%%  Problem 3.1.4
% Try different fits on periodic function
% y = exp( sin(t-1) )
%% Part (a), polynomial fit
%Use a polynomial fit to the data.
clear all; format compact;
t = linspace(0,2*pi,200)';  b = exp(sin(t-1));
p = 7;  % degree of polynomial fit is p-1
A = zeros(length(t),p+1);
for j=1:p+1
    A(:,j) = t.^(j-1);
end
disp(' ')
disp('Part (a) coefficients: ')
c = (A'*A)\(A'*b)
p = A*c;  % evaluate polynomial at all t values in one line!

%% Part (b), trig function fit
% Trigonometic function fit uses periodic functions which have that
% property in common with the original data.
A = ones(length(t),5);
A(:,2) = cos(t); 
A(:,3) = sin(t);
A(:,4) = cos(2*t);
A(:,5) = sin(2*t);

disp(' ')
disp('Part (b) coefficients: ')
d = (A'*A)\(A'*b)
trig = A*d;  % again, compute the fits at all t values in one line! 

%% Part (c)
% Plot the results for comparison.

plot(t,p,'-',t,trig,'--',t,b,'-.')
axis([0 2*pi 0 3])
xlabel('t'); 
legend('6th degree poly','trig fit','actual','Location','NorthEast');
title('Problem 3.1.4 -- Fit of Periodic Function');

%% Discussion
% The fit with the trigonometric functions is better near the ends
% because those functions for the interpolation are periodic, like the 
% original function is.  The polynomial does pretty well, but to achieve a 
% specified level of error we expect that using a smaller number of trig 
% functions should do the job.