Min-Max Elementwise Backward Error for Roots of Polynomials and a Corresponding Backward Stable Root Finder
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A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials p(z). Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on the coefficients of p(z) that do not participate much in the overall backward error. By how much these coefficients can be perturbed is determined via an associated max-times polynomial and its tropical roots. An algorithm is designed for computing the roots of p(z). It uses a companion linearization C(z) = A-zB of p(z) to which we added an extra zero leading coefficient, and an appropriate two-sided diagonal scaling that balances A and makes B graded in particular when there is variation in the magnitude of the coefficients of p(z). An implementation of the QZ algorithm with a strict deflation criterion for eigenvalues at infinity is then used to obtain approximations to the roots of p(z). Under the assumption that this implementation of the QZ algorithm exhibits a graded backward error when B is graded, we prove that our new algorithm is min-max elementwise backward stable. Several numerical experiments show the superior performance of the new algorithm compared with the MATLAB roots function. Extending the algorithm to polynomial eigenvalue problems leads to a new polynomial eigensolver that exhibits excellent numerical behaviour compared with other existing polynomial eigensolvers, as illustrated by many numerical tests.
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