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   \approx 1, an hypothesis which is almost never verified for polynomials approximating smooth functions.