On the convergence and divergence of Bairstow's method

SummaryNecessary and sufficient condition of algebraic character is given for the invertibility of the Jacobian matrix in Bairstow's method. This leads to a sufficient condition for local quadratic convergence. Results also yield the rank of the Jacobian, when it is singular. In the second part of the paper we give several examples for two step cyclization and more complicated kind of divergence. Some of the polynomials in divergence examples are proved to be irreducible over the field of rational numbers.