# Maple example for interpolating polynomials -- 3/27
# CS 323 - Doug DeCarlo

# Lagrange interpolation
L0 := x -> (x-x1)*(x-x2)/((x0-x1)*(x0-x2));
L1 := x -> (x-x0)*(x-x2)/((x1-x0)*(x1-x2));
L2 := x -> (x-x0)*(x-x1)/((x2-x0)*(x2-x1));
P2 := x -> L0(x)*y0 + L1(x)*y1 + L2(x)*y2;

collect(P2(x), x);

# Let's look at the quadratic coefficient
simplify(coeff(P2(x), x, 2));

# ----

# Newton divided differences for interpolation

dd01 := (y1 - y0)/(x1-x0);
dd12 := (y2 - y1)/(x2-x1);
dd012 := (dd12-dd01)/(x2-x0);

P1n := x -> y0 + dd01*(x-x0);
P2n := x -> P1n(x) + dd012*(x-x0)*(x-x1);

# Are they the same?  (Yes -- this produces zero)
simplify(P2(x) - P2n(x));