How to compute cross terms

Is there a function to compute cross terms of a vector? For example, if I have (symbolically) a = [ a1, a2, a3];

Then the cross terms of a for order 2 would be:

[ a1^2, a2^2, a3^2, a1*a2, a1*a3, a2*a3]

And cross terms for order 3 would be:

[ a1^3, a1^2*a2, a1^2*a3, a1*a2^2, a1*a2*a3, a1*a3^2, a1^2*a2, a1*a2^2, a1*a2*a3, a2^3, a2^2*a3, a2*a3^2, a1^2*a3, a1*a2*a3, a1*a3^2, a2^2*a3, a2*a3^2, a3^3] (this contains duplicates which I want to avoid)

It's very close to the expansion of (a1+a2+a3)^3, but I want the individual terms instead of the sum. I also want it numerically, but symbolically would work if I can convert it.

Is there a Matlab function that takes a vector and computes these cross terms?

Answer 1

[John D'Errico]

Easy enough to do it with a loop. All that you need are the exponents. If you really want the symbolic forms later, trivial for that too.

function XT = xterms(N,nv)
% compute all cross terms of order N for nv variables.
XT = zeros(1,nv);
for n = 1:N
  XTnew = zeros(0,nv);
  for m = 1:nv
    XTm = XT;
    XTm(:,m) = XTm(:,m) + 1;
    XTnew = [XTnew;XTm];
  end
  XT = unique(XTnew,'rows');
end

So:

xt = xterms(2,3)
xt =
     0     0     2
     0     1     1
     0     2     0
     1     0     1
     1     1     0
     2     0     0

If you want a symbolic form for each term, (it is a REALLY bad idea to create numbered variables, so I do not advise it.) But this is trivial too.

syms a1 a2 a3
prod(repmat([a1 a2 a3],size(xt,1),1).^xt,2)
ans =
  a3^2
 a2*a3
  a2^2
 a1*a3
 a1*a2
  a1^2

As I said, the exponent array is all that you really need for any computations.

[Eugene]

Yes, thank you. I was thinking along these lines but you're observation that I only need the exponents was the best for avoiding working with numerical data (since I can't do a unique on numbers).

This put me on the correct path, so thanks!