**for**

**Theorem (****Least-Squares
Polynomial Curve
Fitting****).**** **
Given the data
points , the
least squares polynomial of
degree m of the form

that fits the n data points is
obtained by solving the following linear system

for the m+1 coefficients . These
equations are referred to as the "normal equations".

**Proof ****Least
Squares Polynomials** **Least
Squares Polynomials**

One thing is certain, to find the least
squares polynomial the above linear system must be solved. There are
various linear system solvers that could be used for this
task. However, since this is such an important
computation, most mathematical software programs have a built-in
subroutine for this purpose. In *Mathematica* it is
called the " **Fit**" procedure. **Fit[data, funs,
vars]** finds a leastsquares fit to a list of data as a
linear combination of the functions **funs** of variables
vars.

We will check the "closeness of fit" with the **Root
Mean Square or RMS** measure for the "error in the
fit."

**Computer
Programs ****Least
Squares Polynomials** **Least
Squares Polynomials**

**Mathematica Subroutine (Least Squares
Parabola).**

**Example 1.** Find the
standard "least squares parabola" for
the data points .

Use the subroutine **LSParabola** to find the
line. Compare with the line obtained with
*Mathematica*'s **Fit** procedure.

**Solution
1.**

**Example 2.** Find the
polynomial curve fit of degree = 2 for the points
.

Use *Mathematica* to find the "Least Square Quadratic", and find
the RMS
error.

**Solution
2.**

**Example 3.** Find the
polynomial curve fit of degree = 3 for the points
.

Use *Mathematica* to find the "Least Square Cubic", and find the
RMS
error.

**Solution
3.**

**Example 4.** Find the
polynomial curve fit of degree = 4 for the points
.

Use *Mathematica* to find the "Least Square Quartic", and find
the RMS
error.

**Solution
4.**

**Example 5.** Find the
polynomial curve fit of degree = 5 for the points
.

Use *Mathematica* to find the "Least Square Quintic", and find
the RMS
error.

**Solution
5.**

**Example 6.** Why is
the RMS
error for essentially
zero ?

**Solution
6.**

**Caution for polynomial curve fitting.
**

Something goes radically wrong if the data is
radically "NOT polynomial." This phenomenon is called
"polynomial wiggle." The next example illustrates this
concept.

**Example 7.** Find the
least squares polynomial fits of degree n = 2, 3, 4, 5 for the points
.

**Solution
7.**

**Linear Least Squares**

The linear least-squares problem is stated as
follows. Suppose that
data points and
a set of
linearly independent functions
are given. We want to fine
coefficients so
that the function given
by the linear combination

will minimize the sum of the squares of the errors

.

**Theorem (Linear Least
Squares).** The solution to the linear least
squares problem is found by creating the matrix whose
elements are

The coefficients
are found by solving the linear system

where
and .

**Proof ****Least
Squares Polynomials** **Least
Squares Polynomials**

**Example 8.** Use the
linear least squares method to find the polynomial curve fit of
degree = 3 for the points .

**Solution
8.**

**Old Lab Project (****Least
Squares Polynomials** **Least
Squares
Polynomials****).** Internet
hyperlinks to an old lab project.

**Research Experience for
Undergraduates**

**Least
Squares Polynomials** **Least
Squares Polynomials** Internet hyperlinks to
web sites and a bibliography of articles.

**Download this
Mathematica Notebook****
****Least Squares
Poynomials**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004