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

[peter dalgaard]

> 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.

    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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-- 
Peter Dalgaard, Professor,
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