# Feature overview¶

Arb builds upon FLINT, which deals with efficient computation over exact domains such as the rational numbers and finite fields. Arb extends FLINT to cover computations with real and complex numbers. The problem when computing with real and complex numbers is that approximations (typically floating-point numbers) must be used, potentially leading to incorrect results.

Ball arithmetic, also known as mid-rad interval arithmetic, is an extension of floating-point arithmetic in which an error bound is attached to each variable. This allows computing rigorously with real and complex numbers.

With plain floating-point arithmetic, the user must do an error analysis to guarantee that results are correct. Manual error analysis is time-consuming and bug-prone. Ball arithmetic effectively makes error analysis automatic.

In traditional (inf-sup) interval arithmetic, both endpoints of an interval $$[a,b]$$ are full-precision numbers, which makes interval arithmetic twice as expensive as floating-point arithmetic. In ball arithmetic, only the midpoint m of an interval $$[m \pm r]$$ is a full-precision number, and a few bits suffice for the radius r. At high precision, ball arithmetic is therefore not more expensive than plain floating-point arithmetic.

Joris van der Hoeven’s paper [Hoe2009] is a good introduction to the subject.

Other implementations of ball arithmetic include iRRAM and Mathemagix. Arb differs from earlier implementations in technical aspects of the implementation, which makes certain computations more efficient. It also provides a more comprehensive low-level interface, giving the user full access to the internals. Finally, it implements a wider range of transcendental functions, covering a large portion of the special functions in standard reference works such as [NIST2012].

Arb is designed for computer algebra and computational number theory, but may be useful in any area demanding reliable or precise numerical computing. Arb scales seamlessly from tens of digits up to billions of digits. Efficiency is achieved by low level optimizations and use of asymptotically fast algorithms.

Arb contains:

• A module (arf) for correctly rounded arbitrary-precision floating-point arithmetic. Arb’s floating-point numbers have a few special features, such as arbitrary-size exponents (useful for combinatorics and asymptotics) and dynamic allocation (facilitating implementation of hybrid integer/floating-point and mixed-precision algorithms).

• A module (mag) for representing magnitudes (error bounds) more efficiently than with an arbitrary-precision floating-point type.

• A module (arb) for real ball arithmetic, where a ball is implemented as an arf midpoint and a mag radius.

• A module (acb) for complex numbers in rectangular form, represented as pairs of real balls.

• Modules (arb_poly, acb_poly) for polynomials or power series over the real and complex numbers, implemented using balls as coefficients, with asymptotically fast polynomial multiplication and many other operations.

• Modules (arb_mat, acb_mat) for matrices over the real and complex numbers, implemented using balls as coefficients. At the moment, only rudimentary linear algebra operations are provided.

• Functions for high-precision evaluation of various mathematical constants and special functions, implemented using ball arithmetic with rigorous error bounds.

Arb 1.x used a different set of numerical base types (fmpr, fmprb and fmpcb). These types had a slightly simpler internal representation, but generally had worse performance. All methods for the Arb 1.x types have now been ported to faster equivalents for the Arb 2.x types. The last version to include both the Arb 1.x and Arb 2.x types and methods was Arb 2.2. As of Arb 2.9, only a small set of fmpr methods are left for fallback and testing purposes.

Arb uses GMP / MPIR and FLINT for the underlying integer arithmetic and various utility functions. Arb also uses MPFR for testing purposes and internally to evaluate some functions.