**Chapter 7 Approximation
Theory**

**Least Square Approximation**

The general problem of fitting the best least squares line to a collection of data involves minimizing with respect to the parameters and . For a minimum to occur, it it necessary that

,

and .

These equations simplify to
the **normal equations:**

and .

The solution to this system of equations is

,

and

.

The general problem of approximating a set of data, , with an algebraic polynomial of degree using the least squares procedure is handled in a similar manner. It requires choosing the constants to minimize the least squares error

As in the linear case, for to be minimized, it is necessary that for each . Thus, for each ,

.

This gives **normal
equations** in the unknowns, ,

, .

It is helpful to write the equations as follows:

It can be shown that the normal equations have a unique solution provided that the , are distinct.

Occasionally it is appropriate to assume that the data are exponentially related. This requires the approximating function to be of the form

(1)

or

(2)

for some constants and . Considering the logarithm of the approximating equations:

, in the case of Eq. (1)

and , in the case of Eq. (2).

In either case, a linear problem now appears and solutions for and can be obtained by appropriately modifying the normal equations (1) and (2).

In fitting a function to data points , a linear combination of any known functions, including polynomials, may be used:

(3)

where are prescribed functions, are undetermined coefficients, and is the total number of prescribed functions. By fitting Eq. (3) to each data point, an over-determined equation is written as

,

with

,

, and , where .

References:

【1】
R. L. Burden and J. D. Faires, *Numerical Analysis*, PWS,

【2】
S. Nakamura, *Numerical
Analysis and Graphic Visualization with MATLAB*,