%% Assignment 4
% Name: Claudio Maggioni
%
% Date: 19/3/2019
%
% This is a template file for the first assignment to get started with running
% and publishing code in Matlab.  Each problem has its own section (delineated
% by |%%|) and can be run in isolation by clicking into the particular section
% and pressing |Ctrl| + |Enter| (evaluate current section).
%
% To generate a pdf for submission in your current directory, use the following
% three lines of code at the command window:
%
%  >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment3.m', options)
%

%% Problem 3
clear;

n=10;
f = @(x) (exp(-(x^2)/2)/sqrt(2*pi));
x_5 = computeEquidistantXs(5);
x_10 = computeEquidistantXs(10);
x_5c = computeChebyshevXs(5);
x_10c = computeChebyshevXs(10);

xe = (-1:0.01:1)';
y = zeros(size(xe));
for i = 1:size(xe, 1)
    y(i) = f(xe(i));
end

p_5 = computePolypoints(f, xe, x_5, 5);
p_10 = computePolypoints(f, xe, x_10, 10);

p_5c = computePolypoints(f, xe, x_5c, 5);
p_10c = computePolypoints(f, xe, x_10c, 10);

figure;
subplot(1,2,1);
plot(xe, p_5, xe, p_10, xe, y);
subplot(1,2,2);
plot(xe, p_5c, xe, p_10c, xe, y);

function x = computeEquidistantXs(n)
    x = zeros(2*n+1,1);
    for i = 1:2*n+1
        x(i) = (i-n-1)/n;
    end
end


function x = computeChebyshevXs(n)
    x = zeros(2*n+1,1);
    for i = 1:2*n+1
        x(i) = cos((2*i - 1) *pi / (4*n + 2));
    end
end


function p = computePolypoints(f, xe, x, n)
    p = zeros(size(xe));

    for i = 1:(2*n+1)
        e_i = zeros(2*n+1, 1);
        e_i(i) = 1;
        N = NewtonInterpolation(x, e_i);
        p = p + f(x(i)) * HornerNewton(N, x, xe);
    end
end

% Assuming x and y are column vectors with the same length
function N = NewtonInterpolation (x,y)
    n = size(x, 1);
    N = y;
    for i = 1:n
        N(n:-1:i+1) = (N(n:-1:i+1) - N(n-1:-1:i)) ./ (x(n:-1:i+1) - x(n-i:-1:1));
    end
end

% N is the array of coefficients
% 
% xi evaluation points
function p = HornerNewton(N, x, xe)
    n = size(x, 1);
    p = ones(size(xe, 1), 1) * N(n);
    for i = n-1:-1:1 
        p = p .* (xe - x(i)) + N(i);
    end
end