Least Squares Fitting--Polynomial

Generalizing from a straight line (i.e., first degree polynomial) to a th degree polynomial

(1)

the residual is given by

(2)

The partial derivatives (again dropping superscripts) are

			(3)
			(4)
			(5)

These lead to the equations

			(6)
			(7)
			(8)

or, in matrix form

(9)

This is a Vandermonde matrix. We can also obtain the matrix for a least squares fit by writing

(10)

Premultiplying both sides by the transpose of the first matrix then gives

(11)

so

(12)

As before, given  points  and fitting with polynomial coefficients , ...,  gives

(13)

In matrix notation, the equation for a polynomial fit is given by

(14)

This can be solved by premultiplying by the transpose ,

(15)

This matrix equation can be solved numerically, or can be inverted directly if it is well formed, to yield the solution vector

(16)

Setting  in the above equations reproduces the linear solution.