A semi-iterative process for evaluating arctangents

The technique of obtaining arctangents by inverse interpolation [1] is a relatively long process not suitable for a subroutine. The Taylor series expansion for arguments less than unity converges rather slowly for those near unity. The method of small increments of the argument [2] is again inconvenient for a subroutine. A more rapid series expansion in terms of Chebyshev polynomial [3] is given in terms of a new argument, which is less than 0.1989. This method requires the storage of &pgr;, √2 - 1 and seven coefficients and is perhaps widely used. However, for multiple precision not only the coefficients have to be evaluated to the precision desired, but more must be used. Therefore the following alternative method may prove to be convenient and efficient.

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