Re: Calculation of critical p-, z-, t- and F-values


<iandjmsmith@xxxxxxx> wrote in message
news:1195043423.480673.177500@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

I've got it. The problem was you asked for
Quantile[StudentTDistribution[100000000], 0.975] and the 0.975 causes
it to work to machine precision. The method of calculation is very
poor and hence only delivers a results with relative error of 3e-8.

That is correct. As I said, when there is a 'numeric' value in the
expression, this will force the computation to be done in non-symbolic.


You can ask for exact calculations with
Quantile[StudentTDistribution[100000000], 975/1000]


Yes.

According to
http://reference.wolfram.com/mathematica/ref/N.html?q=N&lang=en
N[%, n] attempts to give a result with n-digit precision.


Yes.

It is not clear what is stopping it giving rather than attempting to
give a result with n-digit precision. The + operation is going to lose
about 8 figures so InverseBetaRegularized[1, -(19/20), 50000000, 1/2]
must be calculated to about 58 digits to give 50 digit accuracy. I am
still lost as to why it has printed out 68 digits. Maybe that is how
many figures it did the calculations to.

Ian Smith


Well, Mathematica floating point model is a little hard for me to comprehend
without spending more time on it, and I am no expert at this aspect of
Mathematica, may be a Mathematica expert can comment on this. I know it
sometimes uses "significance arithmetic" and is implemented in software.

I copied a reply I wrote sometime ago in another newsgroup which has a link
to a detailed paper about Mathematica floating point model if anyone is
interested in some reading over their coffee break :)

"There is this paper that goes into all the details you ever wanted to know
about Mathematica handling of floating point arithmetic (LONG URL)

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W8D-4FFGJ35-1&_coverDate=07%2F31%2F2005&_alid=458876682&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6652&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=d64b2c0097f2daf6c879b7f57ad45ab2

"M. Sofroniou and G. Spaletta. Precise numerical computation. The Journal of
Logic and Algebraic Programming 64:113-134. 2005."

There is a 1999 version of the paper that anyone can download for free ( I
am not sure what is the difference):
http://citeseer.ist.psu.edu/sofroniou99precise.html

Any way, I am trying to understand it. But related to significant
arithmetic use in Mathematica, the paper says that is not done all the time,
here is the quote:

"Indeed Mathematica uses fixed-precision arithmetic instead of significance
arithmetic in its large scale numerical routines, such as in linear algebra
and the numerical solution of differential equations; the error bounds
provided by these numerical methods are well studied and provide much
tighter bounds than those based on assumptions of independent error
accumulation."

There is also standard way to tell Mathematica to use hardware floating
point arithmatics as well if you want by setting some global options at the
start of the session.

Nasser "

Nasser


.

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