When is a singularity not a singularity?

Abstract

In approximating a function with a pole‐like singularity at a non‐rational point a* by a rational polynomial, the approximating package will be robust (i.e. undetectably erroneous to a user) if the rational polynomial includes a factor which cancels out the pole to the full accuracy of the machine in use. Naturally in order to be portable, the approximation a to a* must be to the full machine accuracy of the receiving machine.