I understand the concept of Lagrange Interpolation but am having issues understanding how to interpret the following general equation (which I will be provided) for n points. For example, how would you get the equation for n = 4 points from the general equation below?
Thanks!
Let $x_1,x_2,x_3,x_4$ be mutually distinct numbers and we need to fit the polynomial of degree 3 passing through $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. Then $$p(x)=L_1(x)y_1+L_2(x)y_2+L_3(x)y_3+L_4(x)y_4,$$ where $$L_1(x)=\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)},$$ $$L_2(x)=\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)},$$ $$L_3(x)=\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)},$$ $$L_4(x)=\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}.$$