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Re: Re: Re: Re: smallest


Carl Woll wrote:

> Artur wrote:
>
>> If we want to find rational fraction f =p/q such that 
>> 113/355<f<106/333 and sum p+q is minimal
>> anyone procedure proposed  up to now doesn't work
>> good result should be
>> {137563,{p->13215,q->104348}}
>> but isn't
>>  
>>
> Your good result isn't so good, consider:
>
> In[36]:= 113/355 < 219/688 < 106/333
>
> Out[36]= True
>
> One idea (similar to your Recognize approach) is to use Rationalize or 
> RootApproximant with SetPrecision:
>
> In[71]:= Rationalize[SetPrecision[(106/333 + 113/355)/2, 6], 0]
>
> Out[71]= 219/688
>
> In[72]:= RootApproximant[SetPrecision[(106/333 + 113/355)/2, 6], 1]
>
> Out[72]= 219/688
>
> I'm not sure of the correct method to determine the precision to use. 
> It could be something like:
>
> Choose largest prec such that:
>
> IntervalMemberQ[Interval[{lo, hi}], SetPrecision[midpoint, prec]]
>
> is still True.
>
> Carl Woll
> Wolfram Research

I should add that Daniel Lichtblau's minFraction just needs to be 
tweaked a bit to find this result:

minFraction[lo_Rational, hi_Rational] /; 0 < lo < hi :=
 Minimize[{p + q, {Denominator[lo]*p - Numerator[lo]*q > 0, 
Denominator[hi]*p - Numerator[hi]*q < 0, p >= 1, q >= 1}}, {p, q}, Integers]

In[81]:= minFraction[113/355, 106/333]

Out[81]= {907,{p->219,q->688}}

Carl Woll
Wolfram Research

>
>> ARTUR
>>
>> Artur pisze:
>>  
>>
>>> If value p/q is known
>>> smallest Abs[p]+Abs[q ] should be
>>> << NumberTheory`Recognize`
>>> Recognize[p/q,1,x]
>>>
>>> see also
>>> http://www.research.att.com/~njas/sequences/A138335
>>>
>>> Best wishes,
>>> Artur
>>>
>>> Curtis Osterhoudt pisze:
>>>  
>>>   
>>>
>>>>  I doubt this is in the spirit of the problem, but if p and q 
>>>> (assumed integers) aren't restricted to be _positive_, then taking 
>>>> them both to be very large negative numbers would both fit the p/q 
>>>> in I requirement, and p+q as "small" as possible.
>>>>               C.O.
>>>>
>>>> On Thursday 20 March 2008 01:57:30 masmoudi wrote:
>>>>  
>>>>        
>>>>
>>>>> hi
>>>>>
>>>>> suppose that we have an interval  I belong to [0,1]
>>>>>
>>>>> I want to know how to calculate a fraction p/q
>>>>> belong to I and p+q is the smallest possible
>>>>>               
>>>>
>>>>  
>>>>        
>>>
>>> __________ NOD32 Informacje 2701 (20071204) __________
>>>
>>> Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
>>> http://www.nod32.com lub http://www.nod32.pl
>>>
>>>
>>>  
>>>   
>>
>
>



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