Added routine for computing weights and nodes of the Gauss Legendre quadrature.

matlab/gauss_legendre_weights_and_nodes.m

+function [nodes,weights] = gauss_legendre_weights_and_nodes(n,a,b)
+% Computes the weights and nodes for a Legendre Gaussian quadrature rule.
+
+%@info:
+%! @deftypefn {Function File} {@var{nodes}, @var{weights} =} gauss_hermite_weights_and_nodes (@var{n})
+%! @anchor{gauss_legendre_weights_and_nodes}
+%! @sp 1
+%! Computes the weights and nodes for a Legendre Gaussian quadrature rule. designed to approximate integrals
+%! on the finite interval (-1,1) of an unweighted smooth function.
+%! @sp 2
+%! @strong{Inputs}
+%! @sp 1
+%! @table @ @var
+%! @item n
+%! Positive integer scalar, number of nodes (order of approximation).
+%! @item a
+%! Double scalar, lower bound.
+%! @item b
+%! Double scalar, upper bound.
+%! @end table
+%! @sp 1
+%! @strong{Outputs}
+%! @sp 1
+%! @table @ @var
+%! @item nodes
+%! n*1 vector of doubles, the nodes (roots of an order n Legendre polynomial)
+%! @item weights
+%! n*1 vector of doubles, the associated weights.
+%! @end table
+%! @sp 2
+%! @strong{Remarks:}
+%! Only the first input argument (the number of nodes) is mandatory. The second and third input arguments
+%! are used if a change of variables is necessary (ie if we need nodes over the interval [a,b] instead of
+%! of the default interval [-1,1]).
+%! @sp 2
+%! @strong{This function is called by:}
+%! @sp 2
+%! @strong{This function calls:}
+%! @sp 2
+%! @end deftypefn
+%@eod:
+
+% Copyright (C) 2012 Dynare Team
+%
+% This file is part of Dynare.
+%
+% Dynare is free software: you can redistribute it and/or modify
+% it under the terms of the GNU General Public License as published by
+% the Free Software Foundation, either version 3 of the License, or
+% (at your option) any later version.
+%
+% Dynare is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+% GNU General Public License for more details.
+%
+% You should have received a copy of the GNU General Public License
+% along with Dynare.  If not, see <http://www.gnu.org/licenses/>.
+
+% AUTHOR(S) stephane DOT adjemian AT univ DASH lemans DOT fr
+
+bb = sqrt(1./(4-(1./transpose(1:n-1)).^2));
+aa = zeros(n,1);
+JacobiMatrix = diag(bb,1)+diag(aa)+diag(bb,-1);
+[JacobiEigenVectors,JacobiEigenValues] = eig(JacobiMatrix);
+[nodes,idx] = sort(diag(JacobiEigenValues));
+JacobiEigenVector = JacobiEigenVectors(1,:);
+JacobiEigenVector = transpose(JacobiEigenVector(idx));
+weights = 2*JacobiEigenVector.^2;
+
+if nargin==3
+    weights = .5*(b-a)*weights;
+    nodes = .5*(nodes+1)*(b-a)+a;
+end
+
+%@test:1
+%$ [n2,w2] = gauss_legendre_weights_and_nodes(2);
+%$ [n3,w3] = gauss_legendre_weights_and_nodes(3);
+%$ [n4,w4] = gauss_legendre_weights_and_nodes(4);
+%$ [n5,w5] = gauss_legendre_weights_and_nodes(5);
+%$ [n7,w7] = gauss_legendre_weights_and_nodes(7);
+%$
+%$
+%$ % Expected nodes (taken from Judd (1998, table 7.2)).
+%$ e2 = .5773502691; e2 = [-e2; e2];
+%$ e3 = .7745966692; e3 = [-e3; 0 ; e3];
+%$ e4 = [.8611363115; .3399810435]; e4 = [-e4; flipud(e4)];
+%$ e5 = [.9061798459; .5384693101]; e5 = [-e5; 0; flipud(e5)];
+%$ e7 = [.9491079123; .7415311855; .4058451513]; e7 = [-e7; 0; flipud(e7)];
+%$
+%$ % Expected weights (taken from Judd (1998, table 7.2) and http://en.wikipedia.org/wiki/Gaussian_quadrature).
+%$ f2 = [1; 1];
+%$ f3 = [5; 8; 5]/9;
+%$ f4 = [18-sqrt(30); 18+sqrt(30)]; f4 = [f4; flipud(f4)]/36;
+%$ f5 = [322-13*sqrt(70); 322+13*sqrt(70)]/900; f5 = [f5; 128/225; flipud(f5)];
+%$ f7 = [.1294849661; .2797053914; .3818300505]; f7 = [f7; .4179591836; flipud(f7)];
+%$
+%$ % Check the results.
+%$ t(1) = dyn_assert(e2,n2,1e-9);
+%$ t(2) = dyn_assert(e3,n3,1e-9);
+%$ t(3) = dyn_assert(e4,n4,1e-9);
+%$ t(4) = dyn_assert(e5,n5,1e-9);
+%$ t(5) = dyn_assert(e7,n7,1e-9);
+%$ t(6) = dyn_assert(w2,f2,1e-9);
+%$ t(7) = dyn_assert(w3,f3,1e-9);
+%$ t(8) = dyn_assert(w4,f4,1e-9);
+%$ t(9) = dyn_assert(w5,f5,1e-9);
+%$ t(10) = dyn_assert(w7,f7,1e-9);
+%$ T = all(t);
+%@eof:1
+
+%@test:2
+%$ nmax = 50;
+%$
+%$ t = zeros(nmax,1);
+%$
+%$ for i=1:nmax
+%$     [n,w] = gauss_legendre_weights_and_nodes(i);
+%$     t(i) = dyn_assert(sum(w),2,1e-12);
+%$ end
+%$
+%$ T = all(t);
+%@eof:2
+
+%@test:3
+%$ [n,w] = gauss_legendre_weights_and_nodes(9,pi,2*pi);
+%$ % Check that the 
+%$ t(1) = all(n>pi);
+%$ t(2) = all(n<2*pi);
+%$ t(3) = dyn_assert(sum(w),pi,1e-12);
+%$ T = all(t);
+%@eof:3
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