%(1) Plot the original points in the correct part, top left
%(2) When sorting points, add the "ind" to index y to not change them matching.
%Index y with "ind" and store in y
%(3) Change the both floor commands to ceil and also multiply the
%accuracyFactor instead of adding it because it is a factor.
%(4) When calling lagrangeInterp, We are finding for yy(i) not xx, and also change yy(i) to xx(i).
%switch y and x
%(5)Make the line blue for our spline
%(6)When defining splineYY, switch the x and xx
%(7)Add the color of the line to be blue when plotting matlabs spline
%(8)Make the entire bottem part the graph by telling subplot(2,2,3:4)
%(9)start interpY at 0 not 1
%(10)When defining product, change the part after prod(x(i) to -[x(1:i-1), x(i+1:end)]
function [xx, yy] = mySplinePlot(x, y, accuracyFactor)

% Plot the original points
subplot(2,2,1) %%%(1)This should go on the top left
hold on
plot(x,y,'ro')
title('mySplinePlot')

% Sort the points by increasing x-coordinates
[x ind] = sort(x); %%%(2)Get the positions before changed
y = y(ind); %%%(2)index y with ind


% Find the number of points based on the max difference between x-values
% and create an xx vector of this number of points, evenly spaced.  The max
% difference between adjacent points should be rounded up, as should the
% result after applying the accuracy factor.
pointsPerInterval = ceil(ceil(max(diff(x))).*accuracyFactor); %%%(3)change floor to ceil in both cases and also multiply
%accuracyFactor instead of adding
totalPoints = (length(x))*pointsPerInterval + length(x);
xx = linspace(x(1), x(end), totalPoints);


% Interpolate to find a yy vector using the Lagrange Interpolation
% Polynomial. Initialize yy at the right size for improved efficiency.
yy = zeros(1,totalPoints);
for i = 1:totalPoints
    yy(i) = lagrangeInterp(x, y, xx(i));  %%%(4)We are finding for yy(i) not xx, and also change yy(i) to xx(i). switch y and x

end

% Plot the vectors just calculated.
plot(xx, yy, 'b') %%%(5)Make the line blue




% Call Matlab's spline and plot it as described in the problem statement
splineYY = spline(x, y, xx); %%%(6)switch the x and xx
subplot(2,2,2)
hold on
plot(x,y,'ro')
plot(xx, splineYY , 'b') %%%(7)Add the color of the line to be blue
title('spline')


% Make a plot as described in the problem statement and as shown in the
% solution figures comparing mySplinePlot with spline
subplot(2,2,3:4) %%%(8)Make the entire bottem part the graph
plot(xx, yy, 'r')
hold on
plot(xx, splineYY, 'k--')
legend('mySplinePlot','spline')
title('mySplinePlot vs. spline')
plot(x,y,'ro')
hold off

end


% This function performs the actual Lagrange Interpolation, returning an
% interpolated y-value for a single given x-value. Do NOT change this
% function header.
function interpY = lagrangeInterp(x, y, interpX)

% Initialize interpY and determine the number of terms in the product The
% new interpolated Y value will then be increased based on the requirements
% of the Lagrange Interpolation.
interpY = 0; %%%(9)start interpY at 0 not 1
n = length(x);

% Make an estimation for each known x-coordinate using the formula given.
for i = 1:n

    % If this makes more sense to you in a for loop, you can include a
    % nested for loop here. This is where the actual product from the
    % Lagrange Interpolation formula comes in.
   % P(x) = y_k*PRODUCT((x-x_k)/(x_i-x_k), where [1<=<=n & i~=k])

    product = prod(interpX - [x(1:i-1), x(i+1:end)])/prod(x(i) - [x(1:i-1), x(i+1:end)]);
    %%%(10)change the part after prod(x(i) to -[x(1:i-1), x(i+1:end)]
    % Update the total for interpY based on the value found for the ith
    % coordinate
    interpY = interpY + product * y(i);
end
end