% legendre_expansion.m
% Legendre EXpansion over [-1,1]
% P0 = 1; P1 = x; P2 = (3/2)*x^2-(1/2)
% mu_j = (Pj, Pj) = 2/(2j+1) = 2, 2/3, 2/5
% p(x) = sum_{j=0}^2 a_j*P_j
% a_j = (f,P_j)/(P_j.P_j) = (f, P_j)/mu_j
function main(n);
f = @(x) exp(x);
xx = linspace(-1,1,1000); % Graphical points
fx = f(xx);
Px = zeros(1,length(xx));
for k=0:n
pk=Legen_poly(k);
g = @(x) f(x).*polyval(pk,x);
ak = quad(g,-1,1)/(2/(2*k+1));
Px = Px + ak*polyval(pk,xx);
end
fprintf('Maximum Error of |f(x)-Pn(x)| = %12.4e \n', max(abs(fx-Px)))
plot(xx,fx,xx,Px);
title(['Legendre Expansion of degree ', num2str(n)] )
legend('f(x)', 'p_n(x)')
%%%%%%%% Legendre polynomial of degree n %%%%%%%%
function pn = Legen_poly(n);
pbb = [1]; if n==0, pn=pbb; return; end
pb = [1 0]; if n==1, pn=pb; return; end
for i=2:n;
pn = ( (2*i-1)*[pb,0] - (i-1)*[0, 0, pbb] )/i;
pbb = pb; pb=pn;
end
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