Higher order numerical differentiation on the Infinity Computer

TL;DR

This paper introduces a novel method for higher order numerical differentiation using the Infinity Computer, capable of automatically and accurately computing derivatives of arbitrary order for functions represented by computer procedures.

Contribution

It presents the Infinity Computer as a new computational paradigm that can automatically and precisely compute higher order derivatives for a broad class of functions.

Findings

Infinity Computer can compute derivatives of arbitrary order.

The method is accurate to working precision.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

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.