Specific "scientific" data structures, and their processing

TL;DR

This paper introduces a functional programming approach using Haskell to handle complex, specialized data structures in physics and mathematics, enabling natural manipulation of infinite and semi-numerical data.

Contribution

It demonstrates how lazy evaluation and Haskell datatypes facilitate working with advanced mathematical data structures in a flexible, embedded language framework.

Findings

Enables natural manipulation of infinite sequences.

Supports semi-numerical and symbolic data processing.

Integrates complex data structures into object-oriented frameworks.

Abstract

Programming physicists use, as all programmers, arrays, lists, tuples, records, etc., and this requires some change in their thought patterns while converting their formulae into some code, since the "data structures" operated upon, while elaborating some theory and its consequences, are rather: power series and Pad\'e approximants, differential forms and other instances of differential algebras, functionals (for the variational calculus), trajectories (solutions of differential equations), Young diagrams and Feynman graphs, etc. Such data is often used in a [semi-]numerical setting, not necessarily "symbolic", appropriate for the computer algebra packages. Modules adapted to such data may be "just libraries", but often they become specific, embedded sub-languages, typically mapped into object-oriented frameworks, with overloaded mathematical operations. Here we present a functional approach to this philosophy. We show how the usage of Haskell datatypes and - fundamental for our tutorial - the application of lazy evaluation makes it possible to operate upon such data (in particular: the "infinite" sequences) in a natural and comfortable manner.