Structured matrix methods for the computation of multiple roots of a polynomial

This paper considers the application of structured matrix methods for the computation of multiple roots of a polynomial. In particular, the given polynomial f(y) is formed by the addition of noise to the coefficients of its exact form f̂(y), and the noise causes multiple roots of f̂(y) to break up into simple roots. It is shown that structured matrix methods enable the simple roots of f(y) that originate from the same multiple root of f̂(y) to be 'sewn' together, which therefore allows the multiple roots of f̂(y) to be computed. The algorithm that achieves these results involves several greatest common divisor computations and polynomial deconvolutions, and special care is required for the implementation of these operations because they are ill-posed. Computational examples that demonstrate the theory are included, and the results are compared with the results from MultRoot, which is a suite of Matlab programs for the computation of multiple roots of a polynomial.