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# Re: Math Problem in Mathematica
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.
Daniel:Since this is discrete it can be tackled it directly f[i_,j_,n_]
:= Abs[Sqrt[1-(i/n)^2]-j/n]
s[n_] := Sum[ Min[Table[f[i,j,n]//N,{j,1,n}]],{i,1,n}]
s[20]//Timing
{3.86667 Second,0.262656}
Or by looking at
Solve[(n^2-i^2)==j^2,j]
{{j -> -Sqrt[-i^2 + n^2]}, {j -> Sqrt[-i^2 + n^2]}}
and allowing for the case i = n
s2[n_]:=Sum[Min[N[ f[i,#,n]]&/@({
Max[#,1],Max[#+1,1]}&[Floor[Sqrt[n^2-i^2]]])],{i,1,n}]
s2[20]//Timing
{0.533333 Second,0.262656}
--
Allan Hayes
Training and Consulting
Leicester, UK
hay@haystack.demon.co.uk
http://www.haystack.demon.co.uk
voice: +44 (0)116 271 4198
fax: +44 (0)116 271 8642
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