%% Q1(a) answer
clear
% Testing trigseries
x=linspace(0,pi/2,10);
[y,err]=trigseries(x);
plot(x,y(:,1),'ko',x,y(:,2),'k*');xlabel('x'); hold on;
x1=linspace(x(1),x(end),100);
plot(x1,sin(x1),'k-',x1,cos(x1),'k--');legend('Sine','Cosine');
print -deps q1_fig2.eps

% Checking with different numbers of input variables
[y,err]=trigseries(x,15)
[y,err]=trigseries(x,15,'sine')
[y,err]=trigseries(x,15,'cosine')
%% Q1(b) answer
n1=1;n2=120;k=1;x1=pi/2; x2=4*x1;
for n=n1:n2
    [y,err]=trigseries(x1,n,'sine');
    err_norm(k,1)=norm(err,inf);
    [y,err]=trigseries(x2,n,'sine');
    err_norm(k,2)=norm(err,inf);
    k=k+1;
end
figure;loglog([n1:1:n2],err_norm(:,1),'ok',[n1:1:n2],err_norm(:,2),'k*');xlabel('number of terms');ylabel('L2 error');
grid on;
format short e;disp(err_norm)
disp('Now plot differences between partial sums');
k=1;
for n = n1:1:n2
     diff(k,1) = x1.^(2*k-1)./factorial(2*k-1);
     diff(k,2) = x2.^(2*k-1)./factorial(2*k-1);
     disp([k, diff(k,1)]);
     k=k+1;
     
end
 
 [y,err]=trigseries(x,n2,'sine')
 hold on;loglog([n1:1:n2],diff(:,1),'k-',[n1:1:n2],diff(:,2),'k-');xlabel('number of terms');ylabel('Difference in partial sums');
 axis([1 200 1e-17 1e3]); 
 legend('error, x','error, 4x','Diff in partial sums');grid on;
 print -deps q1_fig2.eps
 disp('Errors become constant once the relative sizes of the new terms in');
 disp('the series are smaller than the machine epsilon 2.2204e-16.');
 disp('The minimum error is a bit larger than machine precision for x=4pi/2 because the error');
 disp('in the Taylor series is significantly greater than machine epsilon larger just before critical number of terms,');
 disp('at which point the error cannot decrease any more. This is because the Taylor series converges more slowly for larger x.');