# Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $$B = A U$$.
The first is a regular Toeplitz matrix $$A$$, while the second is an upper triangular matrix $$U$$.
I thought to use the QR decomposition on $$A$$ in such a way that $$A U = (Q R) U = Q (R U)$$
Since the product of two upper triangular matrices is an upper triangular matrix I would still get a QR type factorisation. Now I have two questions:

1. Is this the best way to calculate eigenvalues or are there more efficient ways since they are structured matrices?
2. If the proposed way is correct, which is the best algorithm to factorize the Toeplitz A matrix? Is it better to transform the Toeplitz matrix into a Hessenberg matrix?

Thank you very much