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 least�squares 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."

Mathematica Subroutine (Least Squares Parabola).

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.  

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  .