• bivariate rootfinding
• resultant-based method
• Bezout matrix

The aim of this thesis consists in analyzing a method for bivariate rootfinding, that is, computing the common roots of two bivariate polynomials. In particular, we are interested in numerically approximate the common zeros in the domain [-1,1]^2. In order to solve the rootfinding problem, we rely on the resultant-based method in combination with an hidden variable technique. Among the different resultant matrices, we choose the Bézout resultant matrix over the more widely used Sylvester matrix. The Bézout matrix can be expressed as a matrix polynomial. The efficient construction of this resultant matrix is a non-trivial step, for which we analyze two possible approaches. The first technique exploits the more general connection between matrix polynomials and a class of linearizations, known in the literature as double ansatz space. The second approach relies on a three-dimensional interpolation on the grid of Chebyshev-Lobatto points.The core of the resultant-based method consists in converting the rootfinding problem into an eigenvalue problem associated with the matrix polynomial representing the Bézout matrix. In order to solve this eigenvalue problem, we employ the colleague pencil, which is a companion-like matrix pencil designed for matrix polynomials expressed in the Chebyshev basis. We develop an estimate of the forward error associated with each computed eigenvalue of the matrix polynomial. The main goal consists in locating a disk of inclusion for each approximate eigenvalue, by combining bounds for its backward error and its condition number.