.. _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:`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:`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. For convergence, the input polynomial is required to be squarefree. If -refine d is passed, the roots are refined to an absolute 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 c0 c1 ... cn c0 + c1 x + ... + cn x^n where all c:s are specified integers 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 prec=53: 0 isolated roots | cpu/wall(s): 0.42 0.426 prec=106: 0 isolated roots | cpu/wall(s): 1.37 1.368 prec=212: 0 isolated roots | cpu/wall(s): 1.48 1.485 prec=424: 100 isolated roots | cpu/wall(s): 0.61 0.611 done! (1 + 1.7285178043492e-125j) +/- (7.2e-122, 7.2e-122j) (2 + 5.1605530263601e-122j) +/- (3.77e-118, 3.77e-118j) (3 + -2.58115555871665e-118j) +/- (5.72e-115, 5.72e-115j) (4 + 1.02141628524271e-115j) +/- (4.38e-112, 4.38e-112j) (5 + 1.61326834094948e-113j) +/- (2.6e-109, 2.6e-109j) ... (95 + 4.15294196875447e-62j) +/- (6.66e-59, 6.66e-59j) (96 + 3.54502401922667e-64j) +/- (7.37e-60, 7.37e-60j) (97 + -1.67755595325625e-65j) +/- (6.4e-61, 6.4e-61j) (98 + 2.04638822325299e-65j) +/- (4e-62, 4e-62j) (99 + -2.73425468028238e-66j) +/- (1.71e-63, 1.71e-63j) (100 + -1.00950111302288e-68j) +/- (3.24e-65, 3.24e-65j) cpu/wall(s): 3.88 3.893 This finds the roots of a Bernoulli polynomial which has both real and complex roots. Note that the program does not attempt to determine that the imaginary parts of the real roots really are zero (this could be done by verifying sign changes):: > build/examples/poly_roots -refine 100 -print 20 b 16 prec=53: 16 isolated roots | cpu/wall(s): 0 0.007 prec=106: 16 isolated roots | cpu/wall(s): 0 0.004 prec=212: 16 isolated roots | cpu/wall(s): 0 0.004 prec=424: 16 isolated roots | cpu/wall(s): 0 0.004 done! (-0.94308706466055783383 + -5.512272663168484603e-128j) +/- (2.2e-125, 2.2e-125j) (-0.75534059252067985752 + 1.937401283040249068e-128j) +/- (1.09e-125, 1.09e-125j) (-0.24999757119077421009 + -4.5347924422246038692e-130j) +/- (3.6e-127, 3.6e-127j) (0.24999757152512726002 + 4.2191300761823281708e-129j) +/- (4.98e-127, 4.98e-127j) (0.75000242847487273998 + 9.0360649917413170142e-128j) +/- (8.88e-126, 8.88e-126j) (1.2499975711907742101 + 7.8804123808107088267e-127j) +/- (2.66e-124, 2.66e-124j) (1.7553405925206798575 + 5.432465269253967768e-126j) +/- (6.23e-123, 6.23e-123j) (1.9430870646605578338 + 3.3035377342500953239e-125j) +/- (7.05e-123, 7.05e-123j) (-0.99509334829256233279 + 0.44547958157103608805j) +/- (5.5e-125, 5.5e-125j) (-0.99509334829256233279 + -0.44547958157103608805j) +/- (5.46e-125, 5.46e-125j) (1.9950933482925623328 + 0.44547958157103608805j) +/- (1.44e-122, 1.44e-122j) (1.9950933482925623328 + -0.44547958157103608805j) +/- (1.43e-122, 1.43e-122j) (-0.92177327714429290564 + -1.0954360955079385542j) +/- (9.31e-125, 9.31e-125j) (-0.92177327714429290564 + 1.0954360955079385542j) +/- (1.02e-124, 1.02e-124j) (1.9217732771442929056 + 1.0954360955079385542j) +/- (9.15e-123, 9.15e-123j) (1.9217732771442929056 + -1.0954360955079385542j) +/- (8.12e-123, 8.12e-123j) cpu/wall(s): 0.02 0.02 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. 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