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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
##### Keywords:
holonomic function
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##### References:
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