A holonomic function is an analytic function, which satisfies a linear differential equation Lf = 0 with polynomial coefficients. In particular, the elementary functions exp, log, sin, Etc., and many special functions such as erf, Si, Bessel functions, etc. Are holonomic Functions. In a previous paper, we have given an asymptotically fast algorithm to evaluate a Holonomic function f at a non-singular poin z' on the Riemann surface of f, up to any Number of decimal digits while estimating the error. However, this algorithm becomes Inefficient, when z' approaches a singularity of f.
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