# Arb

Arb is a C library for arbitrary-precision interval arithmetic. It has full support for both real and complex numbers. The library is thread-safe, portable, and extensively tested. Arb is free software distributed under the GNU Lesser General Public License (LGPL), version 2.1 or later.

Documentation: http://arblib.org

Development updates: http://fredrikj.net/blog/

Author: Fredrik Johansson [email protected]

Bug reports, feature requests and other comments are welcome in private communication, on the GitHub issue tracker, or on the FLINT mailing list [email protected].

## Code example

The following program evaluates `sin(pi + exp(-10000))`

. Since the input to the sine function matches a root to within 4343 digits, at least 4343-digit (14427-bit) precision is needed to get an accurate result. The program repeats the evaluation at 64-bit, 128-bit, ... precision, stopping only when the result is accurate to at least 53 bits.

```
#include "arb.h"
int main()
{
slong prec;
arb_t x, y;
arb_init(x); arb_init(y);
for (prec = 64; ; prec *= 2)
{
arb_const_pi(x, prec);
arb_set_si(y, -10000);
arb_exp(y, y, prec);
arb_add(x, x, y, prec);
arb_sin(y, x, prec);
arb_printn(y, 15, 0); printf("\n");
if (arb_rel_accuracy_bits(y) >= 53)
break;
}
arb_clear(x); arb_clear(y);
flint_cleanup();
}
```

The output is:

```
[+/- 6.01e-19]
[+/- 2.55e-38]
[+/- 8.01e-77]
[+/- 8.64e-154]
[+/- 5.37e-308]
[+/- 3.63e-616]
[+/- 1.07e-1232]
[+/- 9.27e-2466]
[-1.13548386531474e-4343 +/- 3.91e-4358]
```

Each line shows a rigorous enclosure of the exact value of the expression. The program demonstrates how the user can rely on Arb's automatic error bound tracking to get an output that is guaranteed to be accurate -- no error analysis needs to be done by the user.

For more example programs, see: http://arblib.org/examples.html

## Features

Besides basic arithmetic, Arb allows working with univariate polynomials, truncated power series, and matrices over both real and complex numbers.

Basic linear algebra is supported, including matrix multiplication, determinant, inverse, nonsingular solving, matrix exponential, and computation of eigenvalues and eigenvectors.

Support for polynomials and power series is quite extensive, including methods for composition, reversion, product trees, multipoint evaluation and interpolation, complex root isolation, and transcendental functions of power series.

Other features include root isolation for real functions, rigorous numerical integration of complex functions, and discrete Fourier transforms (DFTs).

## Special functions

Arb can compute a wide range of transcendental and special functions, including the gamma function, polygamma functions, Riemann zeta and Hurwitz zeta function, Dirichlet L-functions, polylogarithm, error function, Gauss hypergeometric function 2F1, confluent hypergeometric functions, Bessel functions, Airy functions, Legendre functions and other orthogonal polynomials, exponential and trigonometric integrals, incomplete gamma and beta functions, Jacobi theta functions, modular functions, Weierstrass elliptic functions, complete and incomplete elliptic integrals, arithmetic-geometric mean, Bernoulli numbers, partition function, Barnes G-function, Lambert W function.

## Speed

Arb uses a midpoint-radius (ball) representation of real numbers. At high precision, this allows doing interval arithmetic without significant overhead compared to plain floating-point arithmetic. Various low-level optimizations have also been implemented to reduce overhead at precisions of just a few machine words. Most operations on polynomials and power series use asymptotically fast FFT multiplication based on FLINT. Similarly, most operations on large matrices take advantage of the fast integer matrix multiplication in FLINT.

For basic arithmetic, Arb should generally be around as fast as MPFR (http://mpfr.org), though it can be a bit slower at low precision, and around twice as fast as MPFI (https://perso.ens-lyon.fr/nathalie.revol/software.html).

Transcendental functions in Arb are quite well optimized and should generally be faster than any other arbitrary-precision software currently available. The following table compares the time in seconds to evaluate the Gauss hypergeometric function `2F1(1/2, 1/4, 1, z)`

at the complex number `z = 5^(1/2) + 7^(1/2)i`

, to a given number of decimal digits (Arb 2.8-git and mpmath 0.19 on an 1.90 GHz Intel i5-4300U, Mathematica 9.0 on a 3.07 GHz Intel Xeon X5675).

Digits | Mathematica | mpmath | Arb |
---|---|---|---|

10 | 0.00066 | 0.00065 | 0.000071 |

100 | 0.0039 | 0.0012 | 0.00048 |

1000 | 0.23 | 1.2 | 0.0093 |

10000 | 42.6 | 84 | 0.56 |

## Dependencies, installation, and interfaces

Arb depends on FLINT (http://flintlib.org/), either GMP (http://gmplib.org) or MPIR (http://mpir.org), and MPFR (http://mpfr.org).

See http://arblib.org/setup.html for instructions on building and installing Arb directly from the source code. Arb might also be available (or coming soon) as a package for your Linux distribution.

SageMath (http://sagemath.org/) includes Arb as a standard package and contains a high-level Python interface. See the SageMath documentation for RealBallField (http://doc.sagemath.org/html/en/reference/rings_numerical/sage/rings/real_arb.html) and ComplexBallField (http://doc.sagemath.org/html/en/reference/rings_numerical/sage/rings/complex_arb.html).

Nemo (http://nemocas.org/) is a computer algebra package for the Julia programming language which includes a high-level Julia interface to Arb. The Nemo installation script will create a local installation of Arb along with other dependencies.

A standalone Python interface to FLINT and Arb is also available (https://github.com/fredrik-johansson/python-flint).

A separate wrapper of transcendental functions for use with the C99 `complex double`

type is available (https://github.com/fredrik-johansson/arbcmath).

Other third-party wrappers include:

- A Julia interface: https://github.com/JeffreySarnoff/ArbNumerics.jl
- Another Julia interface: https://github.com/JuliaArbTypes/ArbFloats.jl
- Java wrapper using JNA: https://github.com/crowlogic/arb/