Higher order numerical differentiation on the Infinity Computer
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.
💡 Research Summary
The paper introduces a novel computational paradigm called the Infinity Computer, which is capable of handling finite, infinite, and infinitesimal numbers within a single arithmetic framework. The core of the approach lies in the representation of an infinitesimal unit, denoted ε (the reciprocal of the infinite unit ①), and the definition of arithmetic operations that preserve the distinct nature of ε throughout calculations. By feeding a program‑implemented function f(x) with the argument x + ε, the Infinity Computer evaluates f(x + ε) exactly as a Taylor series:
f(x + ε) = f(x) + f′(x)·ε + ½ f″(x)·ε² + …
Because ε never collapses to zero inside the machine, each coefficient of εⁿ in the result corresponds directly to the n‑th derivative of f at x divided by n!. The machine therefore provides a built‑in “infinitesimal expansion” mechanism that automatically yields derivatives of any order without the need for finite‑difference approximations, symbolic manipulation, or the construction of computational graphs typical of automatic differentiation (AD).
The authors formalize the underlying number system (the “grossone” system) and prove that for any function representable by a finite sequence of elementary operations, loops, and conditionals, the evaluation of f(x + ε) produces a finite grossone‑based expansion whose coefficients are mathematically exact derivatives. They also describe an extraction routine that isolates the ε⁰, ε¹, ε², … components, effectively turning the raw output into a vector of derivative values. This extraction is analogous to reading the mantissa of a floating‑point number but operates on the infinitesimal hierarchy instead of binary digits.
A key theoretical contribution is the demonstration that infinitesimal ε behaves as an algebraic entity distinct from zero, eliminating the classic truncation and round‑off errors associated with traditional finite‑difference schemes. Consequently, the method is numerically stable for any order of differentiation, even when the underlying function exhibits sharp discontinuities or piecewise definitions. The paper also discusses how nested infinitesimal perturbations can be used to compute higher‑order derivatives sequentially, extending the capability of the Infinity Computer beyond the first‑order focus of most AD frameworks.
The experimental section validates the theory on a diverse test suite: simple polynomials, trigonometric and logarithmic functions, and more complex piecewise or loop‑driven algorithms. In all cases, the Infinity Computer delivers derivative values accurate to the working precision (typically double‑precision 64‑bit) and shows orders‑of‑magnitude lower error than forward, central, or complex‑step finite‑difference methods. Notably, third‑ and fourth‑order derivatives, which are notoriously noisy in conventional numerical schemes, are obtained with machine‑exact precision and no observable numerical noise.
Finally, the authors discuss practical implications. The ability to obtain arbitrary‑order derivatives automatically and accurately opens new possibilities in optimization (e.g., higher‑order Newton or tensor methods), sensitivity analysis, and scientific simulation where Jacobians, Hessians, or higher‑order tensors are required. Because the infinitesimal arithmetic is implemented as a logical extension of standard arithmetic, it can be integrated into existing hardware or software stacks without substantial overhead. The paper concludes that the Infinity Computer provides a fundamentally new route to automatic high‑order differentiation, overcoming the limitations of both symbolic differentiation (which can be computationally prohibitive) and traditional numerical differentiation (which suffers from truncation and round‑off errors).