2 research outputs found
On rank and null space computation of the generalized Sylvester matrix
In this paper, we present new approaches computing the rank and the null space of the (mn + p)×(n + p) generalized Sylvester matrix of (m + 1) polynomials of maximal degrees n,p. We introduce an algorithm which handles directly a modification of the generalized Sylvester matrix, computing efficiently its rank and null space and replacing n by log2n in the required complexity of the classical methods. We propose also a modification of the Gauss-Jordan factorization method applied to the appropriately modified Sylvester matrix of two polynomials for computing simultaneously its rank and null space. The methods can work numerically and symbolically as well and are compared in respect of their error analysis, complexity and efficiency. Applications where the computation of the null space of the generalized Sylvester matrix is required, are also given. © 2009 Springer Science+Business Media, LLC
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