Minimax approximations subject to a constraint

Authors:
C. T. Fike and P. H. Sterbenz

Journal:
Math. Comp. **25** (1971), 295-298

MSC:
Primary 41A20

DOI:
https://doi.org/10.1090/S0025-5718-1971-0303176-X

MathSciNet review:
0303176

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Abstract: A class of approximation problems is considered in which a continuous, positive function is approximated by a rational function satisfying some identity. It is proved under certain hypotheses that there is a unique rational approximation satisfying the constraint and yielding minimax relative error and that the corresponding relative-error function does not have an equal-ripple graph. This approximation is, moreover, just the rational approximation to yielding minimax logarithmic error. This approximation, in turn, is just a constant multiple of the rational approximation to yielding minimax relative error but not necessarily satisfying the constraint.

**[1]**W. J. Cody & Anthony Ralston, "A note on computing approximations to the exponential function,"*Comm. ACM*, v. 10, 1967, pp. 53-55.**[2]**I. F. Ganžela and C. T. Fike,*Sterbenz, P. H*, Math. Comp.**23**(1969), 313–318. MR**0245199**, https://doi.org/10.1090/S0025-5718-1969-0245199-6**[3]**W. Kahan, "Library tape functions EXP, TWOXP, and .XPXP.,"*Programmers' Reference Manual*, University of Toronto, 1966. (Mimeographed.)**[4]**W. J. Cody, "Double-precision square root for the CDC-3600,"*Comm. ACM*, v. 7, 1964, pp. 715-718.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0303176-X

Keywords:
Rational approximation,
polynomial approximation,
best approximation,
constrained approximation,
exponential function,
starting approximation for square root

Article copyright:
© Copyright 1971
American Mathematical Society