Added logarithmic reduction algorithm to solve quadratic matrix equation.
This algorithm is a slower alternative to the cyclic reduction algorithm (useful for testing purpose).time-shift
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function [X1, info] = logarithmic_reduction(A,B,C,tol,maxit,check)
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%@info:
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%! @deftypefn {Function File} {[@var{X1}, @var{info}] =} logarithmic_reduction (@var{A},@var{B},@var{C},@var{tol},@var{maxit},@var{check})
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%! @anchor{logarithmic_reduction}
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%! @sp 1
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%! Solves the quadratic matrix equation AX^2 + BX + C = 0.
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%! @sp 2
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%! @strong{Inputs}
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%! @sp 1
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%! @table @ @var
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%! @item A
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%! Square matrix of doubles, n*n.
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%! @item B
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%! Square matrix of doubles, n*n.
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%! @item C
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%! Square matrix of doubles, n*n.
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%! @item tol
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%! Scalar double, tolerance parameter.
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%! @item maxit
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%! Scalar integer, maximum number of iterations.
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%! @item check
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%! Scalar integer, if non zero the solution is checked.
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%! @end table
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%! @sp 1
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%! @strong{Outputs}
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%! @sp 1
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%! @table @ @var
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%! @item X1
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%! Square matrix of doubles, n*n, solution of the matrix equation.
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%! @item info
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%! Scalar integer, if nonzero the algorithm failed in finding the solution of the matrix equation.
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%! @end table
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%! @sp 2
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%! @strong{This function is called by:}
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%! @sp 2
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%! @strong{This function calls:}
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%! @sp 2
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%! @strong{References:}
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%! @sp 1
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%! G. Latouche and V. Ramaswami (1993), "A logarithmic reduction algorithm for Quasi-Birth-Death processes", in Journal of Applied Probability, Vol. 30, No. 3, pp. 650-674.
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%! @sp 2
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%! @end deftypefn
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%@eod:
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% Copyright (C) 2012 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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info = 0;
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n = length(A);
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i = 1:n;
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tmp0 = -B\[A,C];
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X0 = tmp0(:,n+i);
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U0 = tmp0(:,i);
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kk = 0;
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cc = 1;
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while cc>tol && kk<=maxit
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tmp1 = (eye(n)-tmp0*[tmp0(:,n+i);tmp0(:,i)])\[tmp0(:,i)*tmp0(:,i),tmp0(:,n+i)*tmp0(:,n+i)];
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X1 = X0 + U0*tmp1(:,n+i);
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U1 = U0*tmp1(:,i);
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cc = max(max(abs(X1-X0)));
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X0 = X1; U0 = U1;
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tmp0 = tmp1;
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kk = kk+1;
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end
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if kk==maxit
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disp(['logarithmic_reduction:: Convergence not achieved after ' int2str(maxit) ' iterations!']);
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info = 1;
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end
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if nargin>5 && check
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if max(max(abs(A*X1*X1 + B*X1 + C)))>tol
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disp(['logarithmic_reduction:: Algotithm did not converge to the solution of the matrix quadratic equation!']);
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info = 1;
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end
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end
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