(*The energy function is a 10-term sum.  The summands are the functions F and G
discussed in the paper.  This file evaluates the constants a1,a2,a3,a4,a5.  Here aN
is the maximum kth derivative of any of these functions, evaluated at points
corresponding to the TBP, relative to the r^2 potential.*)

(*First we load in the file that computes these derivatives*)

<<derivatives1
<<derivatives2

(*The 4 terms corresponding to the function F trivially satisfy the bounds
    given in the paper.  In this case 
    D_i F=x_i/2.   Since |x_i|<=1, we have |D_i F|<=1/2.  This certainly gives a1<1.
    D_iD_j F = 0 or 1/2.  This certainly gives a2<4.
    all higher derivatives vanish.
    So, we only have to worry about the remaining 6 terms, involving the function G*)

(*This maximizes over the 6 terms.*)

ExactBound[L_]:=Max[Table[ExactBound[m,L],{m,5,10}]];



(*Now we list all possible Nth order derivative indices, for N=1,2,3,4,5.*)

list1=Table[{j},{j,1,4}]

list2=Table[Table[{i,j},{i,j,4}],{j,1,4}];
list2=Flatten[list2,1];

list3=Table[Table[Table[{i,j,k},{i,j,4}],{j,k,4}],{k,1,4}];
list3=Flatten[list3,2];

list4=Table[Table[Table[Table[{i,j,k,l},{i,j,4}],{j,k,4}],{k,l,4}],{l,1,4}]
list4=Flatten[list4,3];

list5=Table[Table[Table[Table[Table[{i,j,k,l,m},{i,j,4}],{j,k,4}],{k,l,4}],{l,m,4}],{m,1,4}]
list5=Flatten[list5,4];



(*Now we print out the maximum derivatives*)

big1=Table[ExactBound[list1[[j]]],{j,1,Length[list1]}];
Print["a1 ",Max[big1]];

big2=Table[ExactBound[list2[[j]]],{j,1,Length[list2]}];
Print["a2 ",Max[big2]];

big3=Table[ExactBound[list3[[j]]],{j,1,Length[list3]}];
Print["a3 ",Max[big3]];

big4=Table[ExactBound[list4[[j]]],{j,1,Length[list4]}];
Print["a4 ",Max[big4]];

big5=Table[ExactBound[list5[[j]]],{j,1,Length[list5]}];
Print["a5 ",Max[big5]];