[R] rational approximations to the normal cdf

[Peter Dalgaard BSA]

"Jason H. Stover" <jstover at gcsu.edu> writes:

> In the R source, nmath/pnorm.c contains the
> code for a rational function approximation 
> for the normal cdf. These constants are listed:
> 
>     const double a[5] = {
>         2.2352520354606839287,
>         161.02823106855587881,
>         1067.6894854603709582,
>         18154.981253343561249,
>         0.065682337918207449113
>     };
>  
> The source file cites a paper by Cody (1969)
> and states that these constants are derived from
> Cody's paper. I've checked the paper, and 
> the constants do not agree, so I assume someone 
> else derived slightly different constants. The source
> also contains a reference to 1993 paper by Cody, 
> but this paper describes software itself, not the 
> derivation of the approximation used.
> 
> Does anyone know of another reference for these
> and other constants used in pnorm.c? Is there a 
> general reference detailing how to derive such 
> approximations?
> 
> Thank-you,
> Jason

pnorm.c is based on ANORM from http://www.netlib.org/toms/715, which
has the same constants and is by Cody himself. The exacts words there
are:

C   The main computation evaluates near-minimax approximations
C   derived from those in "Rational Chebyshev approximations for
C   the error function" by W. J. Cody, Math. Comp., 1969, 631-637.

Notice "derived from". My guess would be that he applied the same
methods but to give optimal approximations over different intervals
from the 1969 paper. 

-- 
   O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907
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