Matlab minimize norm to find coefficients using fminsearch

Question

I am trying to use minimization to calculate coefficients of the polynomial p(x) = 1 + c(1)x + c(2)x^2 to approximate e^x. I need to use points xi = 1 + i/n for natural numbers i on [1,n], first for n=5, then n=10, etc. The approach is to minimize the 1, 2, and inf norm(p(x) - e^x) using fminsearch. So the output should be the 2 coefficients for the 3 p(x)'s. Any suggestions is appreciated.

Answer 1

Well, if anyone was wondering, I did figure the question out finally.

for l = [1 2 inf]
    fprintf('For norm %d\n', l)
    fprintf('Coefficients   c1        c2\n')
    for n = [5 10 100]
        i = 1:n ;
        x = 1 + i/n ;
        c = [1 1] ;

        %Difference function that we want to minimize
        g = @(c) x.*c(1) + x.^2.*c(2) + 1 - exp(x);
        f_norm = @(c) norm(g(c), l) ;
        C = fminsearch(f_norm, c);
        fprintf('n = %d      ', n)
        fprintf('%f    %f\n', C(1), C(2))
        % Compare plot of e^x and p(x).
        p = @(x) C(1)*x + C(2)*x.^2 + 1;
        xx = linspace(1,2,1e5);
        figure;
        plot(xx, p(xx), '--r', xx, exp(xx));
        str = sprintf('Plot with n = %d, for norm %d', n,l);
        title(str,'FontSize',24)
        xlabel('x','FontSize',20)
        ylabel('y','FontSize',20)
        legend('p2 approximation','exponential');
    end
end

This worked in the end to answer the question.