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Fast evaluation of holonomic functions. (English) Zbl 0912.68081
Summary: 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 \(f(z')\).

MSC:
68W30 Symbolic computation and algebraic computation
68W10 Parallel algorithms in computer science
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