mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
Update and rename README to README.md
This commit is contained in:
Fredrik Johansson
parent
befc11cc95
commit
0dd4f82cd9
2 changed files with 151 additions and 19 deletions
19
README
19
README
|
|
@ -1,19 +0,0 @@
|
|||
Arb is a C library for high-performance arbitrary-precision floating-point ball
|
||||
(mid-rad interval) arithmetic. It supports complex numbers, polynomials,
|
||||
matrices, and evaluation of special functions, all with rigorous error
|
||||
bounding.
|
||||
|
||||
Arb depends on FLINT (http://flintlib.org/), MPIR (http://mpir.org)
|
||||
and MPFR (http://mpfr.org). It can be built as a standalone library,
|
||||
or as part of FLINT as an optional extension package.
|
||||
|
||||
For documentation and additional details, see: http://fredrikj.net/arb/
|
||||
|
||||
Development updates are occasionally posted to: http://fredrikj.net/blog/
|
||||
|
||||
Bug reports, feature requests and other comments are welcome on the
|
||||
FLINT mailing list: flint-devel@googlegroups.com
|
||||
|
||||
Author: Fredrik Johansson <fredrik.johansson@gmail.com>
|
||||
|
||||
The build scripts are copied from FLINT and were written by Bill Hart.
|
||||
151
README.md
Normal file
151
README.md
Normal file
|
|
@ -0,0 +1,151 @@
|
|||
# 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 | 0.00090 | 0.00011 |
|
||||
| 100 | 0.0039 | 0.0017 | 0.00075 |
|
||||
| 1000 | 0.23 | 1.5 | 0.019 |
|
||||
| 10000 | 42.6 | 98 | 1.2 |
|
||||
|
||||
## 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/> will include Arb as a standard package
|
||||
in an upcoming version, including a high-level Python interface
|
||||
to Arb as part of the SageMath standard library.
|
||||
|
||||
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.
|
||||
Loading…
Add table
Reference in a new issue