Error analysis of the algorithm for shifting the zeros of a polynomial by synthetic division
G. W. Stewart · Mathematics of Computation · 1971
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 ) g(z) = f(z + s) from the coefficients of the polynomial f . It is shown that if | z + s | ≅ | z | + | s | |z + s| \cong |z| + |s| then the value of the computed polynomial g ∗ ( z ) {g^\ast }(z) differs from g ( z ) g(z) by no more than a bound on the error made in computing f ( z + s ) 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.