Added routine performing reduced rank Cholesky factorization.

matlab/reduced_rank_cholesky.m

+function T = reduced_rank_cholesky(X)
+% Computes the cholesky decomposition of a symetric semidefinite matrix or of a definite positive matrix.
+
+%@info:
+%! @deftypefn {Function File} { @var{T} =} reduced_rank_cholesky (@var{X})
+%! @anchor{reduced_rank_cholesky}
+%! @sp 1
+%! Computes the cholesky decomposition of a symetric semidefinite matrix or of a definite positive matrix.
+%! @sp 2
+%! @strong{Inputs}
+%! @sp 1
+%! @table @ @var
+%! @item X
+%! n*n matrix of doubles to be factorized (X is supposed to be semidefinite positive).
+%! @end table
+%! @sp 2
+%! @strong{Outputs}
+%! @sp 1
+%! @table @ @var
+%! @item T
+%! q*n matrix of doubles such that T'*T = X, where q is the number of positive eigenvalues in X.
+%! @end table
+%! @sp 2
+%! @strong{Remarks}
+%! @sp 1
+%! [1] If X is not positive definite, then X has to be a symetric semidefinite matrix.
+%! @sp 1
+%! [2] The matrix T is upper triangular iff X is positive definite.
+%! @sp 2
+%! @strong{This function is called by:}
+%! @sp 1
+%! @ref{particle/sequential_importance_particle_filter}
+%! @sp 2
+%! @strong{This function calls:}
+%! @sp 2
+%! @end deftypefn
+%@eod:
+
+% Copyright (C) 2009, 2010, 2011 Dynare Team
+%
+% This file is part of Dynare.
+%
+% Dynare 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.
+%
+% Dynare 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 Dynare.  If not, see <http://www.gnu.org/licenses/>.    
+    
+[T,X_is_not_positive_definite] = chol(X);
+
+if X_is_not_positive_definite
+    n = length(X);
+    [U,D] = eig(X);
+    [tmp,max_elements_indices] = max(abs(U),[],1);
+    negloc = (U(max_elements_indices+(0:n:(n-1)*n))<0);
+    U(:,negloc) = -U(:,negloc);
+    D = diag(D);
+    tol = sqrt(eps(max(D))*length(D)*10);
+    t = (abs(D) > tol);
+    D = D(t);
+    if ~(sum(D<0))
+        T = diag(sqrt(D))*U(:,t)';
+    else
+        disp('reduced_rank_cholesky:: Input matrix is not semidefinite positive!')
+        T = NaN;
+    end
+end
+
+%@test:1
+%$ n = 10;
+%$ m = 100;
+%$
+%$ X = randn(n,m);
+%$ X = X*X';
+%$
+%$ t = ones(2,1);
+%$
+%$ try
+%$    T = reduced_rank_cholesky(X);
+%$ catch
+%$    t(1) = 0;
+%$    T = all(t);
+%$    return;
+%$ end
+%$
+%$
+%$ % Check the results.
+%$ t(2) = dyn_assert(T,chol(X),1e-16);
+%$ T = all(t);
+%@eof:1
\ No newline at end of file