89 lines
3.5 KiB
Matlab
89 lines
3.5 KiB
Matlab
function [DP6,DP6inv] = Q6_plication(p)
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% Computes the 6-way duplication Matrix DP6 (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(kron(kron(x,x),x,x),x),x)=DP6*z
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% where z is of dimension np=p*(p+1)*(p+2)*(p+3)*(p+4)*(p+5)/(1*2*3*4*5*6)
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% and is obtained from y by removing each second and later occurence of the
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% same element. 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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% * DP6 [p^6 by np] 6-way duplication matrix
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% * DP6inv [np by np] Moore-Penrose inverse of DP6
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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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% * binom_coef (embedded)
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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)*(p+4)*(p+5)/(1*2*3*4*5*6);
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DP6 = spalloc(p^6,np,p^6);
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counti=1;
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for i1=1:p
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for i2=i1:p
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for i3=i2:p
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for i4=i3:p
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for i5=i4:p
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for i6=i5:p
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idx = uperm([i6 i5 i4 i3 i2 i1]);
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for r = 1:size(idx,1)
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ii1 = idx(r,1); ii2= idx(r,2); ii3=idx(r,3); ii4=idx(r,4); ii5=idx(r,5); ii6=idx(r,6);
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n = ii1 + (ii2-1)*p + (ii3-1)*p^2 + (ii4-1)*p^3 + (ii5-1)*p^4 + (ii6-1)*p^5;
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m = mue(p,i6,i5,i4,i3,i2,i1);
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DP6(n,m)=1;
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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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end
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end
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if nargout==2
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DP6inv = (transpose(DP6)*DP6)\transpose(DP6);
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end
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function m = mue(p,i1,i2,i3,i4,i5,i6)
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% Auxiliary expression, see page 122 of Meijer (2005)
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m = binom_coef(p,6,1) - binom_coef(p,1,i1+1) - binom_coef(p,2,i2+1) - binom_coef(p,3,i3+1) - binom_coef(p,4,i4+1) - binom_coef(p,5,i5+1) - binom_coef(p,6,i6+1);
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m = round(m);
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end
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function N = binom_coef(p,q,i)
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% Auxiliary expression for binomial coefficients, see page 119 of Meijer (2005)
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t = q; r =p+q-i;
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if t==0
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N=1;
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else
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N=1;
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for h = 0:(t-1)
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N = N*(r-h);
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end
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N=N/factorial(t);
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end
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end
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end |