.. _examples:

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

.. highlight:: text

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:`arb_const_pi` function with a
working precision of roughly `n \log_2(10)` bits.

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

    > 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
    [3.14159265358979323846{...999959 digits...}42209010610577945815 +/- 3e-1000000]

The program prints an interval guaranteed to contain `\pi`, and where
all displayed digits are correct up to an error of plus or minus
one unit in the last place (see :func:`arb_printn`).
By default, only the first and last few digits are printed.
Pass 0 as a second argument to print all digits (or pass *m* to
print *m* + 1 leading and *m* trailing digits, as above with
the default *m* = 20).

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:`arb_mat_det`
does not contain zero.

Sample output::

    $ build/examples/hilbert_matrix 200
    prec=20: [+/- 1.32e-335]
    prec=40: [+/- 1.63e-545]
    prec=80: [+/- 1.30e-933]
    prec=160: [+/- 3.62e-1926]
    prec=320: [+/- 1.81e-4129]
    prec=640: [+/- 3.84e-8838]
    prec=1280: [2.955454297e-23924 +/- 8.29e-23935]
    success!
    cpu/wall(s): 8.494 8.513
    virt/peak/res/peak(MB): 134.98 134.98 111.57 111.57

Called with ``-eig n``, instead of computing the determinant,
the program computes the smallest eigenvalue of the Hilbert matrix
(in fact, it isolates all eigenvalues and prints the smallest eigenvalue)::

    $ build/examples/hilbert_matrix -eig 50
    prec=20: nan
    prec=40: nan
    prec=80: nan
    prec=160: nan
    prec=320: nan
    prec=640: [1.459157797e-74 +/- 2.49e-84]
    success!
    cpu/wall(s): 1.84 1.841
    virt/peak/res/peak(MB): 33.97 33.97 10.51 10.51

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::

    > 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

logistic.c
-------------------------------------------------------------------------------

This program computes the *n*-th iterate of the logistic map defined
by `x_{n+1} = r x_n (1 - x_n)` where `r` and `x_0` are given.
It takes the following parameters::

    logistic n [x_0] [r] [digits]

The inputs `x_0`, *r* and *digits* default to 0.5, 3.75 and 10 respectively.
The computation is automatically restarted with doubled precision
until the result is accurate to *digits* decimal digits.

Sample output::

    > build/examples/logistic 10
    Trying prec=64 bits...success!
    cpu/wall(s): 0 0.001
    x_10 = [0.6453672908 +/- 3.10e-11]

    > build/examples/logistic 100
    Trying prec=64 bits...ran out of accuracy at step 18
    Trying prec=128 bits...ran out of accuracy at step 53
    Trying prec=256 bits...success!
    cpu/wall(s): 0 0
    x_100 = [0.8882939923 +/- 1.60e-11]

    > build/examples/logistic 10000
    Trying prec=64 bits...ran out of accuracy at step 18
    Trying prec=128 bits...ran out of accuracy at step 53
    Trying prec=256 bits...ran out of accuracy at step 121
    Trying prec=512 bits...ran out of accuracy at step 256
    Trying prec=1024 bits...ran out of accuracy at step 525
    Trying prec=2048 bits...ran out of accuracy at step 1063
    Trying prec=4096 bits...ran out of accuracy at step 2139
    Trying prec=8192 bits...ran out of accuracy at step 4288
    Trying prec=16384 bits...ran out of accuracy at step 8584
    Trying prec=32768 bits...success!
    cpu/wall(s): 0.859 0.858
    x_10000 = [0.8242048008 +/- 4.35e-11]

    > build/examples/logistic 1234 0.1 3.99 30
    Trying prec=64 bits...ran out of accuracy at step 0
    Trying prec=128 bits...ran out of accuracy at step 10
    Trying prec=256 bits...ran out of accuracy at step 76
    Trying prec=512 bits...ran out of accuracy at step 205
    Trying prec=1024 bits...ran out of accuracy at step 461
    Trying prec=2048 bits...ran out of accuracy at step 974
    Trying prec=4096 bits...success!
    cpu/wall(s): 0.009 0.009
    x_1234 = [0.256445391958651410579677945635 +/- 3.92e-31]

real_roots.c
-------------------------------------------------------------------------------

This program isolates the roots of a function on the interval `(a,b)`
(where *a* and *b* are input as double-precision literals)
using the routines in the :ref:`arb_calc <arb-calc>` module.
The program takes the following arguments::

    real_roots function a b [-refine d] [-verbose] [-maxdepth n] [-maxeval n] [-maxfound n] [-prec n]

The following functions (specified by an integer code) are implemented:

  * 0 - `Z(x)` (Riemann-Siegel Z-function)
  * 1 - `\sin(x)`
  * 2 - `\sin(x^2)`
  * 3 - `\sin(1/x)`
  * 4 - `\operatorname{Ai}(x)` (Airy function)
  * 5 - `\operatorname{Ai}'(x)` (Airy function)
  * 6 - `\operatorname{Bi}(x)` (Airy function)
  * 7 - `\operatorname{Bi}'(x)` (Airy function)

The following options are available:

  * ``-refine d``: If provided, after isolating the roots, attempt to refine
    the roots to *d* digits of accuracy using a few bisection steps followed
    by Newton's method with adaptive precision, and then print them.

  * ``-verbose``: Print more information.

  * ``-maxdepth n``: Stop searching after *n* recursive subdivisions.

  * ``-maxeval n``: Stop searching after approximately *n* function evaluations
    (the actual number evaluations will be a small multiple of this).

  * ``-maxfound n``: Stop searching after having found *n* isolated roots.

  * ``-prec n``: Working precision to use for the root isolation.

With *function* 0, the program isolates roots of the Riemann zeta function
on the critical line, and guarantees that no roots are missed
(there are more efficient ways to do this, but it is a nice example)::

    > build/examples/real_roots 0 0.0 50.0 -verbose
    interval: [0, 50]
    maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
    found isolated root in: [14.111328125, 14.16015625]
    found isolated root in: [20.99609375, 21.044921875]
    found isolated root in: [25, 25.048828125]
    found isolated root in: [30.419921875, 30.4443359375]
    found isolated root in: [32.91015625, 32.958984375]
    found isolated root in: [37.548828125, 37.59765625]
    found isolated root in: [40.91796875, 40.966796875]
    found isolated root in: [43.310546875, 43.3349609375]
    found isolated root in: [47.998046875, 48.0224609375]
    found isolated root in: [49.755859375, 49.7802734375]
    ---------------------------------------------------------------
    Found roots: 10
    Subintervals possibly containing undetected roots: 0
    Function evaluations: 3058
    cpu/wall(s): 0.202 0.202
    virt/peak/res/peak(MB): 26.12 26.14 2.76 2.76

Find just one root and refine it to approximately 75 digits::

    > build/examples/real_roots 0 0.0 50.0 -maxfound 1 -refine 75
    interval: [0, 50]
    maxdepth = 30, maxeval = 100000, maxfound = 1, low_prec = 30
    refined root (0/8):
    [14.134725141734693790457251983562470270784257115699243175685567460149963429809 +/- 2.57e-76]

    ---------------------------------------------------------------
    Found roots: 1
    Subintervals possibly containing undetected roots: 7
    Function evaluations: 761
    cpu/wall(s): 0.055 0.056
    virt/peak/res/peak(MB): 26.12 26.14 2.75 2.75

Find the first few roots of an Airy function and refine them to 50 digits each::

    > build/examples/real_roots 4 -10 0 -refine 50
    interval: [-10, 0]
    maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
    refined root (0/6):
    [-9.022650853340980380158190839880089256524677535156083 +/- 4.85e-52]

    refined root (1/6):
    [-7.944133587120853123138280555798268532140674396972215 +/- 1.92e-52]

    refined root (2/6):
    [-6.786708090071758998780246384496176966053882477393494 +/- 3.84e-52]

    refined root (3/6):
    [-5.520559828095551059129855512931293573797214280617525 +/- 1.05e-52]

    refined root (4/6):
    [-4.087949444130970616636988701457391060224764699108530 +/- 2.46e-52]

    refined root (5/6):
    [-2.338107410459767038489197252446735440638540145672388 +/- 1.48e-52]

    ---------------------------------------------------------------
    Found roots: 6
    Subintervals possibly containing undetected roots: 0
    Function evaluations: 200
    cpu/wall(s): 0.003 0.003
    virt/peak/res/peak(MB): 26.12 26.14 2.24 2.24

Find roots of `\sin(x^2)` on `(0,100)`. The algorithm cannot isolate
the root at `x = 0` (it is at the endpoint of the interval, and in any
case a root of multiplicity higher than one). The failure is reported::

    > build/examples/real_roots 2 0 100
    interval: [0, 100]
    maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
    ---------------------------------------------------------------
    Found roots: 3183
    Subintervals possibly containing undetected roots: 1
    Function evaluations: 34058
    cpu/wall(s): 0.032 0.032
    virt/peak/res/peak(MB): 26.32 26.37 2.04 2.04

This does not miss any roots::

    > build/examples/real_roots 2 1 100
    interval: [1, 100]
    maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
    ---------------------------------------------------------------
    Found roots: 3183
    Subintervals possibly containing undetected roots: 0
    Function evaluations: 34039
    cpu/wall(s): 0.023 0.023
    virt/peak/res/peak(MB): 26.32 26.37 2.01 2.01

Looking for roots of `\sin(1/x)` on `(0,1)`, the algorithm finds many roots,
but will never find all of them since there are infinitely many::

    > build/examples/real_roots 3 0.0 1.0
    interval: [0, 1]
    maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
    ---------------------------------------------------------------
    Found roots: 10198
    Subintervals possibly containing undetected roots: 24695
    Function evaluations: 202587
    cpu/wall(s): 0.171 0.171
    virt/peak/res/peak(MB): 28.39 30.38 4.05 4.05

Remark: the program always computes rigorous containing intervals
for the roots, but the accuracy after refinement could be less than *d* digits.

poly_roots.c
-------------------------------------------------------------------------------

This program finds the complex roots of an integer polynomial
by calling :func:`arb_fmpz_poly_complex_roots`, which in turn calls
:func:`acb_poly_find_roots` with increasing
precision until the roots certainly have been isolated.
The program takes the following arguments::

    poly_roots [-refine d] [-print d] <poly>

    Isolates all the complex roots of a polynomial with integer coefficients.

    If -refine d is passed, the roots are refined to a relative tolerance
    better than 10^(-d). By default, the roots are only computed to sufficient
    accuracy to isolate them. The refinement is not currently done efficiently.

    If -print d is passed, the computed roots are printed to d decimals.
    By default, the roots are not printed.

    The polynomial can be specified by passing the following as <poly>:

    a <n>          Easy polynomial 1 + 2x + ... + (n+1)x^n
    t <n>          Chebyshev polynomial T_n
    u <n>          Chebyshev polynomial U_n
    p <n>          Legendre polynomial P_n
    c <n>          Cyclotomic polynomial Phi_n
    s <n>          Swinnerton-Dyer polynomial S_n
    b <n>          Bernoulli polynomial B_n
    w <n>          Wilkinson polynomial W_n
    e <n>          Taylor series of exp(x) truncated to degree n
    m <n> <m>      The Mignotte-like polynomial x^n + (100x+1)^m, n > m
    coeffs <c0 c1 ... cn>        c0 + c1 x + ... + cn x^n

    Concatenate to multiply polynomials, e.g.: p 5 t 6 coeffs 1 2 3
    for P_5(x)*T_6(x)*(1+2x+3x^2)

This finds the roots of the Wilkinson polynomial with roots at the
positive integers 1, 2, ..., 100::

    > build/examples/poly_roots -print 15 w 100
    computing squarefree factorization...
    cpu/wall(s): 0.001 0.001
    roots with multiplicity 1
    searching for 100 roots, 100 deflated
    prec=32: 0 isolated roots | cpu/wall(s): 0.098 0.098
    prec=64: 0 isolated roots | cpu/wall(s): 0.247 0.247
    prec=128: 0 isolated roots | cpu/wall(s): 0.498 0.497
    prec=256: 0 isolated roots | cpu/wall(s): 0.713 0.713
    prec=512: 100 isolated roots | cpu/wall(s): 0.104 0.105
    done!
    [1.00000000000000 +/- 3e-20]
    [2.00000000000000 +/- 3e-19]
    [3.00000000000000 +/- 1e-19]
    [4.00000000000000 +/- 1e-19]
    [5.00000000000000 +/- 1e-19]
    ...
    [96.0000000000000 +/- 1e-17]
    [97.0000000000000 +/- 1e-17]
    [98.0000000000000 +/- 3e-17]
    [99.0000000000000 +/- 3e-17]
    [100.000000000000 +/- 3e-17]
    cpu/wall(s): 1.664 1.664

This finds the roots of a Bernoulli polynomial which has both real
and complex roots::

    > build/examples/poly_roots -refine 100 -print 20 b 16
    computing squarefree factorization...
    cpu/wall(s): 0.001 0
    roots with multiplicity 1
    searching for 16 roots, 16 deflated
    prec=32: 16 isolated roots | cpu/wall(s): 0.006 0.006
    prec=64: 16 isolated roots | cpu/wall(s): 0.001 0.001
    prec=128: 16 isolated roots | cpu/wall(s): 0.001 0.001
    prec=256: 16 isolated roots | cpu/wall(s): 0.001 0.002
    prec=512: 16 isolated roots | cpu/wall(s): 0.002 0.001
    done!
    [-0.94308706466055783383 +/- 2.02e-21]
    [-0.75534059252067985752 +/- 2.70e-21]
    [-0.24999757119077421009 +/- 4.27e-21]
    [0.24999757152512726002 +/- 4.43e-21]
    [0.75000242847487273998 +/- 4.43e-21]
    [1.2499975711907742101 +/- 1.43e-20]
    [1.7553405925206798575 +/- 1.74e-20]
    [1.9430870646605578338 +/- 3.21e-20]
    [-0.99509334829256233279 +/- 9.42e-22] + [0.44547958157103608805 +/- 3.59e-21]*I
    [-0.99509334829256233279 +/- 9.42e-22] + [-0.44547958157103608805 +/- 3.59e-21]*I
    [1.9950933482925623328 +/- 1.10e-20] + [0.44547958157103608805 +/- 3.59e-21]*I
    [1.9950933482925623328 +/- 1.10e-20] + [-0.44547958157103608805 +/- 3.59e-21]*I
    [-0.92177327714429290564 +/- 4.68e-21] + [-1.0954360955079385542 +/- 1.71e-21]*I
    [-0.92177327714429290564 +/- 4.68e-21] + [1.0954360955079385542 +/- 1.71e-21]*I
    [1.9217732771442929056 +/- 3.54e-20] + [1.0954360955079385542 +/- 1.71e-21]*I
    [1.9217732771442929056 +/- 3.54e-20] + [-1.0954360955079385542 +/- 1.71e-21]*I
    cpu/wall(s): 0.011 0.012

Roots are automatically separated by multiplicity by performing an initial
squarefree factorization::

    > build/examples/poly_roots -print 5 p 5 p 5 t 7 coeffs 1 5 10 10 5 1
    computing squarefree factorization...
    cpu/wall(s): 0 0
    roots with multiplicity 1
    searching for 6 roots, 3 deflated
    prec=32: 3 isolated roots | cpu/wall(s): 0 0.001
    done!
    [-0.97493 +/- 2.10e-6]
    [-0.78183 +/- 1.49e-6]
    [-0.43388 +/- 3.75e-6]
    [0.43388 +/- 3.75e-6]
    [0.78183 +/- 1.49e-6]
    [0.97493 +/- 2.10e-6]
    roots with multiplicity 2
    searching for 4 roots, 2 deflated
    prec=32: 2 isolated roots | cpu/wall(s): 0 0
    done!
    [-0.90618 +/- 1.56e-7]
    [-0.53847 +/- 6.91e-7]
    [0.53847 +/- 6.91e-7]
    [0.90618 +/- 1.56e-7]
    roots with multiplicity 3
    searching for 1 roots, 0 deflated
    prec=32: 0 isolated roots | cpu/wall(s): 0 0
    done!
    0
    roots with multiplicity 5
    searching for 1 roots, 1 deflated
    prec=32: 1 isolated roots | cpu/wall(s): 0 0
    done!
    -1.0000
    cpu/wall(s): 0 0.001

complex_plot.c
-------------------------------------------------------------------------------

This program plots one of the predefined functions over a complex
interval `[x_a, x_b] + [y_a, y_b]i` using domain coloring, at
a resolution of *xn* times *yn* pixels.

The program takes the parameters::

    complex_plot [-range xa xb ya yb] [-size xn yn] <func>

Defaults parameters are `[-10,10] + [-10,10]i` and *xn* = *yn* = 512.

A color function can be selected with -color. Valid options
are 0 (phase=hue, magnitude=brightness) and 1 (phase only,
white-gold-black-blue-white counterclockwise).

The output is written to ``arbplot.ppm``. If you have ImageMagick,
run ``convert arbplot.ppm arbplot.png`` to get a PNG.

Function codes ``<func>`` are:

  * ``gamma``   - Gamma function
  * ``digamma`` - Digamma function
  * ``lgamma``  - Logarithmic gamma function
  * ``zeta``    - Riemann zeta function
  * ``erf``     - Error function
  * ``ai``      - Airy function Ai
  * ``bi``      - Airy function Bi
  * ``besselj`` - Bessel function `J_0`
  * ``bessely`` - Bessel function `Y_0`
  * ``besseli`` - Bessel function `I_0`
  * ``besselk`` - Bessel function `K_0`
  * ``modj``    - Modular j-function
  * ``modeta``  - Dedekind eta function
  * ``barnesg`` - Barnes G-function
  * ``agm``     - Arithmetic geometric mean

The function is just sampled at point values; no attempt is made to resolve
small features by adaptive subsampling.

For example, the following plots the Riemann zeta function around
a portion of the critical strip with imaginary part between 100 and 140::

    > build/examples/complex_plot zeta -range -10 10 100 140 -size 256 512

lvalue.c
-------------------------------------------------------------------------------

This program evaluates Dirichlet L-functions. It takes the following input::

    > build/examples/lvalue
    lvalue [-character q n] [-re a] [-im b] [-prec p] [-z] [-deflate] [-len l]

    Print value of Dirichlet L-function at s = a+bi.
    Default a = 0.5, b = 0, p = 53, (q, n) = (1, 0) (Riemann zeta)
    [-z]       - compute Z(s) instead of L(s)
    [-deflate] - remove singular term at s = 1
    [-len l]   - compute l terms in Taylor series at s

Evaluating the Riemann zeta function and
the Dirichlet beta function at `s = 2`::

    > build/examples/lvalue -re 2 -prec 128
    L(s) = [1.64493406684822643647241516664602518922 +/- 4.37e-39]
    cpu/wall(s): 0.001 0.001
    virt/peak/res/peak(MB): 26.86 26.88 2.05 2.05

    > build/examples/lvalue -character 4 3 -re 2 -prec 128
    L(s) = [0.91596559417721901505460351493238411077 +/- 7.86e-39]
    cpu/wall(s): 0.002 0.003
    virt/peak/res/peak(MB): 26.86 26.88 2.31 2.31

Evaluating the L-function for character number 101 modulo 1009
at `s = 1/2` and `s = 1`::

    > build/examples/lvalue -character 1009 101
    L(s) = [-0.459256562383872 +/- 5.24e-16] + [1.346937111206009 +/- 3.03e-16]*I
    cpu/wall(s): 0.012 0.012
    virt/peak/res/peak(MB): 26.86 26.88 2.30 2.30

    > build/examples/lvalue -character 1009 101 -re 1
    L(s) = [0.657952586112728 +/- 6.02e-16] + [1.004145273214022 +/- 3.10e-16]*I
    cpu/wall(s): 0.017 0.018
    virt/peak/res/peak(MB): 26.86 26.88 2.30 2.30

Computing the first few coefficients in the Laurent series of the
Riemann zeta function at `s = 1`::

    > build/examples/lvalue -re 1 -deflate -len 8
    L(s) = [0.577215664901532861 +/- 5.29e-19]
    L'(s) = [0.072815845483676725 +/- 2.68e-19]
    [x^2] L(s+x) = [-0.004845181596436159 +/- 3.87e-19]
    [x^3] L(s+x) = [-0.000342305736717224 +/- 4.20e-19]
    [x^4] L(s+x) = [9.6890419394471e-5 +/- 2.40e-19]
    [x^5] L(s+x) = [-6.6110318108422e-6 +/- 4.51e-20]
    [x^6] L(s+x) = [-3.316240908753e-7 +/- 3.85e-20]
    [x^7] L(s+x) = [1.0462094584479e-7 +/- 7.78e-21]
    cpu/wall(s): 0.003 0.004
    virt/peak/res/peak(MB): 26.86 26.88 2.30 2.30

Evaluating the Riemann zeta function near the first nontrivial root::

    > build/examples/lvalue -re 0.5 -im 14.134725
    L(s) = [1.76743e-8 +/- 1.93e-14] + [-1.110203e-7 +/- 2.84e-14]*I
    cpu/wall(s): 0.001 0.001
    virt/peak/res/peak(MB): 26.86 26.88 2.31 2.31

    > build/examples/lvalue -z -re 14.134725 -prec 200
    Z(s) = [-1.12418349839417533300111494358128257497862927935658e-7 +/- 4.62e-58]
    cpu/wall(s): 0.001 0.001
    virt/peak/res/peak(MB): 26.86 26.88 2.57 2.57

    > build/examples/lvalue -z -re 14.134725 -len 4
    Z(s) = [-1.124184e-7 +/- 7.00e-14]
    Z'(s) = [0.793160414884 +/- 4.09e-13]
    [x^2] Z(s+x) = [0.065164586492 +/- 5.39e-13]
    [x^3] Z(s+x) = [-0.020707762705 +/- 5.37e-13]
    cpu/wall(s): 0.002 0.003
    virt/peak/res/peak(MB): 26.86 26.88 2.57 2.57

lcentral.c
-------------------------------------------------------------------------------

This program computes the central value `L(1/2)` for each Dirichlet L-function
character modulo *q* for each *q* in the range *qmin* to *qmax*. Usage::

    > build/examples/lcentral
    Computes central values (s = 0.5) of Dirichlet L-functions.

    usage: build/examples/lcentral [--quiet] [--check] [--prec <bits>] qmin qmax

The first few values::

    > build/examples/lcentral 1 8
    3,2: [0.48086755769682862618122006324 +/- 7.35e-30]
    4,3: [0.66769145718960917665869092930 +/- 1.62e-30]
    5,2: [0.76374788011728687822451215264 +/- 2.32e-30] + [0.21696476751886069363858659310 +/- 3.06e-30]*I
    5,4: [0.23175094750401575588338366176 +/- 2.21e-30]
    5,3: [0.76374788011728687822451215264 +/- 2.32e-30] + [-0.21696476751886069363858659310 +/- 3.06e-30]*I
    7,3: [0.71394334376831949285993820742 +/- 1.21e-30] + [0.47490218277139938263745243935 +/- 4.52e-30]*I
    7,2: [0.31008936259836766059195052534 +/- 5.29e-30] + [-0.07264193137017790524562171245 +/- 5.48e-30]*I
    7,6: [1.14658566690370833367712697646 +/- 1.95e-30]
    7,4: [0.31008936259836766059195052534 +/- 5.29e-30] + [0.07264193137017790524562171245 +/- 5.48e-30]*I
    7,5: [0.71394334376831949285993820742 +/- 1.21e-30] + [-0.47490218277139938263745243935 +/- 4.52e-30]*I
    8,5: [0.37369171291254730738158695002 +/- 4.01e-30]
    8,3: [1.10042140952554837756713576997 +/- 3.37e-30]
    cpu/wall(s): 0.002 0.003
    virt/peak/res/peak(MB): 26.32 26.34 2.35 2.35

Testing a large *q*::

    > build/examples/lcentral --quiet --check --prec 256 100000 100000
    cpu/wall(s): 1.668 1.667
    virt/peak/res/peak(MB): 35.67 46.66 11.67 22.61

It is conjectured that the central value never vanishes. Running with ``--check``
verifies that the interval certainly is nonzero. This can fail with
insufficient precision::

    > build/examples/lcentral --check --prec 15 100000 100000
    100000,71877: [0.1 +/- 0.0772] + [+/- 0.136]*I
    100000,90629: [2e+0 +/- 0.106] + [+/- 0.920]*I
    100000,28133: [+/- 0.811] + [-2e+0 +/- 0.501]*I
    100000,3141: [0.8 +/- 0.0407] + [-0.1 +/- 0.0243]*I
    100000,53189: [4.0 +/- 0.0826] + [+/- 0.107]*I
    100000,53253: [1.9 +/- 0.0855] + [-3.9 +/- 0.0681]*I
    Value could be zero!
    100000,53381: [+/- 0.0329] + [+/- 0.0413]*I
    Aborted

integrals.c
-------------------------------------------------------------------------------

This program computes integrals using :func:`acb_calc_integrate`.
Invoking the program without parameters shows usage::

    > build/examples/integrals
    Compute integrals using acb_calc_integrate.
    Usage: integrals -i n [-prec p] [-tol eps] [-twice] [...]

    -i n       - compute integral n (0 <= n <= 23), or "-i all"
    -prec p    - precision in bits (default p = 64)
    -goal p    - approximate relative accuracy goal (default p)
    -tol eps   - approximate absolute error goal (default 2^-p)
    -twice     - run twice (to see overhead of computing nodes)
    -heap      - use heap for subinterval queue
    -verbose   - show information
    -verbose2  - show more information
    -deg n     - use quadrature degree up to n
    -eval n    - limit number of function evaluations to n
    -depth n   - limit subinterval queue size to n

    Implemented integrals:
    I0 = int_0^100 sin(x) dx
    I1 = 4 int_0^1 1/(1+x^2) dx
    I2 = 2 int_0^{inf} 1/(1+x^2) dx   (using domain truncation)
    I3 = 4 int_0^1 sqrt(1-x^2) dx
    I4 = int_0^8 sin(x+exp(x)) dx
    I5 = int_1^101 floor(x) dx
    I6 = int_0^1 |x^4+10x^3+19x^2-6x-6| exp(x) dx
    I7 = 1/(2 pi i) int zeta(s) ds  (closed path around s = 1)
    I8 = int_0^1 sin(1/x) dx  (slow convergence, use -heap and/or -tol)
    I9 = int_0^1 x sin(1/x) dx  (slow convergence, use -heap and/or -tol)
    I10 = int_0^10000 x^1000 exp(-x) dx
    I11 = int_1^{1+1000i} gamma(x) dx
    I12 = int_{-10}^{10} sin(x) + exp(-200-x^2) dx
    I13 = int_{-1020}^{-1010} exp(x) dx  (use -tol 0 for relative error)
    I14 = int_0^{inf} exp(-x^2) dx   (using domain truncation)
    I15 = int_0^1 sech(10(x-0.2))^2 + sech(100(x-0.4))^4 + sech(1000(x-0.6))^6 dx
    I16 = int_0^8 (exp(x)-floor(exp(x))) sin(x+exp(x)) dx  (use higher -eval)
    I17 = int_0^{inf} sech(x) dx   (using domain truncation)
    I18 = int_0^{inf} sech^3(x) dx   (using domain truncation)
    I19 = int_0^1 -log(x)/(1+x) dx   (using domain truncation)
    I20 = int_0^{inf} x exp(-x)/(1+exp(-x)) dx   (using domain truncation)
    I21 = int_C wp(x)/x^(11) dx   (contour for 10th Laurent coefficient of Weierstrass p-function)
    I22 = N(1000) = count zeros with 0 < t <= 1000 of zeta(s) using argument principle
    I23 = int_0^{1000} W_0(x) dx
    I24 = int_0^pi max(sin(x), cos(x)) dx
    I25 = int_{-1}^1 erf(x/sqrt(0.0002)*0.5+1.5)*exp(-x) dx
    I26 = int_{-10}^10 Ai(x) dx
    I27 = int_0^10 (x-floor(x)-1/2) max(sin(x),cos(x)) dx
    I28 = int_{-1-i}^{-1+i} sqrt(x) dx
    I29 = int_0^{inf} exp(-x^2+ix) dx   (using domain truncation)
    I30 = int_0^{inf} exp(-x) Ai(-x) dx   (using domain truncation)
    I31 = int_0^pi x sin(x) / (1 + cos(x)^2) dx

A few examples::

    build/examples/integrals -i 4
    I4 = int_0^8 sin(x+exp(x)) dx ...
    cpu/wall(s): 0.02 0.02
    I4 = [0.34740017265725 +/- 3.95e-15]

    > build/examples/integrals -i 3 -prec 333 -tol 1e-80
    I3 = 4 int_0^1 sqrt(1-x^2) dx ...
    cpu/wall(s): 0.024 0.024
    I3 = [3.141592653589793238462643383279502884197169399375105820974944592307816406286209 +/- 4.24e-79]

    > build/examples/integrals -i 9 -heap
    I9 = int_0^1 x sin(1/x) dx  (slow convergence, use -heap and/or -tol) ...
    cpu/wall(s): 0.019 0.018
    I9 = [0.3785300 +/- 3.17e-8]

.. highlight:: c

