Bivariate polynomial solver by values

The purpose of this thesis is to provide the mathematical tools for computing simple isolated roots of bivariate (and univariate) polynomial equations described by their values or some of the derivatives at some nodes. The polynomial root-finding problem has a long and rich history in computational mathematics, mainly because of its frequent applications in engineering. It may be the case that the polynomial system as it appears naturally in the application formulation is not presented in the familiar monomial basis. The approach we take here avoids conversion between bases. This is important because converting from the basis in which the polynomials are given to the monomial basis (or any other basis) is likely to degrade the conditioning of the problem.;A complete chapter of this thesis is devoted to studying the application of the presented algorithm in Computer Aided Geometric Design. There are also other geometric applications for which our root-finding algorithm can apply. The applications we have introduced here are merely representative.;Our experiments show that the algorithm presented here is numerically stable. While this thesis has not been primarily concerned with efficiency the algorithms presented here are efficient enough to suggest that the effort involved in making them competitive will be worthwhile.;The root-finding algorithm presented in this thesis is a hybrid of a resultant-type algorithm and eigenvalue techniques. Here, we use some new formulations for the Bezout matrix and the companion matrix pencil associated with a pair of bivariate polynomial equations described by their values. These two matrix constructions are the essential tools to compute all the simple isolated roots of a bivariate polynomial system. We use the Bezout matrix to eliminate one of the variables and then we construct the companion matrix pencil to convert our polynomial root-finding problem to a generalized eigenvalue problem. We then use conventional software to compute the eigenvalues.