Structured matrix methods for a polynomial root solver using approximate greatest common divisor computations and approximate polynomial factorisations.

This thesis discusses the use of structure preserving matrix methods for the numerical
approximation of all the zeros of a univariate polynomial in the presence of
noise. In particular, a robust polynomial root solver is developed for the calculation
of the multiple roots and their multiplicities, such that the knowledge of the noise
level is not required. This designed root solver involves repeated approximate greatest
common divisor computations and polynomial divisions, both of which are ill-posed
computations. A detailed description of the implementation of this root solver is
presented as the main work of this thesis. Moreover, the root solver, implemented
in MATLAB using 32-bit floating point arithmetic, can be used to solve non-trivial
polynomials with a great degree of accuracy in numerical examples.