Bairstow s method loses its quadratic rate of convergence when trying to find a quadratic factor of multiplicity greater than one although there are a couple of modifications to it which fix this I do not believe it loses its quadratic convergence to a single quadratic factor based on whether that quadratic factor has distinct or identical roots as it is searching for the quadratic coefficients and not for the roots directly I ve never seen a reference which stated the latter specifically but there is lots of literature on the quadratic multiplicity difficulty and one would conjecture that if double roots also caused a problem they would be mentioned in that