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Re: Re: Re: Re: smallest

• To: mathgroup at smc.vnet.net
• Subject: [mg86870] Re: [mg86833] Re: [mg86828] Re: [mg86792] Re: [mg86771] smallest
• From: Carl Woll <carlw at wolfram.com>
• Date: Mon, 24 Mar 2008 01:45:16 -0500 (EST)
• References: <200803200757.CAA29500@smc.vnet.net> <200803210653.BAA18315@smc.vnet.net> <200803220554.AAA00496@smc.vnet.net> <200803230600.BAA25688@smc.vnet.net>

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

>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 
>>
>>
>>
>>  
>>    
>>

• References:
  • smallest fraction
    • From: masmoudi <mas_atef@yahoo.fr>
  • Re: smallest fraction
    • From: Curtis Osterhoudt <cfo@lanl.gov>
  • Re: Re: smallest fraction
    • From: Artur <grafix@csl.pl>
  • Re: Re: Re: smallest fraction
    • From: Artur <grafix@csl.pl>