From quotient-difference to generalized eigenvalues and sparse polynomial interpolation

Abstract

The numerical quotient-difference algorithm,or the qd-algorithm, can be used for determining the poles of a meromorphic function directly from its Taylor coeffcients. We show that the poles computed in the qd-algorithm, regardless of their multiplicities,are converging to the solution of a generalized eigenvalue problem. In a special case when all the poles are simple,such generalized eigenvalue problem can be viewed as a reformulation of Prony 's method,a method that is closely related to the Ben-Or/Tiwari algorithm for interpolating a multivariate sparse polynomial in computer algebra.