153 lines
4.1 KiB
Matlab
153 lines
4.1 KiB
Matlab
function [X, info] = cycle_reduction(A0, A1, A2, cvg_tol, ch)
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%@info:
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%! @deftypefn {Function File} {[@var{X}, @var{info}] =} cycle_reduction (@var{A0},@var{A1},@var{A2},@var{cvg_tol},@var{ch})
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%! @anchor{cycle_reduction}
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%! @sp 1
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%! Solves the quadratic matrix equation A2*X^2 + A1*X + A0 = 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 A0
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%! Square matrix of doubles, n*n.
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%! @item A1
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%! Square matrix of doubles, n*n.
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%! @item A2
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%! Square matrix of doubles, n*n.
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%! @item cvg_tol
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%! Scalar double, tolerance parameter.
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%! @item ch
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%! Any matlab object, if not empty 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 X
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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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%! D.A. Bini, G. Latouche, B. Meini (2002), "Solving matrix polynomial equations arising in queueing problems", Linear Algebra and its Applications 340, pp. 222-244
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%! @sp 1
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%! D.A. Bini, B. Meini (1996), "On the solution of a nonlinear matrix equation arising in queueing problems", SIAM J. Matrix Anal. Appl. 17, pp. 906-926.
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%! @sp 2
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%! @end deftypefn
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%@eod:
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% Copyright © 2013-2023 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 <https://www.gnu.org/licenses/>.
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max_it = 300;
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it = 0;
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info = 0;
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X = [];
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crit = Inf;
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A0_0 = A0;
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Ahat1 = A1;
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if (nargin == 5 && ~isempty(ch) )
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A1_0 = A1;
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A2_0 = A2;
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end
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n = length(A0);
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id0 = 1:n;
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id2 = id0+n;
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cont = 1;
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while cont
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tmp = ([A0; A2]/A1)*[A0 A2];
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A1 = A1 - tmp(id0,id2) - tmp(id2,id0);
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A0 = -tmp(id0,id0);
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A2 = -tmp(id2,id2);
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Ahat1 = Ahat1 -tmp(id2,id0);
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crit = norm(A0,1);
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if crit < cvg_tol
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% keep iterating until condition on A2 is met
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if norm(A2,1) < cvg_tol
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cont = 0;
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end
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elseif it == max_it
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info(1) = 401;
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info(2) = log(norm(A1,1));
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return
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elseif isnan(crit)
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info(1) = 402;
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info(2) = log(norm(A1,1))
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return
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end
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it = it + 1;
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end
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X = -Ahat1\A0_0;
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if (nargin == 5 && ~isempty(ch) )
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%check the solution
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res = norm(A0_0 + A1_0 * X + A2_0 * X * X, 1);
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if (res > cvg_tol)
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info(1) = 403
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info(2) = log(res)
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dprintf('The norm of the residual is %s whereas the tolerance criterion is %s', num2str(res), num2str(cvg_tol));
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end
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end
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return % --*-- Unit tests --*--
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%@test:1
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t = zeros(3,1);
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% Set the dimension of the problem to be solved.
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n = 500;
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% Set the equation to be solved
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A = eye(n);
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B = diag(30*ones(n,1)); B(1,1) = 20; B(end,end) = 20; B = B - diag(10*ones(n-1,1),-1); B = B - diag(10*ones(n-1,1),1);
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C = diag(15*ones(n,1)); C = C - diag(5*ones(n-1,1),-1); C = C - diag(5*ones(n-1,1),1);
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% Solve the equation with the cycle reduction algorithm
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try
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tic; X1 = cycle_reduction(C,B,A,1e-7); elapsedtime = toc;
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disp(['Elapsed time for cycle reduction algorithm is: ' num2str(elapsedtime) ' (n=' int2str(n) ').'])
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t(1) = 1;
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catch
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% nothing to do here.
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end
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% Solve the equation with the logarithmic reduction algorithm
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try
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tic; X2 = logarithmic_reduction(A,B,C,1e-16,100); elapsedtime = toc;
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disp(['Elapsed time for logarithmic reduction algorithm is: ' num2str(elapsedtime) ' (n=' int2str(n) ').'])
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t(2) = 1;
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catch
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% nothing to do here.
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end
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% Check the results.
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if t(1) && t(2)
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t(3) = dassert(X1,X2,1e-12);
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end
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T = all(t);
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%@eof:1 |