59 lines
2.0 KiB
Matlab
59 lines
2.0 KiB
Matlab
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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 (C) 2011 Dynare Team
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% stephane DOT adjemian AT univ DASH lemans DOT fr
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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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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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