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function d=hess_element(func,element1,element2,args)
% function d=hess_element(func,element1,element2,args)
% returns an entry of the finite differences approximation to the hessian of func
%
% INPUTS
% func [function name] string with name of the function
% element1 [int] the indices showing the element within the hessian that should be returned
% element2 [int]
% args [cell array] arguments provided to func
%
% OUTPUTS
% d [double] the (element1,element2) entry of the hessian
%
% SPECIAL REQUIREMENTS
% none
% Copyright © 2010-2020 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
assert(element1 <= length(args) && element2 <= length(args));
func = str2func(func);
h=1e-6;
p10 = args;
p01 = args;
m10 = args;
m01 = args;
p11 = args;
m11 = args;
p10{element1} = p10{element1} + h;
m10{element1} = m10{element1} - h;
p11{element1} = p11{element1} + h;
m11{element1} = m11{element1} - h;
p01{element2} = p01{element2} + h;
m01{element2} = m01{element2} - h;
p11{element2} = p11{element2} + h;
m11{element2} = m11{element2} - h;
% From Abramowitz and Stegun. Handbook of Mathematical Functions (1965)
% formulas 25.3.24 and 25.3.27 p. 884
if element1==element2
d = (16*func(p10{:})...
+16*func(m10{:})...
-30*func(args{:})...
-func(p11{:})...
-func(m11{:}))/(12*h^2);
else
d = (func(p10{:})...
+func(m10{:})...
+func(p01{:})...
+func(m01{:})...
-2*func(args{:})...
-func(p11{:})...
-func(m11{:}))/(-2*h^2);
end
end
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