clear
% Create the coordinates vector
x = -15 : 0.1 : 15;

% f is the result vector of applying func to each cell in x
f = func(x).';

% Add a random noise to vector f, and store the result in y
% We'll use the same y vector for all sections in this assignment.
y = f + 0.4*randn(size(f));


% Genetare each column of matrix A.
col4 = ones(size(x));
col3 = x;
col2 = x.^2;
col1 = x.^3;

% A is the appropriate Vandermonde matrix for this LS problem
A = ([col1; col2; col3; col4]).';

% Calculate A transpose
A_t = A.';

% Calculate K matrix
K = A_t*A;

% Calculate the inverse matrix of K
K_inv = inv(K);

% Find the optimal solution for the LS problem
% This vector is the coefficient vector for the approximation polynomial
x_opt = KA_t*y;

% Calculate the polynomial from the coefficients vector x_opt, denoted P,
% and apply it to the coordinates vector x, i.e, res_vector = P(x).
res_vector = polyval(x_opt, x);
% Show P(x), y
plot(x,res_vector, 'g');
hold on;
plot(x,y, 'b');
hold off;
legend('P(x)', 'y');

% Calculate the remainder vector in respect to the distorted values, i.e,
% vector y
r1 = y - res_vector;
% Calculate the remainder vector in respect to the original function values
r2 = f - res_vector;

% Calculate the L2 norm
n1 = norm(r1)^2;
n2 = norm(r2)^2;


% Function definitions

function ret = func(x)
% This is function we want to approximate in this exercise.
ret = sin(0.3*x) - cos(0.5*x);
end

% Clear the workspace, except y vector
clearvars -except y
% Create the coordinates vector
x = -15 : 0.1 : 15;

% f is the result vector of applying func to each cell in x
f = func(x).';

% Create a result vector
res_vector = zeros(size(x));

% This for loop calculates the approximation of the function for each
% coordinate:
window_size = 3;
for i = 1:length(x)
    % Calculate the lower and higher index bounds
    L = max([i-window_size 1]);
    H = min([i+window_size length(x)]);
    % Calculate the height of matrix A
    n = H - L + 1;
    % Calculate each column of the Vandermonde matrix
    col4 = ones(1,n);
    % Reduce the x vector to the relevant indices
    x_r = x(L:H);
    col3 = x_r;
    col2 = x_r.^2;
    col1 = x_r.^3;
    % Create A and A transpose
    A_t = ([col1; col2; col3; col4]);
    A = A_t.';
    % Create K and K inverse
    K = A_t*A;
    % Reduce the y vector to the relevant indices
    y_r = y(L:H);
    % Calculate the optimal solution, which is the coefficient vector for
    % the approximation polynomial
    x_opt = KA_t*y_r;
    % Calculate the value of the polynomial in the coordinate x(i), and
    % store the result in res_vector(i)
    t = A(max([i-L 1]), :);
    res_vector(i) = t*x_opt;
end

% Show P(x), y
plot(x,res_vector, 'g');
hold on;
plot(x,y, 'b');
hold off;
legend('P(x)', 'y');

% Calculate the remainder vector in respect to the distorted values, i.e,
% vector y
r1 = y - res_vector;
% Calculate the remainder vector in respect to the original function values
r2 = f - res_vector;

% Calculate the L2 norm
n1 = norm(r1)^2;
n2 = norm(r2)^2;


% Function definitions

function ret = func(x)
% This is function we want to approximate in this exercise.
ret = sin(0.3*x) - cos(0.5*x);
end