
[5] Ker-I Ko, On the Computational Complexity of Best Chebyshev Approximations.
Complexity, Number 2,pp. 65-120 ,1986.
[6] Ker-I Ko, Complexity Theory of Real Functions. Boston: Birkhauser,
1991.
[7] Roland E. Larson, Robert P. Hostetler, Bruch H. Edwards and David E.
Heyd, Calculus with Analytic Geometry. D. C. Heath and Company,
1994.
[8] J. Mason, Chebyshev Polynomials: Theory and Applications. Kluwer
Academic, 1996.
[9] John C. Mason, and David C. Handscomb, Chebyshev Polynomials. CRC
Press Compan, 2003.
[10] Jean-Michel Muller, Elementary Functions: Algorithms and Implementation.
Birkhauser, 1997.
[11] D. Pavlovic and M.H. Escardo, Calculus in Conductive Form. Thirteenth
Annual IEEE Symposium on Logic in Computer Science, pp. 408-
417 ,1998.
[12] Marian Pour-El and J. Ian Richards, Computability in Analysis and
Physics. Berlin: Springer-Verlag, 1989.
[13] Theodore J. Rivlin, Chebyshev Polynomials: from approximation theory
to algebra and number theory. New York, Chichester : Wiley, 1990.
[1] Howard Anton, Calculus With Analytic Geometry, Fourth edition. Anton
Textbooks, Inc., 1992.
[2] Nicolas Brisebarre, Jean-Michel Muller, and Arnaud Tisserand, Computing
Machine-Efficient Polynomial Approximation. ACM Transactions on
Mathematical Software, Number 2, Volume 32,pp. 236-256 ,2006.
[3] B. D-Aguanno, A. Nobile, and E. Roman, CHPACK: A Package For
The Manipulation of Chebyshev Approximations. Computer Physics
Communications, Number 29, pp. 361-374 ,1983.
[4] David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of
Scientific Computing. Brooks/Cole, 2002.
[5] Ker-I Ko, On the Computational Complexity of Best Chebyshev Approximations.
Complexity, Number 2,pp. 65-120 ,1986.
[6] Ker-I Ko, Complexity Theory of Real Functions. Boston: Birkhauser,
1991.
[7] Roland E. Larson, Robert P. Hostetler, Bruch H. Edwards and David E.
Heyd, Calculus with Analytic Geometry. D. C. Heath and Company,
1994.
[8] J. Mason, Chebyshev Polynomials: Theory and Applications. Kluwer
Academic, 1996.
[9] John C. Mason, and David C. Handscomb, Chebyshev Polynomials. CRC
Press Compan, 2003.
[10] Jean-Michel Muller, Elementary Functions: Algorithms and Implementation.
Birkhauser, 1997.
[11] D. Pavlovic and M.H. Escardo, Calculus in Conductive Form. Thirteenth
Annual IEEE Symposium on Logic in Computer Science, pp. 408-
417 ,1998.
[12] Marian Pour-El and J. Ian Richards, Computability in Analysis and
Physics. Berlin: Springer-Verlag, 1989.
[13] Theodore J. Rivlin, Chebyshev Polynomials: from approximation theory
to algebra and number theory. New York, Chichester : Wiley, 1990.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54593", author = "Mohammed A. Abutheraa and David Lester", title = "Computable Function Representations Using Effective Chebyshev Polynomial", abstract = "We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.
", keywords = "Approximation Theory, Chebyshev Polynomial, Computable Functions, Computable Real Arithmetic, Integration, Numerical Analysis.", volume = "1", number = "7", pages = "301-7", }
