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Re: 2-D Chebyshev Polynomial Regression

Chris Farr wrote:

> Has anyone created a function in Mathematica to approximate a function
> of two variables using 2-D Chebyshev Polynomial Regression?
> That is, has someone created a Mathematica algorithm which takes as its
> input a real valued function f(x,y) defined on [a,b] X [c,d] and
> returns a Chebyshev polynomial approximation p(x,y)?

The following code may do what you want. It takes a real valued function
f[x,y] defined on [-1,1] X [-1,1] and returns a Chebyshev polynomial
approximation p[f][x,y]. 

Extending the one-dimensional Chebyshev formula to two dimensions in an
obvious fashion,

In[1]:= c[i_, j_, m_, n_][f_] := c[i, j, m, n][f] = 
   Chop[N[4/(n m) Sum[f[Cos[Pi (k-1/2)/m], Cos[Pi (l-1/2)/n]]*
        Cos[Pi i (k-1/2)/m] Cos[Pi j (l-1/2)/n],{k,1,m},{l,1,n}]]]

and defining a short-hand notation for the Chebyshev T polynomials (with
the correction for the zeroth coefficient incorporated into the
polynomial itself to simplify the notation),

In[2]:= t[n_][x_]=ChebyshevT[n,x];
In[3]:= t[0][x_] = 1/2; 

then the (m,n)-th order polynomial approximation for f reads

In[4]:= p[m_,n_][f_]:= p[m,n][f] = 
   Function[{x,y},Sum[c[k,l,m,n][f] t[k][x]

For example, with the (pure) function

In[5]:= f = Function[{x,y},(x y^2)/(x + y + 1)]; 

the (3,3)-approximation reads

In[6]:= p[3,3][f][x,y]
                      2                              2 0.375 y x + 0.125
(2 y  - 1) x + 0.125 x + 0.125 (2 x  - 1) - 
            2                              2          2
  0.375 (2 x  - 1) y - 0.375 y + 0.125 (2 x  - 1) (2 y  - 1) + 
  0.125 (2 y  - 1) + 0.125

This code uses dynamic programming (i.e., g[x_]:= g[x] = ...) to store
intermediate computations and pure functions (so that computation for
new functions f is immediate).

The transformation from [a,b] X [c,d] to [-1,1] X [-1,1] is

	x -> (2x-a-b)/(b-a)
	y -> (2y-c-d)/(d-c)

with inverse

	x -> ((b-a)x+a+b)/2
	y -> ((d-c)y+c+d)/2

so only a simple modification of the above code is required to
generalize it to [a,b] X [c,d].

Note also that Mathematica includes a number of packages for related
computations including NumericalMath`Approximations` and

Hope that this helps.


Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul at  AUSTRALIA                   

            God IS a weakly left-handed dice player

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