TR-2006007: Root-Finding with Eigen-Solving
نویسندگان
چکیده
We survey and extend the recent progress in polynomial root-finding based on reducing the problem to eigen-solving for highly structured generalized companion matrices. In particular we cover the selection of the eigen-solvers and the matrices and the resulting benefits based on exploiting matrix structure. No good estimates for global convergence of the basic eigen-solvers have been proved, but according to ample empirical evidence, they require a constant number of iteration steps per eigenvalue. If we assume such a bound, then the resulting root-finders are optimal up to a constant factor because they use linear arithmetic time per step and perform with a constant (double) precision. Some eigen-solvers also supply useful auxiliary results for polynomial root-finding, e.g., new proximity tests for the roots and new technical support for polynomial factorization. The algorithms can be extended to solving secular equations.
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