Geodesic
Root-squaring with DPR1 matrices
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 189-193

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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 new effective framework. Our approach is distinct from the earlier fast methods for squaring companion matrices. Bibl. – 13 titles. 
@article{ZNSL_2009_373_a11,
     author = {V. Y. Pan},
     title = {Root-squaring with {DPR1} matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--193},
     publisher = {mathdoc},
     volume = {373},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a11/}
}
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V. Y. Pan. Root-squaring with DPR1 matrices. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 189-193. http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a11/