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function grdd = fn_gradcd(fcn,x0,grdh,P1,P2,P3,P4,P5,P6,P7,P8,P9,...
P10,P11,P12,P13,P14,P15,P16,P17,P18,P19,P20)
%function grdd = fn_gradcd(fcn,x0,grdh,P1,P2,P3,P4,P5,P6,P7,P8,P9,...
% P10,P11,P12,P13,P14,P15,P16,P17,P18,P19,P20)
% computes numerical gradient of a single-valued function or Jacobian
% matrix of a vector-valued function using a central difference with
% function grdd = gradcd(fcn,x0,grdh)
%
% fcn: a string naming a vector-valued function (f:n-by-1 -> k-by-1).
% x0: a column vector n-by-1, at which point the hessian is evaluated.
% grdh: step size, n*1. Set as follows
% step = eps^(1/3);
% %step = 1e-04;
% grdh = step * (max([abs(x) ones(length(x),1)]'))' .* (abs(x) ./ x);
%--------------------
% grdd: n-by-k Jacobian matrix (gradients).
%
% Written by Tao Zha, 2002.
%
% 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
n = length(x0);
k = length(f0); % dimension of "fcn"
% ** 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 ones(n,1)]'))' .* dax0;
end
xdh = x0 + dh;
dh = xdh - x0; % This increases precision slightly
%
argplus = x0(:,ones(n,1));
argminus = argplus;
dnum = 1:n+1:n^2; % positions of diagonals in vec(argplus).
argplus(dnum) = xdh; % replace the diagonals of "argplus" by "xdh".
argminus(dnum) = x0-dh; % replace the diagonals of "argplus" by "xdh".
grdd = zeros(k,n); % preallocate to speed the loop.
i = 0;
while i ~= n
i = i+1;
fp = eval([fcn '(argplus(:,i)' tailstr]);
fm = eval([fcn '(argminus(:,i)' tailstr]);
grdd(:,i) = fp - fm;
end
dhm = dh(:,ones(k,1));
dhm = dhm'; % k*n
grdd = grdd ./ (2*dhm); % k-by-n.
grdd = grdd'; % n-by-k.
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