(*This file computes the partial derivatives at the minimizer, the TBP.   You can specify the
exponent you would like to use. We are interested in the case when EXP=1 or EXP =2.
For these exponents we use a modified version of the Cholesky Decomposition to check
that all eigenvalues are greater than 1/10.*)

(*inverse stereographic projection*)
s[x_, y_] := {(2*x)/(1 + x*x + y*y), (2*y)/(1 + x*x + y*y), 1 - 2/(1 + x*x + y*y)};


(*These are the two basic functions making up the energy potential function*)

F[x1_,y1_,EXP_]:=(
a=s[x1,y1];
b={0,0,1};
c=1/(a-b).(a-b);
Power[c,EXP/2])

G[x1_,y1_,x2_,y2_,EXP_]:=(
a=s[x1,y1];
b=s[x2,y2];
c=1/(a-b).(a-b);
Power[c,EXP/2])

(*Here is the energy function*)

ff[x0_,x1_,y1_,x2_,y2_,x3_,y3_,EXP_]:=(
list1={F[x0,0,EXP],
F[x1,y1,EXP],
F[x2,y2,EXP],
F[x3,y3,EXP],
G[x0,0,x1,y1,EXP],
G[x0,0,x2,y2,EXP],
G[x0,0,x3,y3,EXP],
G[x1,y1,x2,y2,EXP],
G[x1,y1,x3,y3,EXP],
G[x2,y2,x3,y3,EXP]};
Sum[list1[[j]],{j,1,10}])



(**this function computes the second partial derivatives**)

D2[i_,j_,EXP_]:=(
(*The basic expression*)
A=ff[x0,x1,y1,x2,y2,x3,y3,EXP];
(*The variable list*)
VAR={x0,x1,y1,x2,y2,x3,y3};
(*The point corresponding to the TBP*)
special={1,0,0,-1/2,Sqrt[3]/2,-1/2,-Sqrt[3]/2};
(*suitable second derivative*)
a1=D[A,VAR[[i]],VAR[[j]]];
{x0,x1,y1,x2,y2,x3,y3}={1,0,0,-1/2,Sqrt[3]/2,-1/2,-Sqrt[3]/2};
a2=a1;
Clear[x0,x1,y1,x2,y2,x3,y3];
a2)

(*Now we imput the file that does the modified cholesky decomposition*)
<<cholesky

(*Now we compute the Hessian Matrix and check that the lowest eigenvalue
    exceeds 1/10*)

(*The exponent 1*)
c1=Table[Table[D2[i,j,1],{i,1,7}],{j,1,7}]
Print["test 1 ",CheckLowestEigenvalue[c1,1/10]]

c2=Table[Table[D2[i,j,2],{i,1,7}],{j,1,7}]
Print["test 2 ",CheckLowestEigenvalue[c2,1/10]]