Univariate polynomial root-finding by arming with constraints

• Authors: Victor Y. Pan
• Venue: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
• Year: 2012

Elimination methods are highly effective for the solution of linear and nonlinear systems of equations, but there are important examples where reversing the elimination idea can improve global convergence of iterative methods, that is their convergence from the start. We believe that these examples reveal a direction that the algorithm designers should explore systematically, and we show this principle at work for the approximation of a single root (zero) of a univariate polynomial of a degree n. We reduce the latter task to the solution of a multivariate polynomial system of n equations with n unknowns and achieve faster and more consistent global convergence of Newton's iteration applied to the system rather than a single equation. Global convergence behavior of our algorithms is similar to that of Durand--Kerner's (Weierstrass') iteration, but we direct convergence to a single root and use linear arithmetic time per iteration loop. Having m processors one can apply the algorithms concurrently, to approximate up to m roots within the same parallel time. Technically the computations boil down to solving Sylvester or generalized Sylvester linear systems of equations, linked to partial fraction decompositions. By solving these tasks efficiently, we arrive at effective algorithms for polynomial root-finding.