(**This file computes the modified Cholesky decomposition of a symmetric real matrix.
      We use this routine to show that the hessian of energy has lowest eigenvalue at least
      1/10, relative to the potential function 1/r and 1/r^2.*)


ModifiedCholesky[a_]:=(
len=Length[a];
l=Table[Table[0,{len}],{len}];
d=Table[0,{len}];
d[[1]]=a[[1,1]];
For[i=1,i<=len,i=i+1,l[[i,i]]=1];

For[j=1,j<=len,j=j+1, {
For[i=j+1,i<=len,i=i+1, {
l[[i,j]]=(a[[i,j]]-Sum[l[[i,k]] l[[j,k]] d[[k]],{k,1,j-1}])/d[[j]];
d[[i]]=a[[i,i]]-Sum[l[[i,k]] l[[i,k]] d[[k]],{k,1,i-1}];
}]}];
dd=Table[Table[0,{len}],{len}];
For[j=1,j<=len,j=j+1,dd[[j,j]]=d[[j]]];
{l,dd})


(*This is a check that a real symmetric matrix is positive definite.  
    1 means "yes" and 0 means "no".*)

CheckPositiveDefinite[A_]:=(
d1=ModifiedCholesky[A];
d2=d1[[2]];
(*Get the diagonal elements*)
d3=Table[d2[[j,j]],{j,1,Length[d2]}];
d4=Min[d3];
If[d4>0,1,0])

(*This checks if the lowest positive eigenvalue of a real symmetric matrix
    exceeds r*)

CheckLowestEigenvalue[A_,r_]:=(
AA=A-r IdentityMatrix[Length[A]];
CheckPositiveDefinite[AA])