Root-squaring with DPR1 matrices

Recent progress in polynomial root-finding relies on employing the associated companion and generalized companion DPR1 matrices. (“DPR1” stands for “diagonal plus rank-one.”) We propose an algorithm that uses nearly linear arithmetic time to square a DPR1 matrix. Consequently, the algorithm squares the roots of the associated characteristic polynomial. This incorporates the classical techniques of polynomial root-finding by means of root-squaring into a new effective framework. Our approach is distinct from the earlier fast methods for squaring companion matrices. Bibliography: 13 titles.