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.<\/p>\r\n","references":"[1] Howard Anton, Calculus With Analytic Geometry, Fourth edition. Anton\r\nTextbooks, Inc., 1992.\r\n[2] Nicolas Brisebarre, Jean-Michel Muller, and Arnaud Tisserand, Computing\r\nMachine-Efficient Polynomial Approximation. ACM Transactions on\r\nMathematical Software, Number 2, Volume 32,pp. 236-256 ,2006.\r\n[3] B. D-Aguanno, A. Nobile, and E. Roman, CHPACK: A Package For\r\nThe Manipulation of Chebyshev Approximations. Computer Physics\r\nCommunications, Number 29, pp. 361-374 ,1983.\r\n[4] David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of\r\nScientific Computing. Brooks\/Cole, 2002.\r\n[5] Ker-I Ko, On the Computational Complexity of Best Chebyshev Approximations.\r\nComplexity, Number 2,pp. 65-120 ,1986.\r\n[6] Ker-I Ko, Complexity Theory of Real Functions. Boston: Birkhauser,\r\n1991.\r\n[7] Roland E. Larson, Robert P. Hostetler, Bruch H. Edwards and David E.\r\nHeyd, Calculus with Analytic Geometry. D. C. Heath and Company,\r\n1994.\r\n[8] J. Mason, Chebyshev Polynomials: Theory and Applications. Kluwer\r\nAcademic, 1996.\r\n[9] John C. Mason, and David C. Handscomb, Chebyshev Polynomials. CRC\r\nPress Compan, 2003.\r\n[10] Jean-Michel Muller, Elementary Functions: Algorithms and Implementation.\r\nBirkhauser, 1997.\r\n[11] D. Pavlovic and M.H. Escardo, Calculus in Conductive Form. Thirteenth\r\nAnnual IEEE Symposium on Logic in Computer Science, pp. 408-\r\n417 ,1998.\r\n[12] Marian Pour-El and J. Ian Richards, Computability in Analysis and\r\nPhysics. Berlin: Springer-Verlag, 1989.\r\n[13] Theodore J. Rivlin, Chebyshev Polynomials: from approximation theory\r\nto algebra and number theory. New York, Chichester : Wiley, 1990.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 7, 2007"}