Algorithms for Computing Selected Solutions of Polynomial Equations

We present eecient and accurate algorithms to compute solutions of zero-dimensional multivariate polynomial equations in a given domain. The total number of solutions correspond to the Bezout bound for dense polynomial systems or the Bernstein bound for sparse systems. In most applications the actual number of solutions in the domain of interest is much lower than the Bezout or Bernstein bound. Our approach is based on global symbolic formulation of the problem using resultants and matrix computations and localizing it to nd selected solutions based on numerical computations. The problem of nding roots is reduced to computing eigenvalues of a generalized companion matrix and we use the structure of the matrix to compute the solutions in the domain of interest only. The resulting algorithm combines symbolic preprocessing with numerical iterations and works well in practice. We discuss its performance on a number of applications.