1function T = reduced_rank_cholesky(X)
2% Computes the cholesky decomposition of a symetric semidefinite matrix or of a definite positive matrix.
3
4%@info:
5%! @deftypefn {Function File} { @var{T} =} reduced_rank_cholesky (@var{X})
6%! @anchor{reduced_rank_cholesky}
7%! @sp 1
8%! Computes the cholesky decomposition of a symetric semidefinite matrix or of a definite positive matrix.
9%! @sp 2
10%! @strong{Inputs}
11%! @sp 1
12%! @table @ @var
13%! @item X
14%! n*n matrix of doubles to be factorized (X is supposed to be semidefinite positive).
15%! @end table
16%! @sp 2
17%! @strong{Outputs}
18%! @sp 1
19%! @table @ @var
20%! @item T
21%! q*n matrix of doubles such that T'*T = X, where q is the number of positive eigenvalues in X.
22%! @end table
23%! @sp 2
24%! @strong{Remarks}
25%! @sp 1
26%! [1] If X is not positive definite, then X has to be a symetric semidefinite matrix.
27%! @sp 1
28%! [2] The matrix T is upper triangular iff X is positive definite.
29%! @sp 2
30%! @strong{This function is called by:}
31%! @sp 1
32%! @ref{particle/sequential_importance_particle_filter}
33%! @sp 2
34%! @strong{This function calls:}
35%! @sp 2
36%! @end deftypefn
37%@eod:
38
39% Copyright (C) 2009-2017 Dynare Team
40%
41% This file is part of Dynare.
42%
43% Dynare is free software: you can redistribute it and/or modify
44% it under the terms of the GNU General Public License as published by
45% the Free Software Foundation, either version 3 of the License, or
46% (at your option) any later version.
47%
48% Dynare is distributed in the hope that it will be useful,
49% but WITHOUT ANY WARRANTY; without even the implied warranty of
50% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
51% GNU General Public License for more details.
52%
53% You should have received a copy of the GNU General Public License
54% along with Dynare.  If not, see <http://www.gnu.org/licenses/>.
55
56[T,X_is_not_positive_definite] = chol(X);
57
58if X_is_not_positive_definite
59    n = length(X);
60    [U,D] = eig(X);
61    [tmp,max_elements_indices] = max(abs(U),[],1);
62    negloc = (U(max_elements_indices+(0:n:(n-1)*n))<0);
63    U(:,negloc) = -U(:,negloc);
64    D = diag(D);
65    tol = sqrt(eps(max(D))*length(D)*10);
66    t = (abs(D) > tol);
67    D = D(t);
68    if ~(sum(D<0))
69        T = diag(sqrt(D))*U(:,t)';
70    else
71        disp('reduced_rank_cholesky:: Input matrix is not semidefinite positive!')
72        T = NaN;
73    end
74end
75
76%@test:1
77%$ n = 10;
78%$ m = 100;
79%$
80%$ X = randn(n,m);
81%$ X = X*X';
82%$
83%$ t = ones(2,1);
84%$
85%$ try
86%$    T = reduced_rank_cholesky(X);
87%$ catch
88%$    t(1) = 0;
89%$    T = all(t);
90%$    return
91%$ end
92%$
93%$
94%$ % Check the results.
95%$ t(2) = dassert(T,chol(X),1e-16);
96%$ T = all(t);
97%@eof:1