Added routine computing Gaussian cubature weights and nodes (implementation of algorithms described in Stroud 1971).
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function [nodes, weights] = cubature_with_gaussian_weight(d,n,method)
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%@info:
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%! @deftypefn {Function File} {@var{nodes}, @var{weights} =} cubature_with_gaussian_weight (@var{d}, @var{n})
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%! @anchor{cubature_with_gaussian_weight}
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%! @sp 1
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%! Computes nodes and weights for a n-order cubature with gaussian weight.
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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 d
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%! Scalar integer, dimension of the region of integration.
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%! @item n
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%! Scalar integer equal to 3 or 5, approximation order.
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%! @end table
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%! @sp 2
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%! @strong{Outputs}
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%! @sp 1
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%! @table @ @var
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%! @item nodes
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%! n*m matrix of doubles, the m nodes where the integrated function has to be evaluated. The number of nodes, m, is equal to 2*@var{d} is @var{n}==3 or 2*@var{d}^2+1 if @var{n}==5.
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%! @item weights
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%! m*1 vector of doubles, weights associated to the nodes.
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%! @end table
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%! @sp 2
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%! @strong{Remarks}
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%! @sp 1
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%! The routine returns nodes and associated weights to compute a multivariate integral of the form:
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%!
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%! \int_D f(x)*\exp(-<x,x>) dx
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%!
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%!
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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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% AUTHOR(S) stephane DOT adjemian AT univ DASH lemans DOT fr
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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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%V = sqrt(pi)^d;
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r = sqrt(d/2);
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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,'Stroud') && isequal(n,5)
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r = sqrt((d+2)/2);
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s = sqrt((d+2)/4);
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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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%@test:1
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%$ % Set problem
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%$ d = 4;
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%$
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%$ t = zeros(5,1);
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%$
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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);
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%$ t(1) = 1;
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%$ catch exception
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%$ t = t(1);
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%$ T = all(t);
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%$ LOG = getReport(exception,'extended');
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%$ return
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%$ end
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%$
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%$ % Check the results.
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%$ nodes = sqrt(2)*nodes;
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%$
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%$ % Compute (approximated) first order moments.
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%$ m1 = nodes*weights;
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%$
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%$ % Compute (approximated) second order moments.
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%$ m2 = nodes.^2*weights;
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%$
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%$ % Compute (approximated) third order moments.
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%$ m3 = nodes.^3*weights;
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%$
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%$ % Compute (approximated) fourth order moments.
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%$ m4 = nodes.^4*weights;
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%$
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%$ t(2) = dyn_assert(m1,zeros(d,1),1e-12);
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%$ t(3) = dyn_assert(m2,ones(d,1),1e-12);
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%$ t(4) = dyn_assert(m3,zeros(d,1),1e-12);
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%$ t(5) = dyn_assert(m4,d*ones(d,1),1e-10);
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%$ T = all(t);
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%@eof:1
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%@test:2
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%$ % Set problem
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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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%$
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%$ t = zeros(5,1);
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%$
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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);
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%$ t(1) = 1;
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%$ catch exception
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%$ t = t(1);
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%$ T = all(t);
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%$ LOG = getReport(exception,'extended');
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%$ return
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%$ end
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%$
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%$ % Check the results.
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%$ nodes = sqrt(2)*Omega*nodes;
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%$
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%$ % Compute (approximated) first order moments.
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%$ m1 = nodes*weights;
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%$
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%$ % Compute (approximated) second order moments.
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%$ m2 = nodes.^2*weights;
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%$
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%$ % Compute (approximated) third order moments.
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%$ m3 = nodes.^3*weights;
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%$
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%$ % Compute (approximated) fourth order moments.
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%$ m4 = nodes.^4*weights;
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%$
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%$ t(2) = dyn_assert(m1,zeros(d,1),1e-12);
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%$ t(3) = dyn_assert(m2,transpose(1:d),1e-12);
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%$ t(4) = dyn_assert(m3,zeros(d,1),1e-12);
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%$ t(5) = dyn_assert(m4,d*transpose(1:d).^2,1e-10);
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%$ T = all(t);
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%@eof:2
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%@test:3
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%$ % Set problem
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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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%$
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%$ t = zeros(4,1);
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%$
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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);
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%$ t(1) = 1;
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%$ catch exception
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%$ t = t(1);
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%$ T = all(t);
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%$ LOG = getReport(exception,'extended');
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%$ return
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%$ end
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%$
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%$ % Check the results.
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%$ nodes = sqrt(2)*Omega*nodes;
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%$
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%$ % Compute (approximated) first order moments.
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%$ m1 = nodes*weights;
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%$
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%$ % Compute (approximated) second order moments.
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%$ m2 = bsxfun(@times,nodes,transpose(weights))*transpose(nodes);
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%$
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%$ t(2) = dyn_assert(m1,zeros(d,1),1e-12);
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%$ t(3) = dyn_assert(diag(m2),transpose(1:d),1e-12);
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%$ t(4) = dyn_assert(m2(:),vec(diag(diag(m2))),1e-12);
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%$ T = all(t);
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%@eof:3
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%@test:4
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%$ % Set problem
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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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%$
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%$ t = zeros(4,1);
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%$
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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);
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%$ t(1) = 1;
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%$ catch exception
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%$ t = t(1);
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%$ T = all(t);
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%$ LOG = getReport(exception,'extended');
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%$ return
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%$ end
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%$
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%$ % Correct nodes for the covariance matrix
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%$ for i=1:length(weights)
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%$ nodes(:,i) = sqrt(2)*Omega*nodes(:,i);
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%$ end
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%$
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%$ % Check the results.
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%$
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%$ % Compute (approximated) first order moments.
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%$ m1 = nodes*weights;
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%$
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%$ % Compute (approximated) second order moments.
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%$ m2 = bsxfun(@times,nodes,transpose(weights))*transpose(nodes);
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%$
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%$ % Compute (approximated) third order moments.
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%$ m3 = nodes.^3*weights;
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%$
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%$ t(2) = dyn_assert(m1,zeros(d,1),1e-12);
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%$ t(3) = dyn_assert(m2(:),vec(Sigma),1e-12);
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%$ t(4) = dyn_assert(m3,zeros(d,1),1e-12);
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%$ T = all(t);
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%@eof:4
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%@test:5
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%$ % Set problem
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%$ d = 5;
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%$
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%$ t = zeros(6,1);
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%$
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%$ % Call the tested routine
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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 exception
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%$ t = t(1);
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%$ T = all(t);
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%$ LOG = getReport(exception,'extended');
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%$ return
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%$ end
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%$
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%$ % Check the results.
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%$ nodes = sqrt(2)*nodes;
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%$
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%$ % Compute (approximated) first order moments.
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%$ m1 = nodes*weights;
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%$
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%$ % Compute (approximated) second order moments.
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%$ m2 = nodes.^2*weights;
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%$
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%$ % Compute (approximated) third order moments.
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%$ m3 = nodes.^3*weights;
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%$
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%$ % Compute (approximated) fourth order moments.
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%$ m4 = nodes.^4*weights;
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%$
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%$ % Compute (approximated) fifth order moments.
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%$ m5 = nodes.^5*weights;
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%$
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%$ t(2) = dyn_assert(m1,zeros(d,1),1e-12);
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%$ t(3) = dyn_assert(m2,ones(d,1),1e-12);
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%$ t(4) = dyn_assert(m3,zeros(d,1),1e-12);
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%$ t(5) = dyn_assert(m4,3*ones(d,1),1e-12);
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%$ t(6) = dyn_assert(m5,zeros(d,1),1e-12);
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%$ T = all(t);
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%@eof:5
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