Newton Iteration: From Numerics to Combinatorics, and Back

Abstract

The talk will explore a variety of old and recent algorithms whose efficiency boils down to the fast convergence of Newton iteration. Numerically, and close to the root, the number of correct digits is doubled at each iteration. When working with power series, the problem of picking a good initial point disappears and the number of coefficients is doubled at each iteration. This observation, coupled with fast multiplication, leads to fast algorithms in a variety of problems of symbolic computation, ranging from classical results on algebraic series to more recent ones on systems of differential equations.