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equation solving - Find a single max value 


w[n_] := Expand[Sum[Binomial[n - k - 1, k]*(-1)^k*A^(n - 2*k - 1), {k, 0, n - 1}]]
f[x_, y_, z_] :=PolynomialRemainder[(w[z] - 1)*(w[y] - 1), (w[x] - 1), A]

For[i = 3, i < 450, i++,For[j = 3, j < 450, j++,For[k = 3, k < 450, k++,If[i < j < k, 
Print[{i, j, k}, 
 N[Max[FindInstance[{Abs[f[i, j, k]] - Abs[w[i] - 1]} == 0 && 

     3 <= A, {A}]]] ]]] ]]

Above code gives max A for all each cases. My aim is to find max A for all cases.


Q2: How can i reduce the time for above code? Thank you.



Answer



Artes’ comment suggests there is a problem with the concept of optimum you are trying to find. Let me make the more general point about how your code may be improved.




  • First, if you have a triply nested For loop, you are probably doing it wrong. Consider Table instead, or some of the other constructs mentioned here.





  • Second, you don't need to go through every combination of {i, j, k} since you only want the cases where i. I think you can get this set of combinations using Subsets[Range[3,450],{3}] which gives the ordered subsets of exactly length 3.




  • Third, functional programming is usually more efficient. Create the list of indices and then Map the desired function to the elements of that list.




  • Fourth, Print doesn't store anything. You need to save the results for each case as part of a list somewhere and then find the maximum value of that list. The mapping- onto-a-list approach will actually store the results, if only temporarily.





Putting all of this together, something like this should work:


Max[yourfunction /@ Subsets[Range[3, 450], {3}]]

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