Answer to Question #265804 in Quantitative Methods for Terro

Question

Is there any relation between the degree of the polynomial and the number of data points in polynomial curve fitting? If yes, write the relation. If no, write a brief (no more than two lines) explanation.

Answer 1

In general, whenever you have $n$ data points, you can always find a polynomial of degree $n-1$ which passes through all of them: 

$$\begin{bmatrix} 1 & x_1 &x_1^2 & ... & x_1^{n-1} \\ 1 & x_2 &x_2^2 & ... & x_2^{n-1} \\ 1 & x_3 &x_3^2 & ... & x_3^{n-1} \\ \vdots & \vdots & \vdots & ... & \vdots \\ 1 & x_n &x_n^2 & ... & x_n^{n-1} \\ \end{bmatrix}\begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ \vdots \\ c_n \end{bmatrix}=\begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ \vdots \\ y_n \end{bmatrix}$$