Accurate solution of polynomial equations using Macaulay resultant matrices

We propose an algorithm for solving two polynomial equations in two variables. Our algorithm is based on the Macaulay resultant approach, combined with new techniques including randomization to make the algorithm accurate in the presence of roundoff error. The ultimate computation is the solution of a generalized eigenvalue problem via the QZ method. We analyze the error due to roundoff of the method, showing that with high probability the roots are computed accurately, assuming that the input data (that is, the two polynomials) are well-conditioned. Our analysis requires a novel combination of algebraic and numerical techniques.