125 lines
3.7 KiB
Matlab
125 lines
3.7 KiB
Matlab
function [nodes,weights] = gauss_hermite_weights_and_nodes(n)
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% Computes the weights and nodes for an Hermite Gaussian quadrature rule.
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%@info:
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%! @deftypefn {Function File} {@var{nodes}, @var{weights} =} gauss_hermite_weights_and_nodes (@var{n})
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%! @anchor{gauss_hermite_weights_and_nodes}
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%! @sp 1
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%! Computes the weights and nodes for an Hermite Gaussian quadrature rule. designed to approximate integrals
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%! on the infinite interval (-\infty,\infty) of an unweighted smooth function.
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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 n
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%! Positive integer scalar, number of nodes (order of approximation).
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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 nodes
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%! n*1 vector of doubles, the nodes (roots of an order n Hermite polynomial)
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%! @item weights
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%! n*1 vector of doubles, the associated weights.
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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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%! @end deftypefn
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%@eod:
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% Copyright © 2011-2017 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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% Original author: stephane DOT adjemian AT univ DASH lemans DOT fr
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b = sqrt([1:n-1]/2);
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JacobiMatrix = diag(b,1)+diag(b,-1);
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[JacobiEigenVectors,JacobiEigenValues] = eig(JacobiMatrix);
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[nodes,idx] = sort(diag(JacobiEigenValues));
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JacobiEigenVector = JacobiEigenVectors(1,:);
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JacobiEigenVector = transpose(JacobiEigenVector(idx));
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weights = JacobiEigenVector.^2;
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nodes = sqrt(2)*nodes;
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%@test:1
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%$ n = 5;
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%$ [nodes,weights] = gauss_hermite_weights_and_nodes(n);
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%$
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%$ sum_of_weights = sum(weights);
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%$
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%$ % Expected nodes (taken from Judd (1998, table 7.4).
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%$ enodes = [-2.020182870; -0.9585724646; 0; 0.9585724646; 2.020182870];
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%$
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%$ % Check the results.
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%$ t(1) = dassert(1.0,sum_of_weights,1e-12);
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%$ t(2) = dassert(enodes,nodes/sqrt(2),1e-8);
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%$ T = all(t);
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%@eof:1
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%@test:2
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%$ n = 9;
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%$ [nodes,weights] = gauss_hermite_weights_and_nodes(n);
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%$
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%$ sum_of_weights = sum(weights);
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%$ expectation = sum(weights.*nodes);
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%$ variance = sum(weights.*(nodes.^2));
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%$
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%$ % Check the results.
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%$ t(1) = dassert(1.0,sum_of_weights,1e-12);
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%$ t(2) = dassert(1.0,variance,1e-12);
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%$ t(3) = dassert(0.0,expectation,1e-12);
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%$ T = all(t);
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%@eof:2
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%@test:3
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%$ n = 9;
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%$ [nodes,weights] = gauss_hermite_weights_and_nodes(n);
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%$
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%$ NODES = cartesian_product_of_sets(nodes,nodes);
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%$ WEIGHTS = cartesian_product_of_sets(weights,weights);
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%$ WEIGHTS = prod(WEIGHTS,2);
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%$
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%$ sum_of_weights = sum(WEIGHTS);
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%$ expectation = transpose(WEIGHTS)*NODES;
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%$ variance = transpose(WEIGHTS)*NODES.^2;
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%$
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%$ % Check the results.
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%$ t(1) = dassert(1.0,sum_of_weights,1e-12);
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%$ t(2) = dassert(ones(1,2),variance,1e-12);
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%$ t(3) = dassert(zeros(1,2),expectation,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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%$ n = 9; sigma = .1;
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%$ [nodes,weights] = gauss_hermite_weights_and_nodes(n);
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%$
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%$ sum_of_weights = sum(weights);
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%$ expectation = sum(weights.*nodes*.1);
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%$ variance = sum(weights.*((nodes*.1).^2));
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%$
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%$ % Check the results.
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%$ t(1) = dassert(1.0,sum_of_weights,1e-12);
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%$ t(2) = dassert(.01,variance,1e-12);
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%$ t(3) = dassert(0.0,expectation,1e-12);
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%$ T = all(t);
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%@eof:4
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