Calcium: computing in exact real and complex fields

TL;DR

Calcium is a C library enabling exact real and complex number computations using algebraic and symbolic methods, allowing rigorous equality checks for a broad class of numbers.

Contribution

It introduces a practical computational model for real and complex numbers using algebraic extensions with automatic relation discovery and certification.

Findings

Efficient field operations for algebraic and transcendental extensions

Automatic discovery and certification of algebraic relations

Decidable equality for a large class of numbers

Abstract

Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields $\mathbb{Q}(a_1,\ldots,a_n)$ where the extensions numbers $a_k$ may be algebraic or transcendental. The system combines efficient field operations with automatic discovery and certification of algebraic relations, resulting in a practical computational model of $\mathbb{R}$ and $\mathbb{C}$ in which equality is rigorously decidable for a large class of numbers.