Re: Re: Re: Re: smallest

*To*: mathgroup at smc.vnet.net*Subject*: [mg86871] Re: [mg86833] Re: [mg86828] Re: [mg86792] Re: [mg86771] smallest*From*: Carl Woll <carlw at wolfram.com>*Date*: Mon, 24 Mar 2008 01:45:27 -0500 (EST)*References*: <200803200757.CAA29500@smc.vnet.net> <200803210653.BAA18315@smc.vnet.net> <200803220554.AAA00496@smc.vnet.net> <200803230600.BAA25688@smc.vnet.net> <47E65E16.4010608@wolfram.com>

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

**Follow-Ups**:**Re: Re: Re: Re: Re: smallest***From:*Andrzej Kozlowski <akoz@mimuw.edu.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>