Numerical Solution of Partial Differential Equations

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function [x,w,v] = chebpts(n, varargin)
%CHEBPTS Compute the second kind chebyshev points.
%   [X,W,V] = CHEBPTS(N) retuns N Chebyshev points of the second kind in [-1, 1].
%   X is the second kind Chebshev points, W is the weights of Clenshaw-Curtis
%   quadrature, and V is the barycentric weights corresponding to the  Chebyshev
%   points X.
%
%   [X,W,V] = CHEBPTS(N,[a,b]) retuns N Chebyshev points of the second kind in
%   [a, b].
%
% Example:
%
%   [x, w] = chebpts(10, [-2,2]);
%   w*x.^2  % quadrature of x^2 in [-2,2]
%
% This programe is adapted from Chebfun/chebtech2.chebpts, http://www.chebfun.org/


if n == 1
    x = 0;
    w = 2;
    v = 1;
else
    % Chebyshev points:
    m = n - 1;
    x = sin(pi*(-m:2:m)/(2*m)).';  % (Use of sine enforces symmetry.)

    % quadratrue weights
    c = 2./[1, 1-(2:2:(m)).^2];    % Exact integrals of T_k (even)
    c = [c, c(floor(n/2):-1:2)];   % Mirror for DCT via FFT
    w = ifft(c);                   % Interior weights
    w([1,n]) = w(1)/2;             % Boundary weights

    % barycentric interpolate weights
    v = [0.5;ones(n-1,1)];        % Note v(1) is positive.
    v(2:2:end) = -1;
    v(end) = .5*v(end);
end

if nargin == 2
    [x, w] =  rescale(x, w, varargin{1});
end

end


function [x, w] = rescale(x, w, dom)
%RESCALE   Rescale the quadrature points and weights.

a = dom(1);
b = dom(2);

w = .5*w*(b-a);
x = .5*(a+b) + .5*(b-a).*x;


end