# Figure 6.19: Polynomial fitting

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    % Section 6.5.3
    % Boyd & Vandenberghe "Convex Optimization"
    % Original by Lieven Vandenberghe
    % Adapted for CVX by Joelle Skaf - 10/03/05
    % (a figure is generated)
    %
    % Given data u_1,...,u_m and v_1,...,v_m in R, the goal is to fit to the
    % data a polynomial of the form
    % p(u) = x_1 + x_2*u + ... + x_n*u^{n-1}
    % i.e. solve the problem:  minimize ||Ax - v||
    % where A is the Vandermonde matrix s.t. Aij = u_i^{j-1}
    % Two cases are considered: L2-norm and Linfty-norm

    % Input data
    n=6;
    m=40;
    randn('state',0);
    % generate 50 ponts ui, vi
    u = linspace(-1,1,m);
    v = 1./(5+40*u.^2) + 0.1*u.^3 + 0.01*randn(1,m);

    % LS fit polynomial x_1 + x_2*u + ... + x_n*u^(n-1) to (ui,vi)
    fprintf(1,'Computing optimal polynomial in the case of L2-norm...');

    A = vander(u');
    A = A(:,m-n+[1:n]);     % last n columns of A
    x = A\(v');             % coefficients of the polynomial in the following
                            % order: x = [x_n x_(n-1) ... x_2 x_1]'

    fprintf(1,'Done! \n');

    % L-infty fit
    fprintf(1,'Computing optimal polynomial in the case of Linfty-norm...');

    cvx_begin quiet
        variable x1(n)
        minimize (norm(A*x1 - v', inf))
    cvx_end

    fprintf(1,'Done! \n');

    % generates 1000 points in  [-1,1]
    u2 = linspace(-1.1,1.1,1000);

    % evaluate the interpolating polynomial using Horner's method
    vpol = x(1)*ones(1,1000);
    vpoll1 = x1(1)*ones(1,1000);
    for i = 2:n
      vpol = vpol.*u2 + x(i);
      vpoll1 = vpoll1.*u2 + x1(i);
    end;

    figure
    % plot function and interpolating polynomial
    plot(u2, vpol,'-', u, v, 'o', u2, vpoll1,'--');
    xlabel('u');
    ylabel('p(u)');
    title('Fitting of data points with two polynomials of degree 5');
    legend('L_2 norm','data points','L_{\infty} norm', 'Location','Best');
    % print -deps polapprox.eps

    Computing optimal polynomial in the case of L2-norm...Done!
    Computing optimal polynomial in the case of Linfty-norm...Done!