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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 © 2010-2025 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
return % --*-- Unit tests --*--
%@test:1
% Test polynomial function: f(x,y) = x^3 + 2*x^2*y + 3*y^2 + 4*x + 5*y + 6
% Analytical derivatives:
% df/dx = 3*x^2 + 4*x*y + 4
% df/dy = 2*x^2 + 6*y + 5
% d2f/dx2 = 6*x + 4*y
% d2f/dy2 = 6
% d2f/dxdy = d2f/dydx = 4*x
% Test at point (x,y) = (2, 3)
x0 = 2;
y0 = 3;
% Analytical Hessian at (2,3):
% H = [ 6*2 + 4*3, 4*2 ] = [ 24, 8 ]
% [ 4*2, 6 ] [ 8, 6 ]
try
h11 = hess_element('test_poly_2vars', 1, 1, {x0, y0});
h22 = hess_element('test_poly_2vars', 2, 2, {x0, y0});
h12 = hess_element('test_poly_2vars', 1, 2, {x0, y0});
h21 = hess_element('test_poly_2vars', 2, 1, {x0, y0});
t(1) = true;
catch
t(1) = false;
end
if t(1)
t(2) = abs(h11 - 24) < 5e-2;
t(3) = abs(h22 - 6) < 5e-2;
t(4) = abs(h12 - 8) < 5e-2;
t(5) = abs(h21 - 8) < 5e-2;
t(6) = abs(h12 - h21) < 1e-10; % Verify symmetry
end
T = all(t);
%@eof:1
%@test:2
% Test at origin (0,0) for simpler verification
x0 = 0;
y0 = 0;
% Analytical Hessian at (0,0):
% H = [ 0, 0 ]
% [ 0, 6 ]
t=false(5,1);
try
h11 = hess_element('test_poly_2vars', 1, 1, {x0, y0});
h12 = hess_element('test_poly_2vars', 1, 2, {x0, y0});
h21 = hess_element('test_poly_2vars', 2, 1, {x0, y0});
h22 = hess_element('test_poly_2vars', 2, 2, {x0, y0});
t(1) = true;
catch
t(1) = false;
end
if t(1)
t(2) = abs(h11 - 0) < 1e-4;
t(3) = abs(h12 - 0) < 1e-3;
t(4) = abs(h21 - 0) < 1e-3;
t(5) = abs(h22 - 6) < 1e-3;
end
T = all(t);
%@eof:2
%@test:3
% Test negative values
x0 = -1;
y0 = -2;
% Analytical Hessian at (-1,-2):
% H = [ 6*(-1) + 4*(-2), 4*(-1) ] = [ -14, -4 ]
% [ 4*(-1), 6 ] [ -4, 6 ]
t=false(5,1);
try
h11 = hess_element('test_poly_2vars', 1, 1, {x0, y0});
h12 = hess_element('test_poly_2vars', 1, 2, {x0, y0});
h21 = hess_element('test_poly_2vars', 2, 1, {x0, y0});
h22 = hess_element('test_poly_2vars', 2, 2, {x0, y0});
t(1) = true;
catch
t(1) = false;
end
if t(1)
t(2) = abs(h11 - (-14)) < 5e-3;
t(3) = abs(h12 - (-4)) < 1e-3;
t(4) = abs(h21 - (-4)) < 1e-3;
t(5) = abs(h22 - 6) < 1e-3;
end
T = all(t);
%@eof:3
function f = test_poly_2vars(x, y)
% Test polynomial: f(x,y) = x^3 + 2*x^2*y + 3*y^2 + 4*x + 5*y + 6
f = x^3 + 2*x^2*y + 3*y^2 + 4*x + 5*y + 6;
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