Computation of a specified root of a polynomial system of equations using eigenvectors

Abstract

We propose new techniques and algorithms for the solution of a polynomial system of equations by matrix methods. For such a system, we seek its specified root, at which a fixed polynomial takes its maximum or minimum absolute value on the set of roots. We unify several known approaches and simplify the solution substantially, in particular in the case of an overconstrained polynomial system having only a simple root or a few roots. We reduce the solution to the computation of the eigenvector of an associated dense matrix, but we define this matrix implicitly, as a Schur complement in a sparse and structured matrix, and then modify the known methods for sparse eigenvector computation. This enables the acceleration of the solution by roughly factor D, the number of roots. Our experiments show that the computations can be performed numerically, with no increase of the computational precision, and the iteration converges to the specified root quite fast.