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MathGroup Archive 1998

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Re: Math Problem in Mathematica


  • To: mathgroup@smc.vnet.net
  • Subject: [mg10357] Re: Math Problem in Mathematica
  • From: Paul Abbott <paul@physics.uwa.edu.au>
  • Date: Mon, 12 Jan 1998 04:09:54 -0500
  • Organization: University of Western Australia
  • References: <694bda$kp6@smc.vnet.net>

Daniel wrote:

> The problem I would like to formulate in Mathematica is: Let f[i,j] =
> Abs[Sqrt[1-(i/n)^2]-j/n]. i and j run from  1 to n, and n is a fixed
> integer >=1.  I want to find the  sum S of the minimum of f over j, for
> each i for given n.
> 
> Example: n=7.  Min (i=1, j from 1 to 7)= .01
>                Min (i=2, """          )= .04
>                Min (i=3, ...           = .05
>                etc.
> 
> and the sum S = 0.20.
> 
> FindMinimum seemed like the right idea, but I don't know how to make it
> work for a function of discrete values.

A direct implementation works fine:

In[1]:= f[i_, j_, n_] = Abs[Sqrt[1 - (i/n)^2] - j/n]; 

In[2]:= S[n_] := Sum[Min[Table[f[i, j, n], {j, n}]], {i, 1, n}]

In[3]:= S[7]
Out[3]=
24   4 Sqrt[3]   3 Sqrt[5]   2 Sqrt[6]   2 Sqrt[10] -- - --------- -
--------- - --------- + ---------- -  7        7           7          
7           7
 
  Sqrt[13]   Sqrt[33]
  -------- - --------
     7          7

Note that Mathematica gives you the exact answer to this problem rather
than a numerical approximation:

In[4]:= N[%]
Out[4]= 0.348436

This does not agree with the answer above.  If the sum over is
restricted to 1<=i<n then you do get an answer of approximately 0.20.  

Cheers,
	Paul 

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul@physics.uwa.edu.au  AUSTRALIA                            
http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________



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