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% STK_CHOLCOV [STK internal]
%
% CALL: C = stk_cholcov (A, ...)
%
% returns the result of chol (A, ...) when this succeeds. If chol fails,
% then a small amount of "regularization noise" is added to the diagonal
% of A, in order to make chol succeed (see the code for details).
%
% NOTE: why this function ?
%
% This is a first (rough) attempt at solving numerical problems that
% arise when chol is used with a covariance matrix that is semi-positive
% definite, or positive definite with some very small eigenvalues. See
% tickets #3, #4 and #13 on Sourceforge:
%
% https://sourceforge.net/p/kriging/tickets/3/
% https://sourceforge.net/p/kriging/tickets/4/
% https://sourceforge.net/p/kriging/tickets/13/
%
% See also: chol
% Copyright Notice
%
% Copyright (C) 2018, 2019 CentraleSupelec
% Copyright (C) 2014 SUPELEC
%
% Author: Julien Bect <julien.bect@centralesupelec.fr>
% Copying Permission Statement
%
% This file is part of
%
% STK: a Small (Matlab/Octave) Toolbox for Kriging
% (http://sourceforge.net/projects/kriging)
%
% STK 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.
%
% STK 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 STK. If not, see <http://www.gnu.org/licenses/>.
function [C, epsi] = stk_cholcov (A, varargin)
% try to use a plain "chol"
[C, p] = chol (A, varargin{:});
if p == 0 % OK, A is (numerically) positive definite
epsi = 0;
else
epsi = eps;
u = diag (diag (A));
while p > 0
if epsi > 1 % avoids infinite while loops
if ~ all (isfinite (A(:)))
errmsg = 'A contains NaNs or Infs.';
else
errmsg = 'A is not even close to positive definite';
end
stk_error (errmsg, 'InvalidArgument');
end
epsi = epsi * 10;
[C, p] = chol (A + epsi * u, varargin{:});
end
warning ('STK:stk_cholcov:AddingRegularizationNoise', sprintf ...
('Adding a little bit of noise to help chol succeed (epsi = %.2e)', epsi));
end
end % function
%!shared Q, K, L, U, epsi
%! Q = 0.25 * hadamard(4);
%!test
%! K = Q * diag ([1, 0.1, 0.01, 1e-7]) * Q';
%! [U, epsi] = stk_cholcov (K);
%!assert (istriu (U))
%!assert (epsi == 0)
%!test
%! K = Q * diag ([1, 0.1, 0.01, 1e-7]) * Q';
%! [L, epsi] = stk_cholcov (K, 'lower');
%!assert (istril (L))
%!assert (epsi == 0)
%!test
%! K = Q * diag ([1, 0.1, 0.01, -1e-7]) * Q';
%! [U, epsi] = stk_cholcov (K);
%!assert (istriu (U))
%!assert (epsi > 0)
%!test
%! K = Q * diag ([1, 0.1, 0.01, -1e-7]) * Q';
%! [L, epsi] = stk_cholcov (K, 'lower');
%!assert (istril (L))
%!assert (epsi > 0)
%#ok<*SPWRN>
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