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From "Luc Maisonobe (JIRA)" <j...@apache.org>
Subject [jira] [Commented] (MATH-1185) Tail probability drops to zero beyond 10e-17 ?
Date Mon, 05 Jan 2015 19:04:34 GMT
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[ https://issues.apache.org/jira/browse/MATH-1185?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=14264920#comment-14264920
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Luc Maisonobe commented on MATH-1185:
-------------------------------------

The problem is not the global magnitude. Values of the order of magnitude of 10^-308 can be
represented, and values of the order of 10^+308 can also be represented. The problem is how
you compute the value.

All normal numbers have about 15 significant digits. This means that when you are speaking
of 10^-295, the most significat digit is 10^-295 and the least significant digit if 15 orders
of magnitude lowers, hence 10^-308.

Here, you start from something of the order of magnitude of 1.0, so the most significant digit
is 10^0 and the least significant digit is 10^-15. When you do the subtraction, you cannot
recover digits smaller that 10^-15 because they were not in the number you started from.

I do not think R uses higher precision. I think they do not use 1 - x here, but compute the
value using another direct algorithm/

This is exactly the same problem as computing exp(t) -1 for small t. If you do it the obvious
way, by first computing the exponential function and then subtracting 1, you have very bad
results. This is the reason why all mathemetical libraries provide a function expm1(t) which
does not use the obvious way and therefore avoids the subtraction, allowing extremely high
accuracy here.

I don't know if for the expression you need such a dedicated algorithm exists, but for sure
if you need this level of accuracy, don't use a subtraction with 1.

> Tail probability drops to zero beyond 10e-17 ?
> ----------------------------------------------
>
>                 Key: MATH-1185
>                 URL: https://issues.apache.org/jira/browse/MATH-1185
>             Project: Commons Math
>          Issue Type: Wish
>    Affects Versions: 3.3, 3.4
>            Reporter: Sriram Natarajan
>            Priority: Minor
>
> This could be a  simple question, In which case I can expect a clarification. If this
is the wrong place to post such a question, let me know.
> OS: Windows 8.1, Java 1.8.0_25
> ChiSquaredDistribution chisq = new ChiSquaredDistribution(23)
> 1.0 - chisq.cumulativeProbability(130) is  1.1102230246251565E-16
> 1.0 - chisq.cumulativeProbability(131) is  0.0
> Am pretty sure it is not a formatting issue.  Is there a reason why the tail drops to
zero at this point?

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