arb/doc/source/examples.rst

.. _examples:

Example programs
===============================================================================

The *examples* directory
(https://github.com/fredrik-johansson/arb/tree/master/examples)
contains several complete C programs, which are documented below. Running::

    make examples

will compile the programs and place the binaries in ``build/examples``.

pi.c
-------------------------------------------------------------------------------

This program computes `\pi` to an accuracy of roughly *n* decimal digits
by calling the :func:`fmprb_const_pi` function with a
working precision of roughly `n \log_2(10)` bits.

Sample output, computing `\pi` to one million digits::

    fredrik@lemur:~/src/arb$ build/examples/pi 1000000
    computing pi with a precision of 3321933 bits... cpu/wall(s): 0.58 0.586
    virt/peak/res/peak(MB): 28.24 36.84 8.86 15.56
    3.141592654 +/- 1.335e-1000001

The program prints a decimal approximation of the computed ball,
with the midpoint rounded to a number of decimal digits that can be
passed as a second parameter to the program (default = 10).
In the present implementation (see :func:`fmprb_printd`), the
digits are not guaranteed to be correctly rounded.

hilbert_matrix.c
-------------------------------------------------------------------------------

Given an input integer *n*, this program accurately computes the
determinant of the *n* by *n* Hilbert matrix.
Hilbert matrices are notoriously ill-conditioned: although the
entries are close to unit magnitude, the determinant `h_n`
decreases superexponentially (nearly as `1/4^{n^2}`) as
a function of *n*.
This program automatically doubles the working precision
until the ball computed for `h_n` by :func:`fmprb_mat_det`
does not contain zero.

Sample output::

    fredrik@lemur:~/src/arb$ build/examples/hilbert_matrix 200
    prec=20: 0 +/- 5.5777e-330
    prec=40: 0 +/- 2.5785e-542
    prec=80: 0 +/- 8.1169e-926
    prec=160: 0 +/- 2.8538e-1924
    prec=320: 0 +/- 6.3868e-4129
    prec=640: 0 +/- 1.7529e-8826
    prec=1280: 0 +/- 1.8545e-17758
    prec=2560: 2.955454297e-23924 +/- 6.4586e-24044
    success!
    cpu/wall(s): 9.06 9.095
    virt/peak/res/peak(MB): 55.52 55.52 35.50 35.50

keiper_li.c
-------------------------------------------------------------------------------

Given an input integer *n*, this program rigorously computes numerical
values of the Keiper-Li coefficients
`\lambda_0, \ldots, \lambda_n`. The Keiper-Li coefficients
have the property that `\lambda_n > 0` for all `n > 0` if and only if the
Riemann hypothesis is true. This program was used for the record
computations described in [Joh2013]_ (the paper describes
the algorithm in some more detail).

The program takes the following parameters::

    keiper_li n [-prec prec] [-threads num_threads] [-out out_file]

The program prints the first and last few coefficients. It can optionally
write all the computed data to a file. The working precision defaults
to a value that should give all the coefficients to a few digits of
accuracy, but can optionally be set higher (or lower).
On a multicore system, using several threads results in faster
execution.

Sample output::

    fredrik@lemur:~/src/arb$ build/examples/keiper_li 1000 -threads 2
    zeta: cpu/wall(s): 0.4 0.244
    virt/peak/res/peak(MB): 167.98 294.69 5.09 7.43
    log: cpu/wall(s): 0.03 0.038
    gamma: cpu/wall(s): 0.02 0.016
    binomial transform: cpu/wall(s): 0.01 0.018
    0: -0.69314718055994530941723212145817656807550013436026 +/- 6.5389e-347
    1: 0.023095708966121033814310247906495291621932127152051 +/- 2.0924e-345
    2: 0.046172867614023335192864243096033943387066108314123 +/- 1.674e-344
    3: 0.0692129735181082679304973488726010689942120263932 +/- 5.0219e-344
    4: 0.092197619873060409647627872409439018065541673490213 +/- 2.0089e-343
    5: 0.11510854289223549048622128109857276671349132303596 +/- 1.0044e-342
    6: 0.13792766871372988290416713700341666356138966078654 +/- 6.0264e-342
    7: 0.16063715965299421294040287257385366292282442046163 +/- 2.1092e-341
    8: 0.18321945964338257908193931774721859848998098273432 +/- 8.4368e-341
    9: 0.20565733870917046170289387421343304741236553410044 +/- 7.5931e-340
    10: 0.22793393631931577436930340573684453380748385942738 +/- 7.5931e-339
    991: 2.3196617961613367928373899656994682562101430813341 +/- 2.461e-11
    992: 2.3203766239254884035349896518332550233162909717288 +/- 9.5363e-11
    993: 2.321092061239733282811659116333262802034375592414 +/- 1.8495e-10
    994: 2.3218073540188462110258826121503870112747188888893 +/- 3.5907e-10
    995: 2.3225217392815185726928702951225314023773358152533 +/- 6.978e-10
    996: 2.3232344485814623873333223609413703912358283071281 +/- 1.3574e-09
    997: 2.3239447114886014522889542667580382034526509232475 +/- 2.6433e-09
    998: 2.3246517591032700808344143240352605148856869322209 +/- 5.1524e-09
    999: 2.3253548275861382119812576052060526988544993162101 +/- 1.0053e-08
    1000: 2.3260531616864664574065046940832238158044982041872 +/- 3.927e-08
    virt/peak/res/peak(MB): 170.18 294.69 7.51 7.51