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rational approximations to the normal cdf

2 messages · Jason H. Stover, Peter Dalgaard

Original
Jason H. Stover
# 
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
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Peter Dalgaard
# 
"Jason H. Stover" <jstover at gcsu.edu> writes:
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.