% Copyright (C) 2001 Michel Juillard
%
% computes second order partial derivatives
% uses Abramowitz and Stegun (1965) formulas 25.3.24 and 25.3.27 p. 884
function jacob2_=jacob2(func,x)
global gstep_

func = str2func(func);
n=size(x,1);
%h1=max(abs(x),gstep_*ones(n,1))*eps^(1/3);
h1=max(abs(x),sqrt(gstep_)*ones(n,1))*eps^(1/6);
h_1=h1;
xh1=x+h1;
h1=xh1-x;
xh1=x-h_1;
h_1=x-xh1;
xh1=x;

f0=feval(func,x);
f1=zeros(size(f0,1),n);
f_1=f1;

for i=1:n
	xh1(i)=x(i)+h1(i);
	f1(:,i)=feval(func,xh1);
	xh1(i)=x(i)-h_1(i);
	f_1(:,i)=feval(func,xh1);
	xh1(i)=x(i);
	i=i+1;
end
xh_1=xh1;
jacob2_=spalloc(size(f0,1),n*n,3*n);
for i=1:n
	if i > 1
		k=[i:n:n*(i-1)];
		jacob2_(:,(i-1)*n+1:(i-1)*n+i-1)=jacob2_(:,k);
	end	
	jacob2_(:,(i-1)*n+i)=sparse((f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i)));
	temp=f1+f_1-f0*ones(1,n);
        for j=i+1:n
		xh1(i)=x(i)+h1(i);
		xh1(j)=x(j)+h_1(j);
		xh_1(i)=x(i)-h1(i);
		xh_1(j)=x(j)-h_1(j);
		jacob2_(:,(i-1)*n+j)=sparse(-(-feval(func,xh1)-feval(func,xh_1)+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j)));
		xh1(i)=x(i);
		xh1(j)=x(j);
		xh_1(i)=x(i);
		xh_1(j)=x(j);
		j=j+1;
	end
	i=i+1;
end

% 10/03/02 MJ used the 7 points formula

