Is the polynomial so perfidious? (Q1338801)

Some analyses are presented which indicate that: 1) It might be advantageous to have initial approximations to the root of a polynomial equation which are well away from the region of limiting convergence. 2) Polynomial deflation can be numerically stable when the polynomial is divided by \(z- \widehat\alpha\), and \(\widehat\alpha\) is the numerically computed largest zero of the polynomial, as long as we use a modified deflation algorithm which is akin to backward recursion [cf. \textit{J. H. Wilkinson}, MAA Stud. Math. 24, 1-28 (1984; Zbl 0601.65028)]. 3) Bairstow's method is numerically stable when the approximate quadratic factor \(z^ 2- \alpha z- \beta\) is associated with the largest roots (in modulus) of the given polynomial \(p(z)\) of degree \(n\), provided that a modified algorithm is used. 4) Le Verrier's method can be recast and the instability noted by \textit{J. H. Wilkinson} [The algebraic eigenvalue problem (1965; Zbl 0258.65037)] can be used to advantage as we now have an indicator that multiple precision arithmetic is needed to get more accurate results.