libwignernj: a reusable C/C++/Fortran/Python library for exact Wigner symbols and related coefficients

TL;DR

libwignernj is a versatile, open-source library providing exact evaluations of Wigner symbols and related coefficients in multiple programming languages, ensuring high precision and easy integration for scientific applications.

Contribution

It introduces a new, exact rational-based computational approach for Wigner symbols, with multi-language bindings and no runtime dependencies, enhancing precision and usability.

Findings

Provides exact rational evaluations with correct last-bit precision.

Supports multiple precisions including IEEE 754 binary128 and arbitrary precision.

Includes bindings for C++, Python, and Fortran with easy integration.

Abstract

We describe libwignernj, a freely available, BSD-licensed library that evaluates Wigner 3j, 6j, and 9j symbols, Clebsch--Gordan, Racah $W$, and Fano $X$ coefficients, and Gaunt coefficients over both complex and real spherical harmonics in standards-compliant C99. libwignernj represents factorials by the vector of their signed prime-exponent decomposition - a prime-factorization technique introduced for the angular-momentum coefficients by Dodds and Wiechers (Comput. Phys. Commun. 4, 268 (1972)) and refined in a long line of subsequent work - and combines that representation with the multiword-integer Racah sum of Johansson and Forss\'en (SIAM J. Sci. Comput. 38, A376 (2016)), under which every intermediate quantity is an exact rational and all rounding is confined to the final floating-point conversion. Single-, double-, and long-double-precision results are correct to the last representable bit, and IEEE 754 binary128 evaluation through libquadmath and arbitrary-precision evaluation through the GNU Multiple-Precision Floating-Point Reliable (MPFR) library are optionally exposed. libwignernj has no mandatory runtime dependencies and no caller-side initialization step, making it easy to embed across the atomic, molecular, nuclear, and electromagnetic-scattering applications in which these coefficients arise. C++, CPython, and Fortran 90 bindings ship alongside the C library. Half-integer angular momenta are encoded exactly via integer $2j$ arguments throughout the application programming interface (API). CMake-package and pkg-config files ship for drop-in integration into downstream projects, and a continuous-integration (CI) pipeline runs the full test suite on Linux (shared and static), macOS, and Windows on every push.