%% Annotated Matlab transcript -- 3/20
%% CS 323 - Doug DeCarlo

% Example of a quadratic Lagrange interpolating polynomial

x0 = 1;  x1 = 2;    x2 = 3;
y0 = 1;  y1 = 2.5;  y2 = -1;

x = 0:0.01:4;

% plot L0(x), with the data -- see how L0(x0)=1, and L0(x1)=L0(x2)=0
plot(x,0,x0,y0,'.',x1,y1,'.',x2,y2,'.',x,(x-x1).*(x-x2)./((x0-x1).*(x0-x2)));

% now plot the whole interpolating polynomials
plot(x,0,x0,y0,'.',x1,y1,'.',x2,y2,'.',x,(x-x1).*(x-x2)./((x0-x1).*(x0-x2))*y0 + (x-x0).*(x-x2)./((x1-x0).*(x1-x2))*y1 + (x-x0).*(x-x1)./((x2-x0).*(x2-x1)) * y2);

% we can change the data and see how the result changes
y0 = -6;
plot(x,0,x0,y0,'.',x1,y1,'.',x2,y2,'.',x,(x-x1).*(x-x2)./((x0-x1).*(x0-x2))*y0 + (x-x0).*(x-x2)./((x1-x0).*(x1-x2))*y1 + (x-x0).*(x-x1)./((x2-x0).*(x2-x1)) * y2);
y1 = -10;
plot(x,0,x0,y0,'.',x1,y1,'.',x2,y2,'.',x,(x-x1).*(x-x2)./((x0-x1).*(x0-x2))*y0 + (x-x0).*(x-x2)./((x1-x0).*(x1-x2))*y1 + (x-x0).*(x-x1)./((x2-x0).*(x2-x1)) * y2);

% Now let's make a interpolating polynomial that ends up being a line
% as the three points will be collinear
y1 = (y0+y2)/2;
plot(x,0,x0,y0,'.',x1,y1,'.',x2,y2,'.',x,(x-x1).*(x-x2)./((x0-x1).*(x0-x2))*y0 + (x-x0).*(x-x2)./((x1-x0).*(x1-x2))*y1 + (x-x0).*(x-x1)./((x2-x0).*(x2-x1)) * y2);

% You can even get a constant polynomial as a result
y0 = 2;
y1 = 2;
y2 = 2;
plot(x,0,x0,y0,'.',x1,y1,'.',x2,y2,'.',x,(x-x1).*(x-x2)./((x0-x1).*(x0-x2))*y0 + (x-x0).*(x-x2)./((x1-x0).*(x1-x2))*y1 + (x-x0).*(x-x1)./((x2-x0).*(x2-x1)) * y2);