POLYFIT Fit Polynomial To Data

Usage

The polyfit routine has the following syntax

  p = polyfit(x,y,n)

where x and y are vectors of the same size, and n is the degree of the approximating polynomial. The resulting vector p forms the coefficients of the optimal polynomial (in descending degree) that fit y with x.

Function Internals

The polyfit routine finds the approximating polynomial

$$p(x) = p_1 x^n + p_2 x^{n-1} + \dots + p_n x + p_{n+1}$$

such that

$$\sum_{i} (p(x_i) - y_i)^2$$

is minimized. It does so by forming the Vandermonde matrix and solving the resulting set of equations using the backslash operator. Note that the Vandermonde matrix can become poorly conditioned with large n quite rapidly.

Example

A classic example from Edwards and Penny, consider the problem of approximating a sinusoid with a polynomial. We start with a vector of points evenly spaced on the unit interval, along with a vector of the sine of these points.

--> x = linspace(0,1,20);
--> y = sin(2*pi*x);
--> plot(x,y,'r-')

The resulting plot is shown here

Next, we fit a third degree polynomial to the sine, and use polyval to plot it

--> p = polyfit(x,y,3)
p = 
  <double>  - size: [1 4]
 
Columns 1 to 2
   21.9170418782353         -32.8755628173530        
 
Columns 3 to 4
   11.1897267234139          -0.115602892148147      
--> f = polyval(p,x);
--> plot(x,y,'r-',x,f,'ko');

The resulting plot is shown here

Increasing the order improves the fit, as

--> p = polyfit(x,y,11)
p = 
  <double>  - size: [1 12]
 
Columns 1 to 2
   12.4643875255456         -68.5541313918184        
 
Columns 3 to 4
  130.055546752557          -71.0939749422162        
 
Columns 5 to 6
  -38.2813761899149         -14.1222200242114        
 
Columns 7 to 8
   85.1017727524665          -0.564166385379567      
 
Columns 9 to 10
  -41.2861451730128          -0.00293966260467875    
 
Columns 11 to 12
    6.283246741110284        -1.26090310495983e-09   
--> f = polyval(p,x);
--> plot(x,y,'r-',x,f,'ko');

The resulting plot is shown here