92 lines
3.8 KiB
Matlab
92 lines
3.8 KiB
Matlab
function [QP,QPinv] = quadruplication(p)
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% Computes the Quadruplication Matrix QP (and its Moore-Penrose inverse)
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% such that for any p-dimensional vector x:
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% y=kron(kron(kron(x,x),x),x)=QP*z
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% where z is of dimension np=p*(p+1)*(p+2)*(p+3)/2 and is obtained from y
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% by removing each second and later occurence of the same element.
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% This is a generalization of the Duplication matrix.
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% Reference: Meijer (2005) - Matrix algebra for higher order moments.
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% Linear Algebra and its Applications, 410,pp. 112-134
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% =========================================================================
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% INPUTS
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% * p [integer] size of vector
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% -------------------------------------------------------------------------
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% OUTPUTS
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% * QP [p^4 by np] Quadruplication matrix
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% * QPinv [np by np] Moore-Penrose inverse of QP
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% -------------------------------------------------------------------------
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% This function is called by
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% * pruned_state_space_system.m
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% -------------------------------------------------------------------------
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% This function calls
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% * mue (embedded)
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% * uperm
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% =========================================================================
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% Copyright © 2020 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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% =========================================================================
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np = p*(p+1)*(p+2)*(p+3)/24;
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QP = spalloc(p^4,np,p^4);
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if nargout > 1
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QPinv = spalloc(np,np,p^4);
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end
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counti = 1;
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for l=1:p
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for k=l:p
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for j=k:p
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for i=j:p
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idx = pruned_SS.uperm([i j k l]);
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for r = 1:size(idx,1)
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ii = idx(r,1); jj= idx(r,2); kk=idx(r,3); ll=idx(r,4);
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n = ii + (jj-1)*p + (kk-1)*p^2 + (ll-1)*p^3;
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m = mue(p,i,j,k,l);
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QP(n,m)=1;
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if nargout > 1
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if i==j && j==k && k==l
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QPinv(m,n)=1;
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elseif i==j && j==k && k>l
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QPinv(m,n)=1/4;
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elseif i>j && j==k && k==l
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QPinv(m,n)=1/4;
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elseif i==j && j>k && k==l
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QPinv(m,n) = 1/6;
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elseif i>j && j>k && k==l
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QPinv(m,n) = 1/12;
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elseif i>j && j==k && k>l
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QPinv(m,n) = 1/12;
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elseif i==j && j>k && k>l
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QPinv(m,n) = 1/12;
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elseif i>j && j>k && k>l
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QPinv(m,n) = 1/24;
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end
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end
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end
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counti = counti+1;
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end
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end
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end
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end
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%QPinv = (transpose(QP)*QP)\transpose(QP);
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function m = mue(p,i,j,k,l)
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% Auxiliary expression, see page 118 of Meijer (2005)
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m = i + (j-1)*p + 1/2*(k-1)*p^2 + 1/6*(l-1)*p^3 - 1/2*j*(j-1) + 1/6*k*(k-1)*(k-2) - 1/24*l*(l-1)*(l-2)*(l-3) - 1/2*(k-1)^2*p + 1/6*(l-1)^3*p - 1/4*(l-1)*(l-2)*p^2 - 1/4*l*(l-1)*p + 1/6*(l-1)*p;
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m = round(m);
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end
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end |