Least Squares Fitting--Polynomial

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

the residual is given by

The partial derivatives (again dropping superscripts) are

These lead to the equations

or, in matrix form

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

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

so

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

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

This can be solved by premultiplying by the transpose ,

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

Setting in the above equations reproduces the linear solution.