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From: martin@entropy.UUCP (Don Martin)
Newsgroups: net.math
Subject: Re: Series for normal distribution function
Message-ID: <255@entropy.UUCP>
Date: Wed, 5-Mar-86 05:46:59 EST
Article-I.D.: entropy.255
Posted: Wed Mar  5 05:46:59 1986
Date-Received: Sat, 8-Mar-86 22:05:36 EST
References: <443@cubsvax.UUCP>
Organization: UW MathStat, Seattle
Lines: 40
Summary: Hastings approximation to the normal and references.

Ken Winston writes:
> I have seen the following approximation a few different times now as a way
....... ( more of quote at end )
<>
This looks like one of Hasting's approximations. It is not
a series approximtion but rather an ( approximate ) Chebychev
fit that minimizes the maximum error. Hastings published a
strange little book in about 1956 on numerical approximations.
It is strang in that it has almost no text. It seems to be
slides from lectures. It is quite interesting.

For a rundown on various approximations of the normal see:

Statistical Computing by Kennedy and Gentile, pp.89-96.
Marcel Dekker, Inc.

For this methodology ( min-max approximations ) see :
The Approximation of Elementary Functions ( ? may be inacurate )
Hart et al.
John Wiley ( sponsered by SIAM)
> 
> I have seen the following approximation a few different times now as a way
> of finding N(t) on a computer. It seems to be accurate in the usual range, for
> t<=0:
> 
>     N(t) ~= x*exp(-t^2/2)*(.436184 - .120167*x + .937298*x^2)/sqrt(2*pi),
> 	where x = 1/(1+.3327*|t|)
> 
> For t>0 use N(t) = 1-N(-t).
> 
> My ignorance of numerical analysis is vast (not just half-vast). Does anybody
> know where this comes from? What is the general method that gives rise to this
> series?
> 
> Ken Winston
> ...{cmcl2,rna,cubsvax}!wealth!ken

 Donald C. Martin,  phone (206) 543 1044
 SC-32, Dept. of Biostatistics., U. of Wash, Seattle WA, 98195
 {decvax,ihnp4,ucbvax!lbl-csam}!uw-beaver!entropy!martin