How to fit polynomials using Matlab.

gistfile1.txt

Fit polynomial:

  a*x^2 + b*x + c

to a bunch of data points (x,y).

Stuff the x into a vector. Let's just use a bunch of random points between 0 and 1:

octave-3.6.2.exe:1> x = rand(10,1)
x =

   0.94838
   0.10026
   0.68927
   0.98075
   0.24579
   0.29140
   0.98766
   0.54006
   0.31093
   0.12693

Now we need some y's. I'm just gonna cheat and use a quadratic function with some noise so we can see how good the fit is:

octave-3.6.2.exe:2> y = (x.^2 + 2*x + 1) + 0.1*(rand(size(x)) - 1)
y =

   3.7093
   1.1602
   2.7584
   3.9070
   1.4842
   1.5913
   3.8757
   2.2748
   1.6543
   1.1742

The ".^" there is pointwise exponentiation, the rest should be fairly self-explanatory. There's
".*" (pointwise multiply), "./" (pointwise divide) and so forth too.

Now here's how you solve the over-complete linear system for them:

octave-3.6.2.exe:3> [x.^2 x.^1 x.^0] \ y
ans =

   1.15039
   1.84023
   0.95267

Note how we got our original values, plus some noise, back.

The stuff in square brackets is a matrix with the elements of x squared in the first column,
x in the second column, and x to the 0'th power (i.e. 1) in the third. The "\" is just "right
divide": "A / b" = A * inverse(b), "A \ b" = inverse(A) * b.

If the matrix is over-complete, Matlab/Octave will automatically give you a (linear) least-squares
solution.