271 lines
6.2 KiB
Matlab
271 lines
6.2 KiB
Matlab
function [nodes, weights] = cubature_with_gaussian_weight(d, n, method)
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% Computes nodes and weights for a n-order cubature with gaussian weight.
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%
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% INPUTS
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% - d [integer] scalar, dimension of the region of integration.
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% - n [integer] scalar, approximation order (3 or 5).
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% - method [string] Method of approximation ('Stroud' or 'ScaledUnscentedTransform')
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%
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% OUTPUTS
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% - nodes [double] n×m matrix, with m=2×d if n=3 or m=2×d²+1 if n=5, nodes where the integrated function has to be evaluated.
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% - weights [double] m×1 vector, weights associated to the nodes.
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%
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% REMARKS
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% The routine returns nodes and associated weights to compute a multivariate integral of the form:
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% ∞ -<x,x>
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% ∫ f(x) × e dx
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% -∞
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% Copyright © 2012-2023 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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% Set default.
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if nargin<3 || isempty(method)
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method = 'Stroud';
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end
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if strcmp(method,'Stroud') && isequal(n,3)
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r = sqrt(d);
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nodes = r*[eye(d),-eye(d)];
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weights = ones(2*d,1)/(2*d);
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return
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end
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if strcmp(method,'ScaledUnscentedTransform') && isequal(n,3)
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% For alpha=1 and beta=kappa=0 we obtain the same weights and nodes than the 'Stroud' method (with n=3).
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% For alpha=1, beta=0 and kappa=.5 we obtain sigma points with equal weights.
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alpha = 1;
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beta = 0;
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kappa = 0.5;
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lambda = (alpha^2)*(d+kappa) - d;
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nodes = [ zeros(d,1) ( sqrt(d+lambda).*([ eye(d), -eye(d)]) ) ];
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w0_m = lambda/(d+lambda);
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w0_c = w0_m + (1-alpha^2+beta);
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weights = [w0_c; .5/(d+lambda)*ones(2*d,1)];
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return
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end
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if strcmp(method,'Stroud') && isequal(n,5)
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r = sqrt((d+2));
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s = sqrt((d+2)/2);
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m = 2*d^2+1;
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A = 2/(n+2);
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B = (4-d)/(2*(n+2)^2);
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C = 1/(n+2)^2;
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% Initialize the outputs
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nodes = zeros(d,m);
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weights = zeros(m,1);
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% Set the weight for the first node (0)
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weights(1) = A;
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skip = 1;
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% Set the remaining nodes and associated weights.
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nodes(:,skip+(1:d)) = r*eye(d);
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weights(skip+(1:d)) = B;
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skip = skip+d;
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nodes(:,skip+(1:d)) = -r*eye(d);
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weights(skip+(1:d)) = B;
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skip = skip+d;
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for i=1:d-1
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for j = i+1:d
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nodes(:,skip+(1:4)) = s*ee(d,i,j);
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weights(skip+(1:4)) = C;
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skip = skip+4;
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end
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end
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return
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end
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if strcmp(method,'Stroud')
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error(['cubature_with_gaussian_weight:: Cubature (Stroud tables) is not yet implemented with n = ' int2str(n) '!'])
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end
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function v = e(n,i)
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v = zeros(n,1);
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v(i) = 1;
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function m = ee(n,i,j)
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m = zeros(n,4);
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m(:,1) = e(n,i)+e(n,j);
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m(:,2) = e(n,i)-e(n,j);
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m(:,3) = -m(:,2);
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m(:,4) = -m(:,1);
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return % --*-- Unit tests --*--
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%@test:1
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d = 4;
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t = zeros(5,1);
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try
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[nodes,weights] = cubature_with_gaussian_weight(d,3);
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t(1) = 1;
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catch
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t = t(1);
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T = all(t);
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end
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if t(1)
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m1 = nodes*weights;
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m2 = nodes.^2*weights;
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m3 = nodes.^3*weights;
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m4 = nodes.^4*weights;
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t(2) = dassert(m1,zeros(d,1),1e-12);
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t(3) = dassert(m2,ones(d,1),1e-12);
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t(4) = dassert(m3,zeros(d,1),1e-12);
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t(5) = dassert(m4,d*ones(d,1),1e-10);
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T = all(t);
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end
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%@eof:1
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%@test:2
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d = 4;
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Sigma = diag(1:d);
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Omega = diag(sqrt(1:d));
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t = zeros(5,1);
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try
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[nodes,weights] = cubature_with_gaussian_weight(d,3);
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t(1) = 1;
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catch
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t = t(1);
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T = all(t);
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end
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if t(1)
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nodes = Omega*nodes;
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m1 = nodes*weights;
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m2 = nodes.^2*weights;
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m3 = nodes.^3*weights;
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m4 = nodes.^4*weights;
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t(2) = dassert(m1,zeros(d,1),1e-12);
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t(3) = dassert(m2,transpose(1:d),1e-12);
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t(4) = dassert(m3,zeros(d,1),1e-12);
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t(5) = dassert(m4,d*transpose(1:d).^2,1e-10);
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T = all(t);
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end
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%@eof:2
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%@test:3
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d = 4;
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Sigma = diag(1:d);
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Omega = diag(sqrt(1:d));
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t = zeros(4,1);
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try
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[nodes,weights] = cubature_with_gaussian_weight(d,3);
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t(1) = 1;
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catch
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t = t(1);
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T = all(t);
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end
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if t(1)
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nodes = Omega*nodes;
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m1 = nodes*weights;
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m2 = bsxfun(@times,nodes,transpose(weights))*transpose(nodes);
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t(2) = dassert(m1,zeros(d,1),1e-12);
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t(3) = dassert(diag(m2),transpose(1:d),1e-12);
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t(4) = dassert(m2(:),vec(diag(diag(m2))),1e-12);
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T = all(t);
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end
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%@eof:3
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%@test:4
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d = 10;
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a = randn(d,2*d);
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Sigma = a*a';
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Omega = chol(Sigma,'lower');
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t = zeros(4,1);
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try
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[nodes,weights] = cubature_with_gaussian_weight(d,3);
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t(1) = 1;
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catch
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t = t(1);
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T = all(t);
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end
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if t(1)
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for i=1:length(weights)
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nodes(:,i) = Omega*nodes(:,i);
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end
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m1 = nodes*weights;
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m2 = bsxfun(@times,nodes,transpose(weights))*transpose(nodes);
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m3 = nodes.^3*weights;
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t(2) = dassert(m1,zeros(d,1),1e-12);
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t(3) = dassert(m2(:),vec(Sigma),1e-12);
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t(4) = dassert(m3,zeros(d,1),1e-12);
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T = all(t);
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end
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%@eof:4
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%@test:5
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d = 5;
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t = zeros(6,1);
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try
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[nodes,weights] = cubature_with_gaussian_weight(d,5);
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t(1) = 1;
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catch
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t = t(1);
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T = all(t);
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end
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if t(1)
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nodes = nodes;
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m1 = nodes*weights;
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m2 = nodes.^2*weights;
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m3 = nodes.^3*weights;
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m4 = nodes.^4*weights;
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m5 = nodes.^5*weights;
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t(2) = dassert(m1,zeros(d,1),1e-12);
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t(3) = dassert(m2,ones(d,1),1e-12);
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t(4) = dassert(m3,zeros(d,1),1e-12);
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t(5) = dassert(m4,3*ones(d,1),1e-12);
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t(6) = dassert(m5,zeros(d,1),1e-12);
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T = all(t);
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end
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%@eof:5
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%@test:6
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d = 3;
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t = zeros(4,1);
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% Call the tested routine
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try
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[nodes,weights] = cubature_with_gaussian_weight(d,3,'ScaledUnscentedTransform');
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t(1) = 1;
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catch
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t = t(1);
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T = all(t);
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end
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if t(1)
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m1 = nodes*weights;
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m2 = nodes.^2*weights;
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m3 = nodes.^3*weights;
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t(2) = dassert(m1,zeros(d,1),1e-12);
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t(3) = dassert(m2,ones(d,1),1e-12);
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t(4) = dassert(m3,zeros(d,1),1e-12);
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T = all(t);
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end
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%@eof:6
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