On the Remes Algorithm for Rational Approximations
DOI:
https://doi.org/10.24297/jam.v12i10.106Keywords:
Minimax approximation, Rational functions, Remes algorithm, Nonlinear system of leveling Equations, The dual monomial Vandermond system, The leveled reference error.Abstract
This paper is concerned with the minimax approximation of a discrete data set by rational functions. The second algorithm of Remes is applied. A crucial stage of this algorithm is solving the nonlinear system of leveling equations. In this paper, we will give a new approach for this purpose. In this approach, no initial guesses are required. Illustrative numerical example is presented.Downloads
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References
[1] Barrodale, I. and Mason, J. C. (1970)," Two Simple Algorithms for Discrete Rational Approximation", Maths. of Comp., 26, 877-891.
[2] Barrodale, I. and Philips, C. (1975), "Algorithm 495: Solution of an Overdetermined System of Linear Equations in the Chebyshev Norm", ACM Trans. Maths. Software, 1, 264-270.
[3] BjÓ§rck, A. and Pereyra, V. (1970), "Solution of Vandermonde System of Equations", Maths. of Comp., 24, 893-903.
[4] Cheney, E. W. (1966), "Introduction to Approximation Theory", McGraw-Hill.
[5] Chun, C. and Ham, Y. (2008), "Some Fourth-Order Modifications of Newton's Method", Applied Mathematics and Computation",197, 654-658.
[6] Davis, P. J. (1975), "Interpolation and Approximation", Dover Publications, New York.
[7] Higham, N. J. (1988), "Fast Solution of Vandermonde-like Systems Involving Orthogonal Polynomials", J. Numer.
Math., 8, 473-486.
[8] Kaufman, J. R., Leeming, D. G. and Taylor, G. D. (1980), "A Combined Remes-Differential Correction Algorithm for Rational Approximation", Experimental Results, J. Comp. And Maths. with Appl., 6,155-166.
[9] Lee, C. M. and Roberts, F. D. (1973), "A Comparison of Algorithm for Rational Approximation", Math. of Comp., 26, 111-120.
[10] Meinardus, G. (1967), "Approximation of Functions: Theory and Numerical Methods", Springer, Heidelberg.
[11] Mhaskar, H. N. and Pai, D. V. (2000), "Fundamentals of Approximation Theory", Narosa Publishing House, New Delhi. [12] Pachá½¹n, R. and Trefethen, L.N. (2009), "Barycentric-Remez Algorithms for Test Polynomial Approximation in the Chebfun System", BIT Numer. Math., 49, 721-741.
[13] Powell, M. J. D. (1981), "Approximation Theory and Methods", Cambridge University Press, Cambridge, UK.
[14] Remez, E.Y. (1969), "Fundamentals of Numerical Methods for Chebyshev Approximations", Naukova Dumka, Kiev.
[15] Rice, J. R. (1964), "The Approximation of Functions (Vol. 1)", Addison-Wesley.
[16] Steffens, K. G. (2006), "The History of Approximation Theory: From Euler to Bernstein", Birkhäuser, Boston.
[17] Watson, G. A. (1980), "Approximation Theory and Numerical Methods", John Wiley and Sons.
[2] Barrodale, I. and Philips, C. (1975), "Algorithm 495: Solution of an Overdetermined System of Linear Equations in the Chebyshev Norm", ACM Trans. Maths. Software, 1, 264-270.
[3] BjÓ§rck, A. and Pereyra, V. (1970), "Solution of Vandermonde System of Equations", Maths. of Comp., 24, 893-903.
[4] Cheney, E. W. (1966), "Introduction to Approximation Theory", McGraw-Hill.
[5] Chun, C. and Ham, Y. (2008), "Some Fourth-Order Modifications of Newton's Method", Applied Mathematics and Computation",197, 654-658.
[6] Davis, P. J. (1975), "Interpolation and Approximation", Dover Publications, New York.
[7] Higham, N. J. (1988), "Fast Solution of Vandermonde-like Systems Involving Orthogonal Polynomials", J. Numer.
Math., 8, 473-486.
[8] Kaufman, J. R., Leeming, D. G. and Taylor, G. D. (1980), "A Combined Remes-Differential Correction Algorithm for Rational Approximation", Experimental Results, J. Comp. And Maths. with Appl., 6,155-166.
[9] Lee, C. M. and Roberts, F. D. (1973), "A Comparison of Algorithm for Rational Approximation", Math. of Comp., 26, 111-120.
[10] Meinardus, G. (1967), "Approximation of Functions: Theory and Numerical Methods", Springer, Heidelberg.
[11] Mhaskar, H. N. and Pai, D. V. (2000), "Fundamentals of Approximation Theory", Narosa Publishing House, New Delhi. [12] Pachá½¹n, R. and Trefethen, L.N. (2009), "Barycentric-Remez Algorithms for Test Polynomial Approximation in the Chebfun System", BIT Numer. Math., 49, 721-741.
[13] Powell, M. J. D. (1981), "Approximation Theory and Methods", Cambridge University Press, Cambridge, UK.
[14] Remez, E.Y. (1969), "Fundamentals of Numerical Methods for Chebyshev Approximations", Naukova Dumka, Kiev.
[15] Rice, J. R. (1964), "The Approximation of Functions (Vol. 1)", Addison-Wesley.
[16] Steffens, K. G. (2006), "The History of Approximation Theory: From Euler to Bernstein", Birkhäuser, Boston.
[17] Watson, G. A. (1980), "Approximation Theory and Numerical Methods", John Wiley and Sons.
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Published
2016-11-30
How to Cite
Saad, H. L., & Hassan, N. Y. A. (2016). On the Remes Algorithm for Rational Approximations. JOURNAL OF ADVANCES IN MATHEMATICS, 12(10), 6733–6738. https://doi.org/10.24297/jam.v12i10.106
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