FreeMat
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Section: Optimization and Curve Fitting
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
.
The polyfit
routine finds the approximating polynomial
such that
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.
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
polyval
to plot it
--> p = polyfit(x,y,3) p = 21.9170 -32.8756 11.1897 -0.1156 --> f = polyval(p,x); --> plot(x,y,'r-',x,f,'ko');
The resulting plot is shown here
--> p = polyfit(x,y,11) p = Columns 1 to 7 12.4644 -68.5541 130.0555 -71.0940 -38.2814 -14.1222 85.1018 Columns 8 to 12 -0.5642 -41.2861 -0.0029 6.2832 -0.0000 --> f = polyval(p,x); --> plot(x,y,'r-',x,f,'ko');
The resulting plot is shown here