Generalized rational Chebyshev approximation

Author:
Ichizo Ninomiya

Journal:
Math. Comp. **24** (1970), 159-169

MSC:
Primary 41.17; Secondary 65.00

MathSciNet review:
0261229

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, a generalized rational Chebyshev approximation problem is considered.

The problem is this: To minimize the maximum absolute value of the "criterion function" of the error. By imposing a rather natural restriction on the criterion function, the problem is solved completely; the existence, the uniqueness and the characterization of the best approximation are clarified and interesting relationships between the best approximations corresponding to different criterion functions are found.

The theory is applied to the starting rational approximation for Newton iteration for .

**[1]**N. I. Achieser,*Theory of approximation*, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956. MR**0095369****[2]**Charles B. Dunham,*Transformed rational Chebyshev approximation*, Numer. Math.**10**(1967), 147–152. MR**0217496****[3]**David G. Moursund,*Chebyshev approximation using a generalized weight function*, SIAM J. Numer. Anal.**3**(1966), 435–450. MR**0204936****[4]**David G. Moursund,*Optimal starting values for Newton-Raphson calculation of √𝑥*, Comm. ACM**10**(1967), 430–432. MR**0240952****[5]**W. J. Cody, "Double-precision square root for the CDC-3600,"*Comm. ACM*, v. 7, 1964, pp. 715-718.**[6]**P. L. Chebyshev, "Sur les expressions approchées de la racine carrée d'une variable par des fractions simples," in*Oeuvres*. Vol. 2, Chelsea, New York, pp. 542-558.**[7]**P. H. Sterbenz & C. T. Fike, "Optimal starting approximations for Newton's method,"*Math. Comp.*, v. 23, 1969, 313-318.**[8]**Richard F. King and David L. Phillips,*The logarithmic error and Newton’s method for the square root*, Comm. ACM**12**(1969), 87–88. MR**0285109**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0261229-8

Keywords:
Rational approximation,
criterion function,
weight function,
best approximation,
existence of best approximation,
uniqueness of best approximation,
characterization of best approximation,
relations between best approximations,
starting approximation for square root

Article copyright:
© Copyright 1970
American Mathematical Society