% define function data points
taskfunc = (-5:5)';
taskfunc(:, 2) = [
    -6.5743
    0.9765
    3.1026
    1.8572
    1.3165
    -0.6144
    0.1032
    0.3729
    2.5327
    7.3857
    9.4892
];

% perform the task
for polydeg = 0:3
    % obtain factors of approximating polynomial
    [factors, error, gramcond] = approximate(taskfunc, polydeg);
    
    % print error and condition number
    disp(['Approximation degree ', num2str(polydeg), ':']);
    disp(['Error: ', num2str(error)]);
    disp(['Condition number: ', num2str(gramcond)]);
    
    % plot data points
    figure;
    grid on;
    hold on;
    title(['Polynomial approximation of the function (degree ', ...
        num2str(polydeg), ')']);
    scatter(taskfunc(:, 1), taskfunc(:, 2));
    
    % plot approximation
    x = taskfunc(1):0.05:taskfunc(end, 1);
    y = evalapprox(factors, x);
    plot(x, y);
    
    % finish and print graph
    hold off;
    set(gcf, 'PaperPosition', [0 0 6 4]);
    set(gcf, 'PaperSize', [6 4]);
    %print(['report/approx', num2str(polydeg)], '-dpdf');
end

% find the approximating polynomial of the given degree
function [factors, error, gramcond] = approximate(func, polydeg)
    % define the A matrix used for solving the error minimization problem
    A = zeros(size(func, 1), polydeg + 1);
    
    % calculate cells of A using natural basis
    for row = 1:size(A, 1)
        for col = 1:size(A, 2)            
            A(row, col) = func(row, 1) ^ (col - 1);
        end
    end
    
    % solve least-square problem using QRdash decomposition
    [Q, eqsys, invqtq] = QRDecomposition(A);
    eqsys(:, end + 1) = invqtq * Q' * func(:, 2);
    factors = backSubstitution(eqsys);
    
    % calculate error and condition number of Gram's matrix
    error = norm(func(:, 2) - A * factors);
    gramcond = cond(A' * A);
end

% evaluate the value of an approximation at the given x
function y = evalapprox(factors, xarray)
    y = zeros(1, size(xarray, 2));
    
    for xi = 1:size(xarray, 2)
        for i = 1:size(factors, 1)
            y(xi) = y(xi) + factors(i) * xarray(xi) ^ (i - 1);
        end
    end
end