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