Numerische Methoden der Approximationstheorie / Numerical Methods of Approximation Theory pp 223-244 | Cite as

# Some Approximations for Trigonometrical Functions

## Abstract

It is shown that better approximations of tan(x) are obtained if the singular part is taken out, i.e. if we write tan(x) = S(x) + R(x) where S contains the singular part of tan in the sense that the rest R(x) is finite for x → one or several of the points ±π/2,±3π/2,... For example for S(x):= (π/2 - x)^{-1} + (π/2 + x)^{-1} we get tan(x) over the whole interval]-π/2,π/2 [as an easily computed part S(x) plus a rest R(x) which is finite (even C^{∞}), has a rapidly converging power series (in contrast to tan itself), and with low-degree polynomial and rational approximations for any reasonable accuracy. Such approximations are discussed for 1, 2 and 4 singularities taken out. It is also shown that the usual minmax rational approximations over as short an interval as [0,π/4] “contains” both zeroes and singularities of tan — as many as the degrees of numerator and denominator polynomials allow, and with increasing accuracy as these degrees increase. The approximations are examples of a more general situation where a function f to be approximated is split in two parts f = f_{p} + f_{a}, with f_{p} as a simple “preconditioning” part, which mainly serves to give the “accuracy” part f_{a} certain desirable qualities.

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## References

- [1]Jolley, L.B.W.: Summation of Series. New York, Dover Publications 1961. The table of Bernoulli numbers has been taken from “Report of the British Association for the Advancement of Science, 1877”.Google Scholar
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- [3]Pedersen, Poul Wulff: Rapidly converging series for tan and cot, and some related approximations. To appear.Google Scholar
- [4]Hart, John F. (ed.): Computer Approximations. New York, Wiley 1968. Poul Wulff Pedersen D.T.H., The Technical University of Denmark Dept. of Mathematics, Bygn. 303Google Scholar