Ten Methods to Bound Multiple Roots of Polynomials

Given a univariate polynomial P with a k-fold multiple root or a k-fold root cluster near some e z, we discuss various different methods to compute a disc near e z which either contains exactly or contains at least k roots of P . Many of the presented methods are known, some are new. We are especially interested in rigorous methods, that is taking into account all possible effects of rounding errors. In other words every computed bound for a root cluster shall be mathematically correct. We display extensive test sets comparing the methods under different circumstances. Based on the results we present a hybrid method combining five of the previous methods which, for given e z, i) detects the number k of roots near e z and ii) computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster. Therefore, the resulting discs are numerically nearly optimal.