95 lines
3.1 KiB
Matlab
95 lines
3.1 KiB
Matlab
function hessian_mat = penalty_hessian(func,x,penalty,gstep,varargin) % --*-- Unitary tests --*--
|
|
|
|
% Computes second order partial derivatives with penalty_objective_function
|
|
%
|
|
% INPUTS
|
|
% func [string] name of the function
|
|
% x [double] vector, the Hessian of "func" is evaluated at x.
|
|
% penalty [double] penalty base used if function fails
|
|
% gstep [double] scalar, size of epsilon.
|
|
% varargin [void] list of additional arguments for "func".
|
|
%
|
|
% OUTPUTS
|
|
% hessian_mat [double] Hessian matrix
|
|
%
|
|
% ALGORITHM
|
|
% Uses Abramowitz and Stegun (1965) formulas 25.3.23
|
|
% \[
|
|
% \frac{\partial^2 f_{0,0}}{\partial {x^2}} = \frac{1}{h^2}\left( f_{1,0} - 2f_{0,0} + f_{ - 1,0} \right)
|
|
% \]
|
|
% and 25.3.27 p. 884
|
|
%
|
|
% \[
|
|
% \frac{\partial ^2f_{0,0}}{\partial x\partial y} = \frac{-1}{2h^2}\left(f_{1,0} + f_{-1,0} + f_{0,1} + f_{0,-1} - 2f_{0,0} - f_{1,1} - f_{-1,-1} \right)
|
|
% \]
|
|
%
|
|
% SPECIAL REQUIREMENTS
|
|
% none
|
|
%
|
|
|
|
% Copyright (C) 2001-2014 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 <http://www.gnu.org/licenses/>.
|
|
|
|
if ~isa(func, 'function_handle')
|
|
func = str2func(func);
|
|
end
|
|
n=size(x,1);
|
|
h1=max(abs(x),sqrt(gstep(1))*ones(n,1))*eps^(1/6)*gstep(2);
|
|
h_1=h1;
|
|
xh1=x+h1;
|
|
h1=xh1-x;
|
|
xh1=x-h_1;
|
|
h_1=x-xh1;
|
|
xh1=x;
|
|
f0=penalty_objective_function(x,func,penalty,varargin{:});
|
|
f1=zeros(size(f0,1),n);
|
|
f_1=f1;
|
|
for i=1:n
|
|
%do step up
|
|
xh1(i)=x(i)+h1(i);
|
|
f1(:,i)=penalty_objective_function(xh1,func,penalty,varargin{:});
|
|
%do step up
|
|
xh1(i)=x(i)-h_1(i);
|
|
f_1(:,i)=penalty_objective_function(xh1,func,penalty,varargin{:});
|
|
xh1(i)=x(i);%reset parameter
|
|
end
|
|
xh_1=xh1;
|
|
hessian_mat = zeros(size(f0,1),n*n);
|
|
temp=f1+f_1-f0*ones(1,n); %term f_(1,0)+f_(-1,0)-f_(0,0) used later
|
|
for i=1:n
|
|
if i > 1 %fill symmetric part of Hessian based on previously computed results
|
|
k=[i:n:n*(i-1)];
|
|
hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1)=hessian_mat(:,k);
|
|
end
|
|
hessian_mat(:,(i-1)*n+i)=(f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i)); %formula 25.3.23
|
|
for j=i+1:n
|
|
%step in up direction
|
|
xh1(i)=x(i)+h1(i);
|
|
xh1(j)=x(j)+h_1(j);
|
|
%step in down direction
|
|
xh_1(i)=x(i)-h1(i);
|
|
xh_1(j)=x(j)-h_1(j);
|
|
hessian_mat(:,(i-1)*n+j)=-(-penalty_objective_function(xh1,func,penalty,varargin{:})-penalty_objective_function(xh_1,func,penalty,varargin{:})+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j)); %formula 25.3.27
|
|
%reset grid points
|
|
xh1(i)=x(i);
|
|
xh1(j)=x(j);
|
|
xh_1(i)=x(i);
|
|
xh_1(j)=x(j);
|
|
end
|
|
end
|
|
|