%% Example of projection onto an orthogonal set
% *1* Find the projection of  $\sin x$ onto the orthogonal
% set  $\{1,\cos x, \cos 2x\}$ on $[0,\pi]$.  
%
% *2* Find the distance from
% $\sin x$ to its projection.
%
% *3* Plot the function and its projection on the same axes.
%% Step 1
% Compute the necessary inner product of $\sin x$ with each of the
% functions in the orthogonal set. 
%%
% Here is $\langle \sin(x),1\rangle$.
syms x
ip1 = int(sin(x),x,0,sym(pi))

%%
% Here is $\langle \sin(x),\cos x\rangle$.
ip2 = int(sin(x)*cos(x),x,0,sym(pi))

%%
% Here is $\langle \sin(x),\cos 2x\rangle$.
ip3 = int(sin(x)*cos(2*x),x,0,sym(pi))
%% Step 2
% Compute the necessary lengths.
%%
% $||1||^2$ is 
n1 = int(1,0,sym(pi))
%%
% $||\cos 2x||^2$ is
n3 = int((cos(2*x))^2,0,sym(pi))
%% Step 3
% Write down the Fourier coefficients. 
c1 = ip1/n1
%%
c2 = 0
%%
c3 = ip3/n3
%% Step 4 
% The projection is 
proj = c1*1+c2*cos(x)+c3*cos(2*x)
%%
pretty(proj)
%% Step 5
% The distance squared is  
% 
% $$
% ||\sin(x)||^2 \mbox{--}\left( \frac{\langle \sin x ,1 \rangle}{||1||^2} +\frac{
% \langle \sin x ,\cos 2x\rangle}{||\cos 2x||^2}\right)
% $$
% 
% which is 
distsq = int((sin(x))^2,0,sym(pi)) - (ip1^2/n1 + ip3^2/n3)
%%
% so the distance is 
dist = sqrt(distsq)
%%
% which is approximately 
vpa(dist,8)
%% Step 6
% Plot $\sin x$ and its projection.
%%
% I'll use plot so I can control the color and the thickness. To use plot I
% will turn the projection into a vectorized function and I need a vector
% on which to evaluate the functions I'm plotting.
 
X = 0:.001:pi;
Proj = inline(vectorize(proj))
plot(X,sin(X),'linewidth',2)
hold on
plot(X,Proj(X),'color','g','linewidth',2)
hold off
axis equal
axis([0 pi 0 1.1])