152 lines
4.2 KiB
Matlab
152 lines
4.2 KiB
Matlab
function hessian_mat = hessian(func,x, gstep, varargin)
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% Computes second order partial derivatives
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%
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% INPUTS
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% func [string] name of the function
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% x [double] vector, the Hessian of "func" is evaluated at x.
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% gstep [double] scalar, size of epsilon.
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% varargin [void] list of additional arguments for "func".
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%
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% OUTPUTS
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% hessian_mat [double] Hessian matrix
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%
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% ALGORITHM
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% Uses Abramowitz and Stegun (1965) formulas 25.3.23
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% \[
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% \frac{\partial^2 f_{0,0}}{\partial {x^2}} = \frac{1}{h^2}\left( f_{1,0} - 2f_{0,0} + f_{ - 1,0} \right)
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% \]
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% and 25.3.27 p. 884
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%
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% \[
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% \frac{\partial ^2f_{0,0}}{\partial x\partial y} = \frac{-1}{2h^2}\left(f_{1,0} + f_{-1,0} + f_{0,1} + f_{0,-1} - 2f_{0,0} - f_{1,1} - f_{-1,-1} \right)
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% \]
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%
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% SPECIAL REQUIREMENTS
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% none
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%
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% Copyright © 2001-2023 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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if ~isa(func, 'function_handle')
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func = str2func(func);
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end
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n = size(x,1);
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h1 = max(abs(x), sqrt(gstep(1))*ones(n, 1))*eps^(1/6)*gstep(2);
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h_1 = h1;
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xh1 = x+h1;
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h1 = xh1-x;
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xh1 = x-h_1;
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h_1 = x-xh1;
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xh1 = x;
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f0 = feval(func, x, varargin{:});
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f1 = zeros(size(f0, 1), n);
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f_1 = f1;
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for i=1:n
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%do step up
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xh1(i) = x(i)+h1(i);
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f1(:,i) = feval(func, xh1, varargin{:});
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%do step down
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xh1(i) = x(i)-h_1(i);
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f_1(:,i) = feval(func, xh1, varargin{:});
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%reset parameter
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xh1(i) = x(i);
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end
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xh_1 = xh1;
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temp = f1+f_1-f0*ones(1, n); %term f_(1,0)+f_(-1,0)-f_(0,0) used later
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hessian_mat = zeros(size(f0,1), n*n);
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for i=1:n
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if i > 1
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%fill symmetric part of Hessian based on previously computed results
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k = i:n:n*(i-1);
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hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1) = hessian_mat(:,k);
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end
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hessian_mat(:,(i-1)*n+i) = (f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i)); %formula 25.3.23
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for j=i+1:n
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%step in up direction
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xh1(i) = x(i)+h1(i);
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xh1(j) = x(j)+h_1(j);
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%step in down direction
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xh_1(i) = x(i)-h1(i);
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xh_1(j) = x(j)-h_1(j);
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hessian_mat(:,(i-1)*n+j) =-(-feval(func, xh1, varargin{:})-feval(func, xh_1, varargin{:})+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j)); %formula 25.3.27
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%reset grid points
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xh1(i) = x(i);
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xh1(j) = x(j);
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xh_1(i) = x(i);
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xh_1(j) = x(j);
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end
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end
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return % --*-- Unit tests --*--
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%@test:1
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% Create a function.
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fid = fopen('exfun.m','w+');
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fprintf(fid,'function [f,g,H] = exfun(xvar)\\n');
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fprintf(fid,'x = xvar(1);\\n');
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fprintf(fid,'y = xvar(2);\\n');
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fprintf(fid,'f = x^2* log(y);\\n');
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fprintf(fid,'if nargout>1\\n');
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fprintf(fid,' g = zeros(2,1);\\n');
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fprintf(fid,' g(1) = 2*x*log(y);\\n');
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fprintf(fid,' g(2) = x*x/y;\\n');
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fprintf(fid,'end\\n');
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fprintf(fid,'if nargout>2\\n');
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fprintf(fid,' H = zeros(2,2);\\n');
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fprintf(fid,' H(1,1) = 2*log(y);\\n');
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fprintf(fid,' H(1,2) = 2*x/y;\\n');
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fprintf(fid,' H(2,1) = H(1,2);\\n');
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fprintf(fid,' H(2,2) = -x*x/(y*y);\\n');
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fprintf(fid,' H = H(:);\\n');
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fprintf(fid,'end\\n');
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fclose(fid);
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rehash;
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t = zeros(5,1);
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% Evaluate the Hessian at (1,e)
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try
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H = hessian('exfun',[1; exp(1)],[1e-2; 1]);
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t(1) = 1;
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catch
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t(1) = 0;
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end
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% Compute the true Hessian matrix
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[f, g, Htrue] = exfun([1 exp(1)]);
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% Delete exfun routine from disk.
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delete('exfun.m');
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% Compare the values in H and Htrue
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if t(1)
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t(2) = dassert(abs(H(1)-Htrue(1))<1e-6,true);
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t(3) = dassert(abs(H(2)-Htrue(2))<1e-6,true);
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t(4) = dassert(abs(H(3)-Htrue(3))<1e-6,true);
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t(5) = dassert(abs(H(4)-Htrue(4))<1e-6,true);
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end
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T = all(t);
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%@eof:1
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