Taylor series expansion of prismatic gravitational field

The exact analytical formulae to compute the gravitational field of a rectangular prism suffer from round-off errors when the evaluation point is outside the Brillouin sphere of the prism. The error magnitude grows cubically with respect to the distance from the prism. This phenomenon is eminent in not only the gravitational potential but also the gravity vector and the gravity gradient tensor. Unfortunately, the issue of error increase is not settled efficiently by the rewriting of the formulae using the addition theorems of the transcendental functions. Besides, the computational labour of the formulae is not small since at least 18 transcendental functions are employed for the potential computation each time. In order to solve these problems, we developed up to the 16th order 3-D Taylor series expansion of all the gravitational field quantities for a uniform rectangular prism of arbitrary dimensions. For instance, the eighth order truncation guarantees the eight digit accuracy of the potential computation of a nearly cubic prism at the cost of 11 per cent of that of the fast computation of the exact formula when the distance is more than 2.7 times longer than the Brillouin sphere radius. By using a simple algorithm, we present an adaptive procedure combining the truncated series of various orders and the exact formulae in order to compute precisely and quickly the gravitational field of an assembly of prisms everywhere. Exactly the same approach is applicable to the geomagnetic field computation.