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function grdd = fn_hesscd(fcn,x0,grdh,P1,P2,P3,P4,P5,P6,P7,P8,P9,...
P10,P11,P12,P13,P14,P15,P16,P17,P18,P19,P20)
% Computing numerical hessian using a central difference with
% function grdd = hesscd(fcn,x0,grdh,Passed variables1)
%
% fcn: a string naming the objective function.
% x0: a column vector n*1, at which point the hessian is evaluated.
% grdh: step size.
% grdd: hessian matrix (second derivative), n*n.
%
% Written by Tao Zha, May 1998. Revised June 1002.
%
% Copyright (C) 1997-2012 Tao Zha
%
% This 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.
%
% It 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.
%
% If you did not received a copy of the GNU General Public License
% with this software, see <http://www.gnu.org/licenses/>.
%
stps = eps^(1/3);
% eps: floating point relative accuracy or machine precision: 2.22e-16
% stps: step size recommended by Dennis and Schnabel: 6.006e-6
x0 = x0(:);
tailstr = ')';
for i=nargin-3:-1:1
tailstr=[ ',P' num2str(i) tailstr];
end
f0 = eval([fcn '(x0' tailstr]);
% ** initializations
k = length(x0);
grdd = zeros(k);
% ** Computation of stepsize (dh)
if all(grdh)
dh = grdh;
else
ax0 = abs(x0);
if all(x0)
dax0 = x0 ./ ax0;
else
dax0 = 1;
end
dh = stps * (max([ax0 (1e-2)*ones(k,1)]'))' .* dax0;
end
xdh = x0 + dh;
dh = xdh - x0; % This increases precision slightly
dhm = dh(:,ones(k,1));
ee = eye(k) .* dhm;
i = 1;
while i <= k
j = i;
while j <= k
fune1 = eval([fcn '(x0 + ee(:,i) + ee(:,j)' tailstr]);
fune2 = eval([fcn '(x0 - ee(:,i) + ee(:,j)' tailstr]);
fune3 = eval([fcn '(x0 + ee(:,i) - ee(:,j)' tailstr]);
fune4 = eval([fcn '(x0 - ee(:,i) - ee(:,j)' tailstr]);
grdd(i,j) = (fune1 - fune2 - fune3 + fune4) / (4 * dh(i) * dh(j));
if i ~= j
grdd(j,i) = grdd(i,j);
end
j = j+1;
end
i = i+1;
end
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