132 lines
4.2 KiB
Matlab
132 lines
4.2 KiB
Matlab
function [nodes,weights] = gauss_legendre_weights_and_nodes(n,a,b)
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% Computes the weights and nodes for a Legendre 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_legendre_weights_and_nodes}
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%! @sp 1
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%! Computes the weights and nodes for a Legendre Gaussian quadrature rule. designed to approximate integrals
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%! on the finite interval (-1,1) 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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%! @item a
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%! Double scalar, lower bound.
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%! @item b
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%! Double scalar, upper bound.
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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 Legendre 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{Remarks:}
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%! Only the first input argument (the number of nodes) is mandatory. The second and third input arguments
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%! are used if a change of variables is necessary (ie if we need nodes over the interval [a,b] instead of
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%! of the default interval [-1,1]).
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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 © 2012-2017 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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% AUTHOR(S) stephane DOT adjemian AT univ DASH lemans DOT fr
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bb = sqrt(1./(4-(1./transpose(1:n-1)).^2));
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aa = zeros(n,1);
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JacobiMatrix = diag(bb,1)+diag(aa)+diag(bb,-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 = 2*JacobiEigenVector.^2;
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if nargin==3
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weights = .5*(b-a)*weights;
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nodes = .5*(nodes+1)*(b-a)+a;
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end
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%@test:1
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%$ [n2,w2] = gauss_legendre_weights_and_nodes(2);
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%$ [n3,w3] = gauss_legendre_weights_and_nodes(3);
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%$ [n4,w4] = gauss_legendre_weights_and_nodes(4);
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%$ [n5,w5] = gauss_legendre_weights_and_nodes(5);
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%$ [n7,w7] = gauss_legendre_weights_and_nodes(7);
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%$
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%$
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%$ % Expected nodes (taken from Judd (1998, table 7.2)).
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%$ e2 = .5773502691; e2 = [-e2; e2];
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%$ e3 = .7745966692; e3 = [-e3; 0 ; e3];
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%$ e4 = [.8611363115; .3399810435]; e4 = [-e4; flipud(e4)];
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%$ e5 = [.9061798459; .5384693101]; e5 = [-e5; 0; flipud(e5)];
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%$ e7 = [.9491079123; .7415311855; .4058451513]; e7 = [-e7; 0; flipud(e7)];
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%$
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%$ % Expected weights (taken from Judd (1998, table 7.2) and http://en.wikipedia.org/wiki/Gaussian_quadrature).
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%$ f2 = [1; 1];
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%$ f3 = [5; 8; 5]/9;
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%$ f4 = [18-sqrt(30); 18+sqrt(30)]; f4 = [f4; flipud(f4)]/36;
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%$ f5 = [322-13*sqrt(70); 322+13*sqrt(70)]/900; f5 = [f5; 128/225; flipud(f5)];
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%$ f7 = [.1294849661; .2797053914; .3818300505]; f7 = [f7; .4179591836; flipud(f7)];
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%$
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%$ % Check the results.
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%$ t(1) = dassert(e2,n2,1e-9);
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%$ t(2) = dassert(e3,n3,1e-9);
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%$ t(3) = dassert(e4,n4,1e-9);
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%$ t(4) = dassert(e5,n5,1e-9);
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%$ t(5) = dassert(e7,n7,1e-9);
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%$ t(6) = dassert(w2,f2,1e-9);
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%$ t(7) = dassert(w3,f3,1e-9);
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%$ t(8) = dassert(w4,f4,1e-9);
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%$ t(9) = dassert(w5,f5,1e-9);
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%$ t(10) = dassert(w7,f7,1e-9);
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%$ T = all(t);
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%@eof:1
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%@test:2
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%$ nmax = 50;
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%$
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%$ t = zeros(nmax,1);
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%$
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%$ for i=1:nmax
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%$ [n,w] = gauss_legendre_weights_and_nodes(i);
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%$ t(i) = dassert(sum(w),2,1e-12);
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%$ end
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%$
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%$ T = all(t);
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%@eof:2
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%@test:3
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%$ [n,w] = gauss_legendre_weights_and_nodes(9,pi,2*pi);
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%$ % Check that the
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%$ t(1) = all(n>pi);
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%$ t(2) = all(n<2*pi);
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%$ t(3) = dassert(sum(w),pi,1e-12);
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%$ T = all(t);
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%@eof:3 |