mirror of
https://github.com/vale981/arb
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164 lines
6.2 KiB
Markdown
164 lines
6.2 KiB
Markdown
# Arb
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Arb is a C library for arbitrary-precision interval arithmetic.
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It has full support for both real and complex numbers.
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The library is thread-safe, portable, and extensively tested.
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Arb is free software distributed under the
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GNU Lesser General Public License (LGPL), version 2.1 or later.
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Documentation: http://arblib.org
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Development updates: http://fredrikj.net/blog/
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Author: Fredrik Johansson <fredrik.johansson@gmail.com>
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Bug reports, feature requests and other comments are welcome
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in private communication, on the GitHub issue tracker, or on the FLINT mailing list <flint-devel@googlegroups.com>.
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[](https://travis-ci.org/fredrik-johansson/arb)
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## Code example
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The following program evaluates `sin(pi + exp(-10000))`. Since the
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input to the sine function matches a root to within 4343 digits,
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at least 4343-digit (14427-bit) precision is needed to get an accurate
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result. The program repeats the evaluation
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at 64-bit, 128-bit, ... precision, stopping only when the
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result is accurate to at least 53 bits.
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#include "arb.h"
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int main()
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{
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slong prec;
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arb_t x, y;
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arb_init(x); arb_init(y);
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for (prec = 64; ; prec *= 2)
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{
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arb_const_pi(x, prec);
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arb_set_si(y, -10000);
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arb_exp(y, y, prec);
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arb_add(x, x, y, prec);
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arb_sin(y, x, prec);
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arb_printn(y, 15, 0); printf("\n");
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if (arb_rel_accuracy_bits(y) >= 53)
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break;
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}
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arb_clear(x); arb_clear(y);
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flint_cleanup();
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}
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The output is:
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[+/- 6.01e-19]
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[+/- 2.55e-38]
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[+/- 8.01e-77]
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[+/- 8.64e-154]
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[+/- 5.37e-308]
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[+/- 3.63e-616]
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[+/- 1.07e-1232]
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[+/- 9.27e-2466]
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[-1.13548386531474e-4343 +/- 3.91e-4358]
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Each line shows a rigorous enclosure of the exact value
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of the expression. The program demonstrates how the user
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can rely on Arb's automatic error bound tracking to get an output
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that is guaranteed to be accurate -- no error analysis
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needs to be done by the user.
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For more example programs, see: http://arblib.org/examples.html
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## General features
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Besides basic arithmetic, Arb allows working with univariate
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polynomials, truncated power series, and matrices
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over both real and complex numbers.
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Basic linear algebra is supported, including matrix multiplication,
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determinant, inverse, nonsingular solving and matrix exponential.
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Support for polynomial and power series is quite extensive,
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including methods for composition, reversion, product trees,
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multipoint evaluation and interpolation, complex root isolation,
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and transcendental functions of power series.
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Arb has partial support for automatic differentiation (AD), and includes
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rudimentary functionality for rigorous calculus based on AD
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(including real root isolation and complex integration).
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## Special functions
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Arb can compute a wide range of transcendental and special functions,
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including the gamma function, polygamma functions,
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Riemann zeta and Hurwitz zeta function, Dirichlet L-functions, polylogarithm,
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error function, Gauss hypergeometric function 2F1, confluent
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hypergeometric functions, Bessel functions, Airy functions,
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Legendre functions and other orthogonal polynomials,
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exponential and trigonometric integrals, incomplete gamma function,
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Jacobi theta functions, modular functions, Weierstrass elliptic function,
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complete elliptic integrals, arithmetic-geometric mean,
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Bernoulli numbers, partition function, Barnes G-function.
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## Speed
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Arb uses a midpoint-radius (ball) representation of real numbers.
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At high precision, this allows doing interval arithmetic without
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significant overhead compared to plain floating-point arithmetic.
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Various low-level optimizations have also been implemented
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to reduce overhead at precisions of just a few machine
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words. Most operations on polynomials and power series
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use asymptotically fast FFT multiplication.
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For basic arithmetic, Arb should generally be around as fast
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as MPFR (http://mpfr.org), though it can be a bit slower
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at low precision, and around twice as fast as MPFI
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(https://perso.ens-lyon.fr/nathalie.revol/software.html).
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Transcendental functions in Arb are quite well optimized and
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should generally be faster than any other arbitrary-precision
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software currently available. The following table
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compares the time in seconds to evaluate the Gauss
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hypergeometric function `2F1(1/2, 1/4, 1, z)` at
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the complex number `z = 5^(1/2) + 7^(1/2)i`, to a given
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number of decimal digits (Arb 2.8-git and mpmath 0.19 on
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an 1.90 GHz Intel i5-4300U, Mathematica 9.0 on a 3.07 GHz Intel Xeon X5675).
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| Digits | Mathematica | mpmath | Arb |
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| -------:|:------------|:-----------|:-----------|
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| 10 | 0.00066 | 0.00065 | 0.000071 |
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| 100 | 0.0039 | 0.0012 | 0.00048 |
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| 1000 | 0.23 | 1.2 | 0.0093 |
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| 10000 | 42.6 | 84 | 0.56 |
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## Dependencies, installation, and interfaces
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Arb depends on FLINT (http://flintlib.org/), either
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GMP (http://gmplib.org) or MPIR (http://mpir.org),
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and MPFR (http://mpfr.org).
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See http://arblib.org/setup.html for instructions
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on building and installing Arb directly from the source code.
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Arb might also be available (or coming soon) as a package for
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your Linux distribution.
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SageMath (<http://sagemath.org/>) includes Arb as a standard package
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and contains a high-level Python interface. See the SageMath documentation
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for RealBallField (http://doc.sagemath.org/html/en/reference/rings_numerical/sage/rings/real_arb.html)
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and ComplexBallField (http://doc.sagemath.org/html/en/reference/rings_numerical/sage/rings/complex_arb.html).
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Nemo (<http://nemocas.org/>) is a computer algebra package for
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the Julia programming language which includes a high-level
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Julia interface to Arb. The Nemo installation script will
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create a local installation of Arb along with other dependencies.
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An experimental standalone Python interface to FLINT and Arb is also available
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(<https://github.com/fredrik-johansson/python-flint>).
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A separate wrapper of transcendental functions for use with the
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C99 `complex double` type is available
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(<https://github.com/fredrik-johansson/arbcmath>).
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