[Rd] [R] choose(n, k) as n approaches k

[Duncan Murdoch]

On 14/01/2020 10:50 a.m., peter dalgaard wrote:
> 
> 
>> On 14 Jan 2020, at 16:21 , Duncan Murdoch <murdoch.duncan using gmail.com> wrote:
>>
>> On 14/01/2020 10:07 a.m., peter dalgaard wrote:
>>> Yep, that looks wrong (probably want to continue discussion over on R-devel)
>>> I think the culprit is here (in src/nmath/choose.c)
>>>       if (k < k_small_max) {
>>>          int j;
>>>          if(n-k < k && n >= 0 && R_IS_INT(n)) k = n-k; /* <- Symmetry */
>>>          if (k <  0) return 0.;
>>>          if (k == 0) return 1.;
>>>          /* else: k >= 1 */
>>> if n is a near-integer, then k can become non-integer and negative. In your case,
>>> n == 4 - 1e-7
>>> k == 4
>>> n - k == -1e-7 < 4
>>> n >= 0
>>> R_IS_INT(n) = TRUE (relative diff < 1e-7 is allowed)
>>> so k gets set to
>>> n - k == -1e-7
>>> which is less than 0, so we return 0. However, as you point out, 1 would be more reasonable and in accordance with the limit as n -> 4, e.g.
>>>> factorial(4 - 1e-10)/factorial(1e-10)/factorial(4) -1
>>> [1] -9.289025e-11
>>> I guess that the fix could be as simple as replacing n by R_forceint(n) in the k = n - k step.
>>
>> I think that would break symmetry:  you want choose(n, k) to equal choose(n, n-k) when n is very close to an integer.  So I'd suggest the replacement whenever R_IS_INT(n) is true.
>>
> 
> But choose() very deliberately ensures that k is integer, so choose(n, n-k) is ill-defined for non-integer n.

That's only true if there's a big difference.  I'd be worried about 
cases where n and k are close to integers (within 1e-7).  In those 
cases, k is silently rounded to integer.  As I read your suggestion, n 
would only be rounded to integer if k > n-k.  I think both n and k 
should be rounded to integer in this near-integer situation, regardless 
of the value of k.

I believe that lchoose(n, k) already does this.

Duncan Murdoch

> 
>      double r, k0 = k;
>      k = R_forceint(k);
> ...
>      if (fabs(k - k0) > 1e-7)
>          MATHLIB_WARNING2(_("'k' (%.2f) must be integer, rounded to %.0f"), k0, k);
>    
> 
>> Duncan Murdoch
>>
>>> -pd
>>>> On 14 Jan 2020, at 00:33 , Wright, Erik Scott <ESWRIGHT using pitt.edu> wrote:
>>>>
>>>> This struck me as incorrect:
>>>>
>>>>> choose(3.999999, 4)
>>>> [1] 0.9999979
>>>>> choose(3.9999999, 4)
>>>> [1] 0
>>>>> choose(4, 4)
>>>> [1] 1
>>>>> choose(4.0000001, 4)
>>>> [1] 4
>>>>> choose(4.000001, 4)
>>>> [1] 1.000002
>>>>
>>>> Should base::choose(n, k) check whether n is within machine precision of k and return 1?
>>>>
>>>> Thanks,
>>>> Erik
>>>>
>>>> ***
>>>> sessionInfo()
>>>> R version 3.6.0 beta (2019-04-15 r76395)
>>>> Platform: x86_64-apple-darwin15.6.0 (64-bit)
>>>> Running under: macOS High Sierra 10.13.6
>>>>
>>>> 	[[alternative HTML version deleted]]
>>>>
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>