Higher order numerical differentiation on the Infinity Computer
This work provides a novel computational tool for automatic differentiation in optimization and other fields, though it is limited to the specific Infinity Computer architecture.
The paper introduces a method for computing higher-order numerical derivatives on the Infinity Computer, which handles finite, infinite, and infinitesimal numbers. It proves that the Infinity Computer can automatically compute derivatives of arbitrary order to working precision for a wide class of functions.
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer -- the Infinity Computer -- able to work numerically with finite, infinite, and infinitesimal numbers. It is proved that the Infinity Computer is able to calculate values of derivatives of a higher order for a wide class of functions represented by computer procedures. It is shown that the ability to compute derivatives of arbitrary order automatically and accurate to working precision is an intrinsic property of the Infinity Computer related to its way of functioning. Numerical examples illustrating the new concepts and numerical tools are given.