Fast evaluation of holonomic functions

Abstract

A holonomic function is an analytic function, which satisfies a linear differential equation with polynomial coefficients. In particular, the elementary functions exp, log, sin, etc. and many special functions like erf, Si, Bessel functions, etc. are holonomic functions. Given a holonomic function f (determined by the linear differential equation it satisfies and initial conditions in a non singular point z), we show how to perform arbitrary precision evaluations of f at a non singular point z′ on the Riemann surface of f, while estimating the error. Moreover, if the coefficients of the polynomials in the equation for f are algebraic numbers, then our algorithm is asymptotically very fast: if M( n) is the time needed to multiply two n digit numbers, then we need a time O( M( n log 2 n log log n)) to compute n digits of f( z′).