Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

TL;DR

Arb is a C library that provides efficient arbitrary-precision interval arithmetic using midpoint-radius representation, supporting various mathematical objects and functions with high performance.

Contribution

It introduces a versatile and fast implementation of ball arithmetic for real and complex numbers, polynomials, and special functions, with optimized polynomial arithmetic and error management.

Findings

Competitive performance with MPFR and MPC floating-point types

Supports a wide range of mathematical objects and functions

Efficient polynomial arithmetic with interval coefficients

Abstract

Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic. It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many special functions. The core number types are designed for versatility and speed in a range of scenarios, allowing performance that is competitive with non-interval arbitrary-precision types such as MPFR and MPC floating-point numbers. We discuss the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients.