Before revision:  The (unbalanced) QR iteration used for Chebyshev rootfinding does not guarantee backward stability on the polynomial coefficients, unless the vector of coefficients satisfy  p  \approx 1, an hypothesis which is almost never verified for polynomials approximating smooth functions.
After revision:  The (unbalanced) QR iteration used for Chebyshev rootfinding does not guarantee backward stability on the polynomial coefficients, unless the vector of coefficients satisfy  p \|p\| \approx 1, an hypothesis which is almost never verified for polynomials approximating smooth functions.