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.

Documentation: http://fredrikj.net/arb/

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

Author: Fredrik Johansson fredrik.johansson@gmail.com

Bug reports, feature requests and other comments are welcome in private communication, on the GitHub issue tracker, or on the FLINT mailing list flint-devel@googlegroups.com.

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 several other example programs, see: http://fredrikj.net/arb/examples.html

General 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 and matrix exponential.

Support for polynomial 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.

Arb has partial support for automatic differentiation (AD), and includes rudimentary functionality for rigorous calculus based on AD (including real root isolation and complex integration).

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, 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 function, Jacobi theta functions, modular functions, Weierstrass elliptic function, complete elliptic integrals, arithmetic-geometric mean, Bernoulli numbers, partition function, Barnes G-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.

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://fredrikj.net/arb/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 separate wrapper of transcendental functions for use with the C99 complex double type is available (https://github.com/fredrik-johansson/arbcmath).