MATLAB: Numerical sensitivity of function kron(X,Y)

floating pointkronecker productmatrix

Assume X and Y are both symmetric matrices with double floating-point format. Can kron(X,Y) potentially become asymmetric due to the floating point numerics similar to the situation like X'*Y*X ?

I personally think it would be quite unlikely to have the problem I stated above given the mathematical definition of the Kronecker product. Nevertheless, if anyone has suggestions, please share them.

Thanks

Best Answer

My instinct and my tests say no. There are only scalar multiplications in A=kron(X,Y), so those should be pretty safe. However, if you plan to start multiplying A with other data, the benefits of the symmetry might go out the window pretty fast. Note, for example, that C as computed with,

C=reshape(  kron(X,X)*Y(:)    ,N,N);

should be symmetric in exact arithmetic for symmetric X and Y. However, I find that this fails in most numerical tests, likely due to the very same issue you pointed out with X.'*Y*X.

Related Solutions

MATLAB: Numerical sensitivity of X’YX given X is a permutation matrix

Not a true answer, but IF it were going to happen, you would see it here:

[Q,R,X] = qr(rand(3000));
Y = randn(3000);
Y = Y + Y';
Z = X'*Y*X;
all(all((Z-Z') == 0))
ans =
  logical
   1

Often a simple test is worth a lot, just to verify intuition.

So X is a random permutation matrix. Y a random symmetric matrix. Z is perfectly symmetrical. I chose a large enough matrix so that it had 4 CPUs running on the problem, so the BLAS will be kicking in. Usually when there are problems on something like this, it is the BLAS that are claimed to be the culprit.

I think the important point is (I see David made this in his comment) that there are NO adds involved between two non-zero numbers. All of the adds will always end up being of the form u+0 or 0+u, where there can never be any kind of error introduced into the least significant bits.

So I'll claim that this operation, where Y is symmetric and X a permutation matrix will never introduce an asymmetry into the result.

MATLAB: Relational operator == doesn’t work properly in some cases!

You are seeing the effects of floating-point approximation error.

See for example: Why is 0.3 - 0.2 - 0.1 (or similar) not equal to zero? and Floating-Point Numbers.

Best Answer

Related Solutions

MATLAB: Numerical sensitivity of X’*Y*X given X is a permutation matrix

MATLAB: Relational operator == doesn’t work properly in some cases!

Related Question

MATLAB: Numerical sensitivity of X’YX given X is a permutation matrix