Error analysis of the algorithm for shifting the zeros of a polynomial by synthetic division

An analysis is given of the role of rounding errors in the synthetic division algorithm for computing the coefficients of the polynomial $g(z) = f(z + s)$ from the coefficients of the polynomial f. It is shown that if $\vert z + s\vert \cong \vert z\vert + \vert s\vert$ then the value of the computed polynomial ${g^\ast}(z)$ differs from $g(z)$ by no more than a bound on the error made in computing $f(z + s)$ with rounding error. It may be concluded that well-conditioned zeros of f lying near s will not be seriously disturbed by the shift.