Today I have implemented Aberth's method for complex polynomial root finding. And I have to say I am enchanted about its astonishing performance and its intriguing simplicity. Before I go on believing (having quite some experience with numerics, I should say: naively) that this is a fool-proof all-purpose root finding method, I would like to challenge it a little.
Apart from Wilkinson's polynomial, are there other notorious polynomials that often cause root finding algorithms to fail?
Maybe someone even knows a case, where Aberth's method fails in particular?