Python interface (python-flint)

Introduction

Examples:

>>> from flint2 import *
>>> QQ.bernoulli(50)
495057205241079648212477525/66
>>> _.factor()
(1, [5, 417202699, 47464429777438199, 2, 3, 11], [2, 1, 1, -1, -1, -1])
>>> sign * (primes ** exponents).product()
495057205241079648212477525/66

Types, parents and coercions

>>> ZZ(5)
5
>>> _.parent()
Integer ring (fmpz)
>>> QQ(5)
5
>>> _.parent()
Rational field (fmpq)
>>> ZZ(10) / ZZ(6)
Traceback (most recent call last):
  ...
FlintDomainError: x / y is not an element of {Integer ring (fmpz)} for {x = 10}, {y = 6}
>>> x = QQ(1) / 2; x ** x
Traceback (most recent call last):
  ...
FlintDomainError: x ** y is not an element of {Rational field (fmpq)} for {x = 1/2}, {y = 1/2}

>>> ZZ(10) / QQ(6)
5/3
>>> x = QQbar(1) / 2; x ** x
Root a = 0.707107 of 2*a^2-1

Real and complex numbers

>>> RR.zeta(2)
[1.644934066848226 +/- 4.57e-16]
>>> RR.prec = 128
>>> RR.zeta(2)
[1.64493406684822643647241516664602518922 +/- 2.88e-39]
>>> RR.prec = 53       # restore default

API documentation

class flint_ctypes.FlintDomainError

Raised when an operation does not have a well-defined result in the target domain.

class flint_ctypes.FlintUnableError

Raised when an operation cannot be performed because the algorithm is not implemented or there is insufficient precision, memory, etc.

flint_ctypes.set_num_threads(n)

flint_ctypes.get_num_threads()

class flint_ctypes.flint_rand_struct

data

Structure/Union member

class flint_ctypes.fmpz_struct

val

Structure/Union member

class flint_ctypes.fmpq_struct

den

Structure/Union member

num

Structure/Union member

class flint_ctypes.fmpzi_struct

imag

Structure/Union member

real

Structure/Union member

class flint_ctypes.fmpz_poly_struct

alloc

Structure/Union member

coeffs

Structure/Union member

length

Structure/Union member

class flint_ctypes.fmpq_poly_struct

alloc

Structure/Union member

coeffs

Structure/Union member

den

Structure/Union member

length

Structure/Union member

class flint_ctypes.arf_struct

data

Structure/Union member

class flint_ctypes.acf_struct

imag

Structure/Union member

real

Structure/Union member

class flint_ctypes.arb_struct

data

Structure/Union member

class flint_ctypes.acb_struct

imag

Structure/Union member

real

Structure/Union member

class flint_ctypes.fexpr_struct

alloc

Structure/Union member

data

Structure/Union member

class flint_ctypes.qqbar_struct

enclosure

Structure/Union member

poly

Structure/Union member

class flint_ctypes.ca_struct

data

Structure/Union member

class flint_ctypes.nmod_struct

val

Structure/Union member

class flint_ctypes.nmod_poly_struct

alloc

Structure/Union member

coeffs

Structure/Union member

length

Structure/Union member

n

Structure/Union member

ninv

Structure/Union member

nnorm

Structure/Union member

class flint_ctypes.fq_struct

alloc

Structure/Union member

coeffs

Structure/Union member

length

Structure/Union member

class flint_ctypes.fq_nmod_struct

alloc

Structure/Union member

coeffs

Structure/Union member

length

Structure/Union member

n

Structure/Union member

ninv

Structure/Union member

nnorm

Structure/Union member

class flint_ctypes.fq_zech_struct

n

Structure/Union member

class flint_ctypes.gr_vec_struct

alloc

Structure/Union member

entries

Structure/Union member

length

Structure/Union member

class flint_ctypes.gr_poly_struct

alloc

Structure/Union member

coeffs

Structure/Union member

length

Structure/Union member

class flint_ctypes.gr_series_struct

alloc

Structure/Union member

coeffs

Structure/Union member

error

Structure/Union member

length

Structure/Union member

class flint_ctypes.psl2z_struct

a

Structure/Union member

b

Structure/Union member

c

Structure/Union member

d

Structure/Union member

class flint_ctypes.dirichlet_char_struct

log

Structure/Union member

n

Structure/Union member

class flint_ctypes.perm_struct

entries

Structure/Union member

class flint_ctypes.gr_mat_struct

c

Structure/Union member

entries

Structure/Union member

r

Structure/Union member

rows

Structure/Union member

flint_ctypes.fmpz_to_python_int(xref)

flint_ctypes.fmpq_set_python(cref, x)

exception flint_ctypes.Undecidable

class flint_ctypes.gr_ctx_struct

content

Structure/Union member

class flint_ctypes.Truth(value)

__init__(value)

__bool__()

class flint_ctypes.LogicContext(value)

Handle the result of predicates (experimental):

>>> a = (RR_arb(1) / 3) * 3
>>> a
[1.00000000000000 +/- 3.89e-16]
>>> with strict_logic:
...     a == 1
...
Traceback (most recent call last):
  ...
Undecidable: unable to decide x == y for x = [1.00000000000000 +/- 3.89e-16], y = 1.000000000000000 over Real numbers (arb, prec = 53)
>>> with pessimistic_logic:
...     a == 1
...
False
>>> with optimistic_logic:
...     a == 1
...
True

__init__(value)

flint_ctypes.set_logic(which_logic)

class flint_ctypes.gr_ctx

__init__()

property prec

gen()

Generator of this domain.

>>> QQbar.i()
Root a = 1.00000*I of a^2+1
>>> QQ.i()
Traceback (most recent call last):
  ...
FlintDomainError: i is not an element of {Rational field (fmpq)}

i()

Imaginary unit as an element of this domain.

>>> QQbar.i()
Root a = 1.00000*I of a^2+1
>>> QQ.i()
Traceback (most recent call last):
  ...
FlintDomainError: i is not an element of {Rational field (fmpq)}

pi()

The number pi as an element of this domain.

>>> RR.pi()
[3.141592653589793 +/- 3.39e-16]
>>> QQbar.pi()
Traceback (most recent call last):
  ...
FlintDomainError: pi is not an element of {Complex algebraic numbers (qqbar)}

euler()

Euler’s constant as an element of this domain.

>>> RR.euler()
[0.5772156649015329 +/- 9.00e-17]

We do not know whether Euler’s constant is rational:

>>> QQ.euler()
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute euler in {Rational field (fmpq)}

catalan()

Catalan’s constant as an element of this domain.

>>> RR.catalan()
[0.915965594177219 +/- 1.23e-16]

khinchin()

Khinchin’s constant as an element of this domain.

>>> RR.khinchin()
[2.685452001065306 +/- 6.82e-16]

glaisher()

Khinchin’s constant as an element of this domain.

>>> RR.glaisher()
[1.282427129100623 +/- 6.02e-16]

inv(x)

sqrt(x)

rsqrt(x)

floor(x)

ceil(x)

trunc(x)

nint(x)

abs(x)

conj(x)

re(x)

im(x)

sgn(x)

csgn(x)

mul_2exp(x, y)

exp(x)

exp2(x)

exp10(x)

expm1(x)

exp_pi_i(x)

log(x)

log1p(x)

log_pi_i(x)

sin(x)

cos(x)

sin_cos(x)

tan(x)

cot(x)

sec(x)

csc(x)

sin_pi(x)

cos_pi(x)

sin_cos_pi(x)

tan_pi(x)

cot_pi(x)

sec_pi(x)

csc_pi(x)

sinc(x)

sinc_pi(x)

sinh(x)

cosh(x)

sinh_cosh(x)

tanh(x)

coth(x)

sech(x)

csch(x)

asin(x)

acos(x)

atan(x)

atan2(y, x)

>>> RR.atan2(1,2)
[0.4636476090008061 +/- 6.22e-17]

acot(x)

asec(x)

acsc(x)

asin_pi(x)

acos_pi(x)

atan_pi(x)

acot_pi(x)

asec_pi(x)

acsc_pi(x)

asinh(x)

acosh(x)

atanh(x)

acoth(x)

asech(x)

acsch(x)

erf(x)

>>> RR.erf(1)
[0.842700792949715 +/- 3.28e-16]

erfc(x)

>>> RR.erfc(1)
[0.1572992070502851 +/- 3.71e-17]

erfi(x)

>>> RR.erfi(1)
[1.650425758797543 +/- 4.58e-16]

erfcx(x)

erfinv(x)

>>> RR.erfinv(0.5)
[0.4769362762044698 +/- 7.79e-17]

erfcinv(x)

>>> RR.erfc(RR.erfcinv(0.25))
[0.250000000000000 +/- 1.24e-16]

fresnel_s(x, normalized=False)

>>> RR.fresnel_s(1)
[0.3102683017233811 +/- 2.67e-18]
>>> RR.fresnel_s(1, normalized=True)
[0.4382591473903548 +/- 9.24e-17]

fresnel_c(x, normalized=False)

>>> RR.fresnel_c(1)
[0.904524237900272 +/- 1.46e-16]
>>> RR.fresnel_c(1, normalized=True)
[0.779893400376823 +/- 3.59e-16]

fresnel(x, normalized=False)

>>> RR.fresnel(1)
([0.3102683017233811 +/- 2.67e-18], [0.904524237900272 +/- 1.46e-16])
>>> RR.fresnel(1, normalized=True)
([0.4382591473903548 +/- 9.24e-17], [0.779893400376823 +/- 3.59e-16])

gamma_upper(x, y, regularized=0)

>>> RR.gamma_upper(3, 4)
[0.476206611107089 +/- 5.30e-16]
>>> RR.gamma_upper(3, 4, regularized=True)
[0.2381033055535443 +/- 8.24e-17]

gamma_lower(x, y, regularized=0)

>>> RR.gamma_lower(3, 4)
[1.52379338889291 +/- 2.89e-15]
>>> RR.gamma_lower(3, 4, regularized=True)
[0.76189669444646 +/- 6.52e-15]

beta_lower(a, b, x, regularized=0)

>>> RR.beta_lower(2, 3, 0.5)
[0.0572916666666667 +/- 6.08e-17]
>>> RR.beta_lower(2, 3, 0.5, regularized=True)
[0.687500000000000 +/- 5.24e-16]

exp_integral(x, y)

>>> RR.exp_integral(1, 2)
[0.04890051070806 +/- 2.63e-15]

exp_integral_ei(x)

>>> RR.exp_integral_ei(1)
[1.89511781635594 +/- 5.11e-15]

sin_integral(x)

>>> RR.sin_integral(1)
[0.946083070367183 +/- 1.35e-16]

cos_integral(x)

>>> RR.cos_integral(1)
[0.3374039229009681 +/- 5.63e-17]

sinh_integral(x)

>>> RR.sinh_integral(1)
[1.05725087537573 +/- 2.77e-15]

cosh_integral(x)

>>> RR.cosh_integral(1)
[0.837866940980208 +/- 4.78e-16]

log_integral(x, offset=False)

>>> RR.log_integral(2)
[1.04516378011749 +/- 4.01e-15]
>>> RR.log_integral(2, offset=True)
0

dilog(x)

>>> RR.dilog(1)
[1.644934066848226 +/- 6.45e-16]

bessel_j(x, y)

>>> RR.bessel_j(2, 3)
[0.486091260585891 +/- 4.75e-16]

bessel_y(x, y)

>>> RR.bessel_y(2, 3)
[-0.16040039348492 +/- 5.80e-15]

bessel_i(x, y, scaled=False)

>>> RR.bessel_i(2, 3)
[2.24521244092995 +/- 1.88e-15]
>>> RR.bessel_i(2, 3, scaled=True)
[0.111782545296958 +/- 2.09e-16]

bessel_k(x, y, scaled=False)

>>> RR.bessel_k(2, 3)
[0.06151045847174 +/- 8.87e-15]
>>> RR.bessel_k(2, 3, scaled=True)
[1.235470584796 +/- 5.14e-13]

bessel_j_y(x, y)

airy(x)

airy_ai(x)

[0.1352924163128814 +/- 4.17e-17]

airy_bi(x)

[-0.1591474412967932 +/- 2.95e-17]

airy_ai_prime(x)

[1.207423594952871 +/- 3.27e-16]

airy_bi_prime(x)

[0.932435933392776 +/- 5.83e-16]

airy_ai_zero(n)

>>> RR.airy_ai(RR.airy_ai_zero(1))
[+/- 3.51e-16]

airy_bi_zero(n)

>>> RR.airy_bi(RR.airy_bi_zero(1))
[+/- 2.08e-16]

airy_ai_prime_zero(n)

>>> RR.airy_ai_prime(RR.airy_ai_prime_zero(1))
[+/- 1.44e-16]

airy_bi_prime_zero(n)

>>> RR.airy_bi_prime(RR.airy_bi_prime_zero(1))
[+/- 6.18e-16]

coulomb_f(x, y, z)

>>> CC.coulomb_f(2, 3, 4)
[0.101631502833431 +/- 8.03e-16]

coulomb_g(x, y, z)

>>> CC.coulomb_g(2, 3, 4)
[5.371722466 +/- 6.15e-10]

coulomb_hpos(x, y, z)

>>> CC.coulomb_hpos(2, 3, 4)
([5.371722466 +/- 6.15e-10] + [0.101631502833431 +/- 8.03e-16]*I)

coulomb_hneg(x, y, z)

>>> CC.coulomb_hneg(2, 3, 4)
([5.371722466 +/- 6.15e-10] + [-0.101631502833431 +/- 8.03e-16]*I)

chebyshev_t(n, x)

Chebyshev polynomial of the first kind.

>>> [ZZ.chebyshev_t(n, 2) for n in range(5)]
[1, 2, 7, 26, 97]
>>> RR.chebyshev_t(0.5, 0.75)
[0.935414346693485 +/- 5.18e-16]
>>> ZZx.chebyshev_t(4, [0, 1])
1 - 8*x^2 + 8*x^4

chebyshev_u(n, x)

Chebyshev polynomial of the second kind.

>>> [ZZ.chebyshev_u(n, 2) for n in range(5)]
[1, 4, 15, 56, 209]
>>> RR.chebyshev_u(0.5, 0.75)
[1.33630620956212 +/- 2.68e-15]
>>> ZZx.chebyshev_u(4, [0, 1])
1 - 12*x^2 + 16*x^4

jacobi_p(n, a, b, x)

Jacobi polynomial.

>>> RR.jacobi_p(3, 1, 2, 4)
[602.500000000000 +/- 3.28e-13]

gegenbauer_c(n, m, x)

Gegenbauer polynomial.

>>> RR.gegenbauer_c(3, 2, 4)
[2000.00000000000 +/- 3.60e-12]

laguerre_l(n, m, x)

Associated Laguerre polynomial (or Laguerre function).

>>> RR.laguerre_l(3, 2, 4)
[-0.66666666666667 +/- 5.71e-15]

hermite_h(n, x)

Hermite polynomial (Hermite function).

>>> RR.hermite_h(3, 4)
464.0000000000000

legendre_p(n, m, x, typ=0)

Associated Legendre function of the first kind.

legendre_q(n, m, x, typ=0)

Associated Legendre function of the second kind.

spherical_y(n, m, theta, phi)

Spherical harmonic.

>>> CC.spherical_y(4, 3, 0.5, 0.75)
([0.076036396941350 +/- 2.18e-16] + [-0.094180781089734 +/- 4.96e-16]*I)

legendre_p_root(n, k, weight=False)

Root of Legendre polynomial. With weight=True, also returns the corresponding weight for Gauss-Legendre quadrature.

>>> RR.legendre_p_root(5, 1)
[0.538469310105683 +/- 1.15e-16]
>>> RR.legendre_p(5, 0, RR.legendre_p_root(5, 1))
[+/- 8.15e-16]
>>> RR.legendre_p_root(5, 1, weight=True)
([0.538469310105683 +/- 1.15e-16], [0.4786286704993664 +/- 7.10e-17])

hypgeom_0f1(a, z, regularized=False)

Hypergeometric function 0F1, optionally regularized.

>>> RR.hypgeom_0f1(3, 4)
[3.21109468764205 +/- 5.00e-15]
>>> RR.hypgeom_0f1(3, 4, regularized=True)
[1.60554734382103 +/- 5.20e-15]
>>> CC.hypgeom_0f1(1, 2+2j)
([2.435598449671389 +/- 7.27e-16] + [4.43452765355337 +/- 4.91e-15]*I)

hypgeom_1f1(a, b, z, regularized=False)

Hypergeometric function 1F1, optionally regularized.

>>> RR.hypgeom_1f1(3, 4, 5)
[60.504568913851 +/- 3.82e-13]
>>> RR.hypgeom_1f1(3, 4, 5, regularized=True)
[10.0840948189752 +/- 3.31e-14]

hypgeom_u(a, b, z)

Hypergeometric function U.

>>> RR.hypgeom_u(1, 2, 3)
[0.3333333333333333 +/- 7.04e-17]

hypgeom_2f1(a, b, c, z, regularized=False)

Hypergeometric function 2F1, optionally regularized.

>>> RR.hypgeom_2f1(1, 2, 3, -4)
[0.29882026094574 +/- 8.48e-15]
>>> RR.hypgeom_2f1(1, 2, 3, -4, regularized=True)
[0.14941013047287 +/- 4.24e-15]

hypgeom_pfq(a, b, z, regularized=False)

Generalized hypergeometric function, optionally regularized.

>>> RR.hypgeom_pfq([1,2], [3,4], 0.5)
[1.09002619782383 +/- 4.32e-15]
>>> RR.hypgeom_pfq([1, 2], [3, 4], 0.5, regularized=True)
[0.090835516485319 +/- 2.36e-16]
>>> CC.hypgeom_pfq([1,2], [3,4], 0.5+0.5j)
([1.08239550393928 +/- 2.16e-15] + [0.096660812453003 +/- 5.55e-16]*I)

fac(x)

Factorial.

>>> ZZ.fac(10)
3628800
>>> ZZ.fac(-1)
Traceback (most recent call last):
  ...
FlintDomainError: fac(x) is not an element of {Integer ring (fmpz)} for {x = -1}

Real and complex factorials extend using the gamma function:

>>> RR.fac(10**20)
[1.93284951431010e+1956570551809674817245 +/- 3.03e+1956570551809674817230]
>>> RR.fac(0.5)
[0.886226925452758 +/- 1.78e-16]
>>> CC.fac(1+1j)
([0.652965496420167 +/- 6.21e-16] + [0.343065839816545 +/- 5.38e-16]*I)

Factorials mod N:

>>> ZZmod(10**7 + 19).fac(10**7)
2343096

fac_vec(length)

Vector of factorials.

>>> ZZ.fac_vec(10)
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
>>> QQ.fac_vec(10) / 3
[1/3, 1/3, 2/3, 2, 8, 40, 240, 1680, 13440, 120960]
>>> ZZmod(7).fac_vec(10)
[1, 1, 2, 6, 3, 1, 6, 0, 0, 0]
>>> sum(RR.fac_vec(100))
[9.427862397658e+155 +/- 3.19e+142]

rfac(x)

Reciprocal factorial.

>>> QQ.rfac(5)
1/120
>>> ZZ.rfac(-2)
0
>>> ZZ.rfac(2)
Traceback (most recent call last):
  ...
FlintDomainError: rfac(x) is not an element of {Integer ring (fmpz)} for {x = 2}
>>> RR.rfac(0.5)
[1.128379167095513 +/- 7.02e-16]

rfac_vec(length)

Vector of reciprocal factorials.

>>> QQ.rfac_vec(8)
[1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040]
>>> ZZmod(7).rfac_vec(7)
[1, 1, 4, 6, 5, 1, 6]
>>> ZZmod(7).rfac_vec(8)
Traceback (most recent call last):
  ...
FlintDomainError: rfac_vec(length) is not an element of {Integers mod 7 (_gr_nmod)} for {length = 8}
>>> sum(RR.rfac_vec(20))
[2.71828182845904 +/- 8.66e-15]

rising(x, n)

Rising factorial.

>>> [ZZ.rising(3, k) for k in range(5)]
[1, 3, 12, 60, 360]
>>> ZZx.rising(ZZx([0,1]), 5)
24*x + 50*x^2 + 35*x^3 + 10*x^4 + x^5
>>> RR.rising(1, 10**7)
[1.202423400515903e+65657059 +/- 5.57e+65657043]

falling(x, n)

Falling factorial.

>>> [ZZ.falling(3, k) for k in range(5)]
[1, 3, 6, 6, 0]
>>> ZZx.falling(ZZx([0,1]), 5)
24*x - 50*x^2 + 35*x^3 - 10*x^4 + x^5
>>> RR.log(RR.falling(RR.pi(), 10**7))
[151180898.7174084 +/- 9.72e-8]

bin(x, y)

Binomial coefficient.

>>> [ZZ.bin(5, k) for k in range(7)]
[1, 5, 10, 10, 5, 1, 0]
>>> RR.bin(100000, 50000)
[2.52060836892200e+30100 +/- 5.36e+30085]
>>> ZZmod(1000).bin(10000, 3000)
200
>>> ZZp64.bin(100000, 50000)
5763493550349629692
>>> ZZp64.bin(10**30, 2)
998763921924463582

bin_vec(n, length=None)

Vector of binomial coefficients, optionally truncated to specified length.

>>> ZZ.bin_vec(8)
[1, 8, 28, 56, 70, 56, 28, 8, 1]
>>> ZZmod(5).bin_vec(8)
[1, 3, 3, 1, 0, 1, 3, 3, 1]
>>> ZZ.bin_vec(0)
[1]
>>> ZZ.bin_vec(1000, 3)
[1, 1000, 499500]
>>> ZZ.bin_vec(4, 8)
[1, 4, 6, 4, 1, 0, 0, 0]
>>> QQ.bin_vec(QQ(1)/2, 5)
[1, 1/2, -1/8, 1/16, -5/128]

gamma(x)

lgamma(x)

rgamma(x)

digamma(x)

doublefac(x)

Double factorial (semifactorial).

>>> [ZZ.doublefac(n) for n in range(10)]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
>>> RR.doublefac(2.5)
[2.40706945611604 +/- 5.54e-15]
>>> CC.doublefac(1+1j)
([0.250650779545753 +/- 7.56e-16] + [0.100474421235437 +/- 4.14e-16]*I)

harmonic(x)

Harmonic numbers.

>>> [QQ.harmonic(n) for n in range(6)]
[0, 1, 3/2, 11/6, 25/12, 137/60]
>>> RR.harmonic(10**9)
[21.30048150234794 +/- 8.48e-15]
>>> ZZp64.harmonic(1000)
6514760847963681162

beta(x, y)

Beta function.

>>> RR.beta(3, 4.5)
[0.01243201243201243 +/- 6.93e-18]
>>> CC.beta(1j, 1+1j)
([-1.18807306241087 +/- 5.32e-15] + [-1.31978426013907 +/- 4.09e-15]*I)

barnes_g(x)

Barnes G-function.

>>> RR.barnes_g(7)
34560.00000000000
>>> CC.barnes_g(1+2j)
([0.54596949228965 +/- 7.69e-15] + [-3.98421873125106 +/- 8.76e-15]*I)

log_barnes_g(x)

Logarithmic Barnes G-function.

>>> RR.log_barnes_g(100)
[15258.0613921488 +/- 3.87e-11]
>>> CC.log_barnes_g(10+20j)
([-452.057343313397 +/- 6.85e-13] + [121.014356688943 +/- 2.52e-13]*I)

zeta(s)

hurwitz_zeta(s, a)

stieltjes(n, a=1)

Stieltjes constant.

>>> CC.stieltjes(1)
[-0.0728158454836767 +/- 2.78e-17]
>>> CC.stieltjes(1, a=0.5)
[-1.353459680804942 +/- 7.22e-16]

polylog(s, z)

Polylogarithm.

>>> CC.polylog(2, -1)
[-0.822467033424113 +/- 3.22e-16]

polygamma(s, z)

Polygamma function.

>>> CC.polygamma(2, 3)
[-0.1541138063191886 +/- 7.16e-17]

lerch_phi(z, s, a)

>>> CC.lerch_phi(2, 3, 4)
([-0.00213902437921 +/- 1.70e-15] + [-0.04716836434127 +/- 5.28e-15]*I)

dirichlet_eta(x)

Dirichlet eta function.

>>> CC.dirichlet_eta(1)
[0.6931471805599453 +/- 6.93e-17]
>>> CC.dirichlet_eta(2)
[0.822467033424113 +/- 2.36e-16]

riemann_xi(x)

Riemann xi function.

>>> s = 2+3j; CC.riemann_xi(s); CC.riemann_xi(1-s)
([0.41627125989962 +/- 4.65e-15] + [0.08882330496564 +/- 1.43e-15]*I)
([0.41627125989962 +/- 4.65e-15] + [0.08882330496564 +/- 1.43e-15]*I)

lambertw(x, k=None)

bernoulli(n)

Bernoulli number \(B_n\) as an element of this domain.

>>> QQ.bernoulli(10)
5/66
>>> RR.bernoulli(10)
[0.0757575757575757 +/- 5.97e-17]

>>> ZZ.bernoulli(0)
1
>>> ZZ.bernoulli(1)
Traceback (most recent call last):
  ...
FlintDomainError: bernoulli(n) is not an element of {Integer ring (fmpz)} for {n = 1}

Huge Bernoulli numbers can be computed numerically:

>>> RR.bernoulli(10**20)
[-1.220421181609039e+1876752564973863312289 +/- 4.69e+1876752564973863312273]
>>> RF.bernoulli(10**20)
-1.220421181609039e+1876752564973863312289
>>> QQ.bernoulli(10**20)
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute bernoulli(n) in {Rational field (fmpq)} for {n = 100000000000000000000}

bernoulli_vec(length)

Vector of Bernoulli numbers.

>>> QQ.bernoulli_vec(12)
[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0]
>>> CC_ca.bernoulli_vec(5)
[1, -0.500000 {-1/2}, 0.166667 {1/6}, 0, -0.0333333 {-1/30}]
>>> sum(RR.bernoulli_vec(100))
[1.127124216595034e+76 +/- 6.74e+60]
>>> sum(RF.bernoulli_vec(100))
1.127124216595034e+76
>>> sum(CC.bernoulli_vec(100))
[1.127124216595034e+76 +/- 6.74e+60]

eulernum(n)

Euler number \(E_n\) as an element of this domain.

>>> ZZ.eulernum(10)
-50521
>>> RR.eulernum(10)
-50521.00000000000

Huge Euler numbers can be computed numerically:

>>> RR.eulernum(10**20)
[4.346791453661149e+1936958564106659551331 +/- 8.35e+1936958564106659551315]
>>> RF.eulernum(10**20)
4.346791453661149e+1936958564106659551331
>>> ZZ.eulernum(10**20)
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute eulernum(n) in {Integer ring (fmpz)} for {n = 100000000000000000000}

eulernum_vec(length)

Vector of Euler numbers.

>>> ZZ.eulernum_vec(12)
[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0]
>>> QQ.eulernum_vec(12) / 3
[1/3, 0, -1/3, 0, 5/3, 0, -61/3, 0, 1385/3, 0, -50521/3, 0]
>>> sum(RR.eulernum_vec(100))
[-7.23465655613392e+134 +/- 3.20e+119]
>>> sum(RF.eulernum_vec(100))
-7.234656556133921e+134

fib(n)

Fibonacci number \(F_n\) as an element of this domain.

>>> ZZ.fib(10)
55
>>> RR.fib(10)
55.00000000000000
>>> ZZ.fib(-10)
-55

Huge Fibonacci numbers can be computed numerically and in modular arithmetic:

>>> RR.fib(10**20)
[3.78202087472056e+20898764024997873376 +/- 4.02e+20898764024997873361]
>>> RF.fib(10**20)
3.782020874720557e+20898764024997873376
>>> F = FiniteField_fq(17, 1)
>>> n = 10**20; F.fib(n); F.fib(n-1) + F.fib(n-2)
13
13

fib_vec(length)

Vector of Fibonacci numbers.

>>> ZZ.fib_vec(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
>>> QQ.fib_vec(10) / 3
[0, 1/3, 1/3, 2/3, 1, 5/3, 8/3, 13/3, 7, 34/3]
>>> sum(RR.fib_vec(100))            
[5.7314784401...e+20 +/- ...]
>>> sum(RF.fib_vec(100))
5.731478440138172e+20

stirling_s1u(n, k)

Unsigned Stirling number of the first kind.

>>> ZZ.stirling_s1u(5, 2)
50
>>> QQ.stirling_s1u(5, 2)
50
>>> ZZ.stirling_s1u(50, 21)
33187391298039120738041153829116024033357291261862000
>>> RR.stirling_s1u(50, 21)
[3.318739129803912e+52 +/- 8.66e+36]

stirling_s1(n, k)

Signed Stirling number of the first kind.

>>> ZZ.stirling_s1(5, 2)
-50
>>> QQ.stirling_s1(5, 2)
-50
>>> RR.stirling_s1(5, 2)
-50.00000000000000

stirling_s2(n, k)

Stirling number of the second kind.

>>> ZZ.stirling_s2(5, 2)
15
>>> QQ.stirling_s2(5, 2)
15
>>> RR.stirling_s2(5, 2)
15.00000000000000
>>> RR.stirling_s2(50, 20)
[7.59792160686099e+45 +/- 5.27e+30]

stirling_s1u_vec(n, length=None)

Vector of unsigned Stirling numbers of the first kind, optionally truncated to specified length.

>>> ZZ.stirling_s1u_vec(5)
[0, 24, 50, 35, 10, 1]
>>> QQ.stirling_s1u_vec(5) / 3
[0, 8, 50/3, 35/3, 10/3, 1/3]
>>> RR.stirling_s1u_vec(5, 3)
[0, 24.00000000000000, 50.00000000000000]

stirling_s1_vec(n, length=None)

Vector of signed Stirling numbers of the first kind, optionally truncated to specified length.

>>> ZZ.stirling_s1_vec(5)
[0, 24, -50, 35, -10, 1]
>>> QQ.stirling_s1_vec(5) / 3
[0, 8, -50/3, 35/3, -10/3, 1/3]
>>> RR.stirling_s1_vec(5, 3)
[0, 24.00000000000000, -50.00000000000000]

stirling_s2_vec(n, length=None)

Vector of Stirling numbers of the second kind, optionally truncated to specified length.

>>> ZZ.stirling_s2_vec(5)
[0, 1, 15, 25, 10, 1]
>>> QQ.stirling_s2_vec(5) / 3
[0, 1/3, 5, 25/3, 10/3, 1/3]
>>> RR.stirling_s2_vec(5, 3)
[0, 1.000000000000000, 15.00000000000000]

bellnum(n)

Bell number \(E_n\) as an element of this domain.

>>> ZZ.bellnum(10)
115975
>>> RR.bellnum(10)
115975.0000000000
>>> ZZp64.bellnum(10000)
355901145009109239
>>> ZZmod(1000).bellnum(10000)
635

Huge Bell numbers can be computed numerically:

>>> RR.bellnum(10**20)
[5.38270113176282e+1794956117137290721328 +/- 5.44e+1794956117137290721313]
>>> ZZ.bellnum(10**20)
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute bellnum(n) in {Integer ring (fmpz)} for {n = 100000000000000000000}

bellnum_vec(length)

Vector of Bell numbers.

>>> ZZ.bellnum_vec(10)
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
>>> QQ.bellnum_vec(10) / 3
[1/3, 1/3, 2/3, 5/3, 5, 52/3, 203/3, 877/3, 1380, 7049]
>>> RR.bellnum_vec(100).sum()
[1.67618752079292e+114 +/- 4.30e+99]
>>> RF.bellnum_vec(100).sum()
1.676187520792924e+114
>>> ZZmod(10000).bellnum_vec(10000).sum()
7337

partitions(n)

Partition function \(p(n)\) as an element of this domain.

>>> ZZ.partitions(10)
42
>>> QQ.partitions(10) / 5
42/5
>>> RR.partitions(10)
42.00000000000000
>>> RR.partitions(10**20)
[1.838176508344883e+11140086259 +/- 8.18e+11140086243]

partitions_vec(length)

Vector of partition numbers.

>>> ZZ.partitions_vec(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
>>> QQ.partitions_vec(10) / 3
[1/3, 1/3, 2/3, 1, 5/3, 7/3, 11/3, 5, 22/3, 10]
>>> ZZmod(10).partitions_vec(10)
[1, 1, 2, 3, 5, 7, 1, 5, 2, 0]
>>> sum(ZZmod(10).partitions_vec(100))
6
>>> sum(RR.partitions_vec(100))
1452423276.000000

zeta_zero(n)

Zero of the Riemann zeta function.

>>> CC.zeta_zero(1)
(0.5000000000000000 + [14.13472514173469 +/- 4.71e-15]*I)
>>> CC.zeta_zero(2)
(0.5000000000000000 + [21.02203963877155 +/- 6.02e-15]*I)

zeta_zeros(num, start=1)

Zeros of the Riemann zeta function.

>>> [x.im() for x in CC.zeta_zeros(4)]
[[14.13472514173469 +/- 4.71e-15], [21.02203963877155 +/- 6.02e-15], [25.01085758014569 +/- 7.84e-15], [30.42487612585951 +/- 5.96e-15]]
>>> [x.im() for x in CC.zeta_zeros(2, start=100)]
[[236.5242296658162 +/- 3.51e-14], [237.7698204809252 +/- 5.29e-14]]

zeta_nzeros(t)

Number of zeros of Riemann zeta function up to given height.

>>> CC.zeta_nzeros(100)
29.00000000000000

dirichlet_l(s, chi)

Dirichlet L-function with character chi.

>>> CC.dirichlet_l(2, DirichletGroup(1)(1))
[1.644934066848226 +/- 4.57e-16]
>>> RR.dirichlet_l(2, DirichletGroup(4)(3))
[0.915965594177219 +/- 2.68e-16]
>>> CC.dirichlet_l(2+3j, DirichletGroup(7)(3))
([1.273313649440491 +/- 9.69e-16] + [-0.074323294425594 +/- 6.96e-16]*I)

hardy_theta(s, chi=None)

Hardy theta function.

>>> CC.hardy_theta(10)
[-3.06707439628989 +/- 6.66e-15]
>>> CC.hardy_theta(10, DirichletGroup(4)(3))
[4.64979557270698 +/- 4.41e-15]

hardy_z(s, chi=None)

Hardy Z-function.

>>> CC.hardy_z(2)
[-0.539633125646145 +/- 8.59e-16]
>>> CC.hardy_z(2, DirichletGroup(4)(3))
[1.15107760668266 +/- 5.01e-15]

dirichlet_chi(n, chi)

Value of the Dirichlet character chi(n).

>>> chi = DirichletGroup(5)(3)
>>> [CC.dirichlet_chi(n, chi) for n in range(5)]
[0, 1.000000000000000, -1.000000000000000*I, 1.000000000000000*I, -1.000000000000000]

modular_j(tau)

j-invariant j(tau).

>>> CC.modular_j(1j)
[1728.0000000000 +/- 5.10e-11]

modular_lambda(tau)

Modular lambda function lambda(tau).

>>> CC.modular_lambda(1j)
[0.50000000000000 +/- 2.16e-15]

modular_delta(tau)

Modular discriminant delta(tau).

>>> CC.modular_delta(1j)
[0.0017853698506421 +/- 6.01e-17]

dedekind_eta(tau)

Dedekind eta function eta(tau).

>>> CC.dedekind_eta(1j)
[0.768225422326057 +/- 9.03e-16]

hilbert_class_poly(D, x)

Hilbert class polynomial H_D(x) evaluated at x.

>>> ZZx.hilbert_class_poly(-20, ZZx.gen())
-681472000 - 1264000*x + x^2
>>> CC.hilbert_class_poly(-20, 1+1j)
(-682736000.0000000 - 1263998.000000000*I)
>>> ZZx.hilbert_class_poly(-21, ZZx.gen())
Traceback (most recent call last):
  ...
FlintDomainError: hilbert_class_poly(D, x) is not an element of {Ring of polynomials over Integer ring (fmpz)} for {D = -21}, {x = x}

eisenstein_g(n, tau)

Eisenstein series G_n(tau).

>>> CC.eisenstein_g(2, 1j)
[3.14159265358979 +/- 8.71e-15]
>>> CC.eisenstein_g(4, 1j); RR.gamma(0.25)**8 / (960 * RR.pi()**2)
[3.1512120021539 +/- 3.41e-14]
[3.15121200215390 +/- 7.72e-15]

eisenstein_e(n, tau)

Eisenstein series E_n(tau).

>>> CC.eisenstein_e(2, 1j)
[0.95492965855137 +/- 3.85e-15]
>>> CC.eisenstein_e(4, 1j); 3*RR.gamma(0.25)**8/(64*RR.pi()**6)
[1.4557628922687 +/- 1.32e-14]
[1.45576289226871 +/- 3.76e-15]

eisenstein_g_vec(tau, n)

Vector of Eisenstein series [G_4(tau), G_6(tau), …]. Note that G_2(tau) is omitted.

>>> CC.eisenstein_g_vec(1j, 3)
[[3.1512120021539 +/- 3.41e-14], [+/- 4.40e-14], [4.255773035365 +/- 2.12e-13]]

agm(x, y=None)

Arithmetic-geometric mean.

>>> RR.agm(2)
[1.45679103104691 +/- 3.98e-15]
>>> RR.agm(2, 3)
[2.47468043623630 +/- 4.68e-15]

elliptic_k(m)

elliptic_e(m)

elliptic_pi(n, m)

elliptic_f(phi, m, pi=0)

elliptic_e_inc(phi, m, pi=0)

elliptic_pi_inc(n, phi, m, pi=0)

carlson_rc(x, y, flags=0)

carlson_rf(x, y, z, flags=0)

carlson_rg(x, y, z, flags=0)

carlson_rd(x, y, z, flags=0)

carlson_rj(x, y, z, w, flags=0)

jacobi_theta(z, tau)

Simultaneous computation of the four Jacobi theta functions.

>>> CC.jacobi_theta(0.125, 1j)
([0.347386687929454 +/- 3.21e-16], [0.843115469091413 +/- 8.18e-16], [1.061113709291166 +/- 5.74e-16], [0.938886290708834 +/- 3.52e-16])

jacobi_theta_1(z, tau)

Jacobi theta function.

>>> CC.jacobi_theta_1(0.125, 1j)
[0.347386687929454 +/- 3.21e-16]

jacobi_theta_2(z, tau)

Jacobi theta function.

>>> CC.jacobi_theta_2(0.125, 1j)
[0.843115469091413 +/- 8.18e-16]

jacobi_theta_3(z, tau)

Jacobi theta function.

>>> CC.jacobi_theta_3(0.125, 1j)
[1.061113709291166 +/- 5.74e-16]

jacobi_theta_4(z, tau)

Jacobi theta function.

>>> CC.jacobi_theta_4(0.125, 1j)
[0.938886290708834 +/- 3.52e-16]

elliptic_invariants(tau)

>>> g2, g3 = CC.elliptic_invariants(1j)
>>> CC.weierstrass_p_prime(0.25, 1j)**2; 4*CC.weierstrass_p(0.25, 1j)**3 - g2*CC.weierstrass_p(0.25, 1j) - g3
[15152.862386715 +/- 7.03e-10]
[15152.86238672 +/- 5.09e-9]

elliptic_roots(tau)

>>> e1, e2, e3 = CC.elliptic_roots(1j)
>>> g2, g3 = CC.elliptic_invariants(1j)
>>> 4*e1**3 - g2*e1 - g3
[+/- 3.12e-11]
>>> 4*e2**3 - g2*e2 - g3
[+/- 8.29e-12]
>>> 4*e3**3 - g2*e3 - g3
[+/- 3.14e-11]

weierstrass_p(z, tau)

weierstrass_p_prime(z, tau)

weierstrass_p_inv(z, tau)

Inverse Weierstrass elliptic function.

>>> CC.weierstrass_p(CC.weierstrass_p_inv(0.5, 1j), 1j)
([0.50000000000 +/- 4.61e-12] + [+/- 6.98e-12]*I)

weierstrass_zeta(z, tau)

weierstrass_sigma(z, tau)

class flint_ctypes.gr_elem(val=None, context=None, random=False)

Base class for elements.

__init__(val=None, context=None, random=False)

>>> ZZ(QQ(1))
1
>>> ZZ(QQ(1) / 3)
Traceback (most recent call last):
  ...
FlintDomainError: 1/3 is not defined in Integer ring (fmpz)

parent()

Return the parent object of this element.

>>> ZZ(0).parent()
Integer ring (fmpz)
>>> ZZ(0).parent() is ZZ
True

is_invertible()

Return whether self has a multiplicative inverse in its domain.

>>> 
>>> ZZ(3).is_invertible()
False
>>> ZZ(-1).is_invertible()
True

divides(other)

Return whether self divides other.

>>> ZZ(5).divides(10)
True
>>> ZZ(5).divides(12)
False

gcd(other)

Greatest common divisor.

>>> ZZ(24).gcd(30)
6

lcm(other)

Least common multiple.

>>> ZZ(24).lcm(30)
120

factor()

Returns a factorization of self as a tuple (prefactor, factors, exponents).

>>> ZZ(-120).factor()
(-1, [2, 3, 5], [3, 1, 1])

is_square()

Return whether self is a perfect square in its domain.

>>> ZZ(3).is_square()
False
>>> ZZ(4).is_square()
True
>>> QQbar(3).is_square()
True

inv()

Multiplicative inverse of this element.

>>> QQ(3).inv()
1/3
>>> QQ(0).inv()
Traceback (most recent call last):
  ...
FlintDomainError: inv(x) is not an element of {Rational field (fmpq)} for {x = 0}

sqrt()

Square root of this element.

>>> ZZ(4).sqrt()
2
>>> ZZ(2).sqrt()
Traceback (most recent call last):
  ...
FlintDomainError: sqrt(x) is not an element of {Integer ring (fmpz)} for {x = 2}
>>> QQbar(2).sqrt()
Root a = 1.41421 of a^2-2
>>> (QQ(25)/16).sqrt()
5/4
>>> QQbar(-1).sqrt()
Root a = 1.00000*I of a^2+1
>>> RR(-1).sqrt()
Traceback (most recent call last):
  ...
FlintDomainError: sqrt(x) is not an element of {Real numbers (arb, prec = 53)} for {x = -1.000000000000000}
>>> RF(-1).sqrt()
nan

rsqrt()

Reciprocal square root of this element.

>>> QQ(25).rsqrt()
1/5

floor()

Floor function: closest integer in the direction of \(-\infty\).

>>> (QQ(3) / 2).floor()
1
>>> (QQ(3) / 2).ceil()
2
>>> (QQ(3) / 2).nint()
2
>>> (QQ(3) / 2).trunc()
1

ceil()

Ceiling function: closest integer in the direction of \(+\infty\).

>>> (QQ(3) / 2).ceil()
2

trunc()

Truncate to integer: closest integer in the direction of zero.

>>> (QQ(3) / 2).trunc()
1

nint()

Nearest integer function: nearest integer, rounding to even on a tie.

>>> (QQ(3) / 2).nint()
2

abs()

conj()

Complex conjugate.

>>> QQbar.i().conj()
Root a = -1.00000*I of a^2+1
>>> CC(-2).log().conj()
([0.693147180559945 +/- 4.12e-16] + [-3.141592653589793 +/- 3.39e-16]*I)
>>> QQ(3).conj()
3

re()

Real part.

>>> QQ(1).re()
1
>>> (QQbar(-1) ** (QQ(1) / 3)).re()
1/2

im()

Imaginary part.

>>> QQ(1).im()
0
>>> (QQbar(-1) ** (QQ(1) / 3)).im()
Root a = 0.866025 of 4*a^2-3

sgn()

Sign function.

>>> QQ(-5).sgn()
-1
>>> CC(-10).sqrt().sgn()
1.000000000000000*I

csgn()

Real-valued extension of the sign function: gives the sign of the real part when nonzero, and the sign of the imaginary part when on the imaginary axis.

>>> QQbar(-10).sqrt().csgn()
1
>>> (-QQbar(-10).sqrt()).csgn()
-1

mul_2exp(other)

Exact multiplication by a dyadic number \(2^y\).

>>> QQ(3).mul_2exp(5)
96
>>> QQ(3).mul_2exp(-5)
3/32
>>> ZZ(100).mul_2exp(-2)
25
>>> ZZ(100).mul_2exp(-3)
Traceback (most recent call last):
  ...
FlintDomainError: mul_2exp(x, y) is not an element of {Integer ring (fmpz)} for {x = 100}, {y = -3}

exp()

Exponential function.

>>> RR(1).exp()
[2.718281828459045 +/- 5.41e-16]
>>> RR_ca(1).exp()
2.71828 {a where a = 2.71828 [Exp(1)]}
>>> QQ(0).exp()
1
>>> QQ(1).exp()
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute exp(x) in {Rational field (fmpq)} for {x = 1}

expm1()

Exponential function minus 1.

>>> RR("1e-10").expm1()
[1.000000000050000e-10 +/- 3.86e-26]
>>> CC(RR("1e-10")).expm1()
[1.000000000050000e-10 +/- 3.86e-26]
>>> RF("1e-10").expm1()
1.000000000050000e-10
>>> CF(RF("1e-10")).expm1()
1.000000000050000e-10
>>> QQ(0).expm1()
0
>>> QQ(1).expm1()
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute expm1(x) in {Rational field (fmpq)} for {x = 1}

exp2()

Exponential function with base 2.

>>> QQ(5).exp2()
32
>>> RF(0.5).exp2()
1.414213562373095

exp10()

Exponential function with base 10.

>>> QQ(5).exp2()
32
>>> RF(0.5).exp10()
3.162277660168380

log()

Natural logarithm.

>>> QQ(1).log()
0
>>> QQ(2).log()
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute log(x) in {Rational field (fmpq)} for {x = 2}
>>> RF(2).log()
0.6931471805599453

log1p()

Natural logarithm with one added to the argument.

>>> QQ(0).log1p()
0
>>> RF(-0.5).log1p()
-0.6931471805599453
>>> RR(1).log1p()
[0.693147180559945 +/- 4.12e-16]

sin()

cos()

tan()

sinh()

cosh()

tanh()

atan()

exp_pi_i()

\(\exp(\pi i x)\) evaluated at self.

>>> (QQbar(1) / 3).exp_pi_i()
Root a = 0.500000 + 0.866025*I of a^2-a+1
>>> (QQbar(2).sqrt()).exp_pi_i()
Traceback (most recent call last):
  ...
FlintDomainError: exp_pi_i(x) is not an element of {Complex algebraic numbers (qqbar)} for {x = Root a = 1.41421 of a^2-2}

log_pi_i()

\(\log(x) / (\pi i)\) evaluated at self.

>>> (QQbar(-1) ** (QQbar(7) / 5)).log_pi_i()
-3/5
>>> (QQbar(1) / 2).log_pi_i()
Traceback (most recent call last):
  ...
FlintDomainError: log_pi_i(x) is not an element of {Complex algebraic numbers (qqbar)} for {x = 1/2}

sin_pi()

\(\sin(\pi x)\) evaluated at self.

>>> (QQbar(1) / 3).sin_pi()
Root a = 0.866025 of 4*a^2-3

cos_pi()

\(\cos(\pi x)\) evaluated at self.

>>> (QQbar(1) / 3).cos_pi()
1/2

tan_pi()

\(\tan(\pi x)\) evaluated at self.

>>> (QQbar(1) / 3).tan_pi()
Root a = 1.73205 of a^2-3

cot_pi()

\(\cot(\pi x)\) evaluated at self.

>>> (QQbar(1) / 3).cot_pi()
Root a = 0.577350 of 3*a^2-1

sec_pi()

\(\sec(\pi x)\) evaluated at self.

>>> (QQbar(1) / 3).sec_pi()
2

csc_pi()

\(\csc(\pi x)\) evaluated at self.

>>> (QQbar(1) / 3).csc_pi()
Root a = 1.15470 of 3*a^2-4

asin_pi()

acos_pi()

atan_pi()

acot_pi()

asec_pi()

acsc_pi()

erf()

erfi()

erfc()

gamma()

lgamma()

rgamma()

digamma()

zeta()

class flint_ctypes.IntegerRing_fmpz

__init__()

class flint_ctypes.RationalField_fmpq

__init__()

class flint_ctypes.GaussianIntegerRing_fmpzi

__init__()

class flint_ctypes.ComplexAlgebraicField_qqbar

__init__()

class flint_ctypes.RealAlgebraicField_qqbar

__init__()

class flint_ctypes.gr_arb_ctx

class flint_ctypes.RealField_arb(prec=53)

__init__(prec=53)

class flint_ctypes.ComplexField_acb(prec=53)

__init__(prec=53)

class flint_ctypes.gr_ctx_ca

options()

class flint_ctypes.RealAlgebraicField_ca(**kwargs)

__init__(**kwargs)

class flint_ctypes.ComplexAlgebraicField_ca(**kwargs)

__init__(**kwargs)

class flint_ctypes.RealField_ca(**kwargs)

__init__(**kwargs)

class flint_ctypes.ComplexField_ca(**kwargs)

__init__(**kwargs)

class flint_ctypes.PolynomialRing_gr_poly(coefficient_ring)

__init__(coefficient_ring)

class flint_ctypes.PowerSeriesRing_gr_series(coefficient_ring, prec=6)

__init__(coefficient_ring, prec=6)

class flint_ctypes.PowerSeriesModRing_gr_series(coefficient_ring, mod=6)

__init__(coefficient_ring, mod=6)

class flint_ctypes.fmpz(val=None, context=None, random=False)

is_prime()

class flint_ctypes.fmpq(val=None, context=None, random=False)

class flint_ctypes.fmpzi(val=None, context=None, random=False)

class flint_ctypes.qqbar(val=None, context=None, random=False)

class flint_ctypes.ca(val=None, context=None, random=False)

class flint_ctypes.arb(val=None, context=None, random=False)

class flint_ctypes.acb(val=None, context=None, random=False)

class flint_ctypes.gr_arf_ctx

class flint_ctypes.RealFloat_arf(prec=53)

__init__(prec=53)

class flint_ctypes.ComplexFloat_acf(prec=53)

__init__(prec=53)

class flint_ctypes.arf(val=None, context=None, random=False)

class flint_ctypes.acf(val=None, context=None, random=False)

class flint_ctypes.IntegersMod_nmod(n)

__init__(n)

class flint_ctypes.nmod(val=None, context=None, random=False)

class flint_ctypes.FiniteField_base

prime()

degree()

order()

class flint_ctypes.FiniteField_fq(p, n)

__init__(p, n)

class flint_ctypes.FiniteField_fq_nmod(p, n)

__init__(p, n)

class flint_ctypes.FiniteField_fq_zech(p, n)

__init__(p, n)

class flint_ctypes.fq_elem(val=None, context=None, random=False)

gen()

multiplicative_order()

norm()

trace()

is_primitive()

pth_root()

class flint_ctypes.fq(val=None, context=None, random=False)

class flint_ctypes.fq_nmod(val=None, context=None, random=False)

class flint_ctypes.fq_zech(val=None, context=None, random=False)

class flint_ctypes.gr_poly(val=None, context=None, random=False)

__init__(val=None, context=None, random=False)

>>> ZZ(QQ(1))
1
>>> ZZ(QQ(1) / 3)
Traceback (most recent call last):
  ...
FlintDomainError: 1/3 is not defined in Integer ring (fmpz)

is_monic()

>>> RRx([2,3,4]).is_monic()
False
>>> RRx([2,3,1]).is_monic()
True
>>> RRx([]).is_monic()
False

monic()

Return self rescaled to a monic polynomial.

>>> f = RRx([1,RR.pi()])
>>> f.monic()
[0.318309886183791 +/- 4.43e-16] + 1.000000000000000*x
>>> RRx([]).monic()   # the zero polynomial cannot be made monic
Traceback (most recent call last):
  ...
ValueError
>>> (f - f).monic()   # unknown whether it is the zero polynomial
Traceback (most recent call last):
  ...
NotImplementedError

derivative()

integral()

roots(domain=None)

Computes the roots in the coefficient ring of this polynomial, returning a tuple (roots, multiplicities). If the ring is not algebraically closed, the sum of multiplicities can be smaller than the degree of the polynomial. If domain is given, returns roots in that ring instead.

>>> (ZZx([3,2]) * ZZx([15,1])**2 * ZZx([-10,1])).roots()
([10, -15], [1, 2])
>>> ZZx([1]).roots()
([], [])

We consider roots of the zero polynomial to be ill-defined:

>>> ZZx([]).roots()
Traceback (most recent call last):
  ...
ValueError

We construct an integer polynomial with rational, real algebraic and complex algebraic roots and extract its roots over different domains:

>>> f = ZZx([-2,0,1]) * ZZx([1, 0, 1]) * ZZx([3, 2])**2
>>> f.roots()   # integer roots (there are none)
([], [])
>>> f.roots(domain=QQ)    # rational roots
([-3/2], [2])
>>> f.roots(domain=AA)     # real algebraic roots
([Root a = 1.41421 of a^2-2, Root a = -1.41421 of a^2-2, -3/2], [1, 1, 2])
>>> f.roots(domain=QQbar)     # complex algebraic roots
([Root a = 1.00000*I of a^2+1, Root a = -1.00000*I of a^2+1, Root a = 1.41421 of a^2-2, Root a = -1.41421 of a^2-2, -3/2], [1, 1, 1, 1, 2])
>>> f.roots(domain=RR)      # real ball roots
([[-1.414213562373095 +/- 4.89e-17], [1.414213562373095 +/- 4.89e-17], -1.500000000000000], [1, 1, 2])
>>> f.roots(domain=CC)      # complex ball roots
([[-1.414213562373095 +/- 4.89e-17], [1.414213562373095 +/- 4.89e-17], 1.000000000000000*I, -1.000000000000000*I, -1.500000000000000], [1, 1, 1, 1, 2])
>>> f.roots(RF)     # real floating-point roots
([-1.414213562373095, 1.414213562373095, -1.500000000000000], [1, 1, 2])
>>> f.roots(CF)     # complex floating-point roots
([-1.414213562373095, 1.414213562373095, 1.000000000000000*I, -1.000000000000000*I, -1.500000000000000], [1, 1, 1, 1, 2])

inv_series(n)

Reciprocal of this polynomial viewed as a power series, truncated to length n.

>>> ZZx([1,2,3]).inv_series(10)
1 - 2*x + x^2 + 4*x^3 - 11*x^4 + 10*x^5 + 13*x^6 - 56*x^7 + 73*x^8 + 22*x^9
>>> ZZx([2,3,4]).inv_series(5)
Traceback (most recent call last):
  ...
FlintDomainError: f.inv_series(n) is not an element of {Ring of polynomials over Integer ring (fmpz)} for {f = 2 + 3*x + 4*x^2}, {n = 5}
>>> QQx([2,3,4]).inv_series(5)
(1/2) + (-3/4)*x + (1/8)*x^2 + (21/16)*x^3 + (-71/32)*x^4

div_series(other, n)

log_series(n)

Logarithm of this polynomial viewed as a power series, truncated to length n.

>>> QQx([1,1]).log_series(8)
x + (-1/2)*x^2 + (1/3)*x^3 + (-1/4)*x^4 + (1/5)*x^5 + (-1/6)*x^6 + (1/7)*x^7
>>> RRx([2,1]).log_series(3)
[0.693147180559945 +/- 4.12e-16] + 0.5000000000000000*x - 0.1250000000000000*x^2
>>> RRx([0,0]).log_series(3)
Traceback (most recent call last):
  ...
FlintDomainError: f.log_series(n) is not an element of {Ring of polynomials over Real numbers (arb, prec = 53)} for {f = 0}, {n = 3}

exp_series(n)

Exponential of this polynomial viewed as a power series, truncated to length n.

>>> QQx([0,1]).exp_series(8)
1 + x + (1/2)*x^2 + (1/6)*x^3 + (1/24)*x^4 + (1/120)*x^5 + (1/720)*x^6 + (1/5040)*x^7
>>> QQx([1,1]).exp_series(2)
Traceback (most recent call last):
  ...
FlintUnableError: failed to compute f.exp_series(n) in {Ring of polynomials over Rational field (fmpq)} for {f = 1 + x}, {n = 2}
>>> RRx([1,1]).exp_series(2)
[2.718281828459045 +/- 5.41e-16] + [2.718281828459045 +/- 5.41e-16]*x
>>> RRx([2,3]).log_series(3).exp_series(3)
[2.000000000000000 +/- 6.97e-16] + [3.00000000000000 +/- 1.61e-15]*x + [+/- 1.49e-15]*x^2

pow_series(other, n)

Power of this polynomial viewed as a power series, truncated to length n.

>>> QQx([4,3,2]).pow_series(QQ(1) / 2, 6)
2 + (3/4)*x + (23/64)*x^2 + (-69/512)*x^3 + (299/16384)*x^4 + (2277/131072)*x^5
>>> (QQx([4,3,2]) ** 2).pow_series(QQ(1) / 2, 6)
4 + 3*x + 2*x^2

atan_series(n)

atanh_series(n)

sqrt_series(n)

rsqrt_series(n)

class flint_ctypes.gr_series(val=None, error=None, context=None, random=False)

__init__(val=None, error=None, context=None, random=False)

>>> ZZ(QQ(1))
1
>>> ZZ(QQ(1) / 3)
Traceback (most recent call last):
  ...
FlintDomainError: 1/3 is not defined in Integer ring (fmpz)

class flint_ctypes.ModularGroup_psl2z(**kwargs)

__init__(**kwargs)

generators()

class flint_ctypes.psl2z(val=None, context=None, random=False)

class flint_ctypes.DirichletGroup_dirichlet_char(q, **kwargs)

Group of Dirichlet characters of given modulus.

>>> G = DirichletGroup(10)
>>> G.q
10
>>> len(G)
4
>>> [G(i) for i in [1,3,7,9]]
[chi_10(1, .), chi_10(3, .), chi_10(7, .), chi_10(9, .)]

__init__(q, **kwargs)

class flint_ctypes.dirichlet_char(val=None, context=None, random=False)

class flint_ctypes.SymmetricGroup_perm(n, **kwargs)

__init__(n, **kwargs)

class flint_ctypes.perm(val=None, context=None, random=False)

class flint_ctypes.Mat(element_domain, nrows=None, ncols=None)

Parent class for matrix domains.

There are two kinds of matrix domains:

• Mat(R), the set of matrices of any size over the domain R.

• Mat(R, n, m), the set of n x m matrices over the domain R. If R is a ring and n = m, then this is also a ring.

While Mat(R) may be more convenient, e.g. for representing linear transformations of arbitrary dimension under a single parent, fixed-shape matrix domains have advantages such as allowing automatic conversion from scalars to scalar matrices of the right size.

>>> Mat(ZZ)
Matrices (any shape) over Integer ring (fmpz)
>>> Mat(ZZ, 2)
Ring of 2 x 2 matrices over Integer ring (fmpz)
>>> Mat(ZZ, 2, 3)
Space of 2 x 3 matrices over Integer ring (fmpz)
>>> Mat(ZZ)([[1, 2, 3], [4, 5, 6]])
[[1, 2, 3],
[4, 5, 6]]
>>> Mat(ZZ, 2, 2)(5)
[[5, 0],
[0, 5]]

__init__(element_domain, nrows=None, ncols=None)

flint_ctypes.MatrixRing(element_ring, n)

class flint_ctypes.gr_mat(*args, **kwargs)

__init__(*args, **kwargs)

>>> ZZ(QQ(1))
1
>>> ZZ(QQ(1) / 3)
Traceback (most recent call last):
  ...
FlintDomainError: 1/3 is not defined in Integer ring (fmpz)

nrows()

ncols()

shape()

det(algorithm=None)

pascal(triangular=0)

stirling(kind=0)

hilbert()

hadamard()

charpoly(R=None, algorithm=None)

transpose()

is_scalar()

Return whether this matrix is a scalar matrix.

is_diagonal()

Return whether this matrix is a diagonal matrix.

is_upper_triangular()

Return whether this matrix is upper triangular.

is_lower_triangular()

Return whether this matrix is lower triangular.

hessenberg(algorithm=None)

Return this matrix reduced to upper Hessenberg form:

>>> B = Mat(QQ, 3, 3)([[4, 2, 3], [-1, 5, -3], [-4, 1, 2]]);
>>> B.hessenberg()
[[4, 14, 3],
[-1, -7, -3],
[0, 37, 14]]

Options: - algorithm: None (default), "gauss" or "householder"

is_hessenberg()

Return whether this matrix is in upper Hessenberg form.

eigenvalues(domain=None)

Computes the eigenvalues in the coefficient ring of this matrix, returning a tuple (eigenvalues, multiplicities). If the ring is not algebraically closed, the sum of multiplicities can be smaller than the dimension of the matrix. If domain is given, returns eigenvalues in that ring instead.

>>> Mat(ZZ)([[1,2],[3,4]]).eigenvalues()
([], [])
>>> Mat(ZZ)([[1,2],[3,-4]]).eigenvalues()
([2, -5], [1, 1])
>>> Mat(ZZ)([[1,2],[3,4]]).eigenvalues(domain=QQbar)
([Root a = 5.37228 of a^2-5*a-2, Root a = -0.372281 of a^2-5*a-2], [1, 1])
>>> Mat(ZZ)([[1,2],[3,4]]).eigenvalues(domain=RR)
([[-0.3722813232690143 +/- 3.01e-17], [5.372281323269014 +/- 3.31e-16]], [1, 1])

The matrix must be square:

>>> Mat(ZZ)([[1,2,3],[4,5,6]]).eigenvalues()
Traceback (most recent call last):
  ...
ValueError

diagonalization()

Matrix diagonalization: returns (D, L, R) where D is a vector of eigenvalues, LAR = diag(D) and LR = 1.

>>> A = Mat(QQ)([[1,2],[-1,4]])
>>> D, L, R = A.diagonalization()
>>> L*A*R
[[3, 0],
[0, 2]]
>>> D
[3, 2]
>>> L*R
[[1, 0],
[0, 1]]

>>> A = Mat(CC)([[1,2],[-1,4]])
>>> D, L, R = A.diagonalization()
>>> D
[([2.00000000000000 +/- 1.86e-15] + [+/- 1.86e-15]*I), ([3.00000000000000 +/- 2.90e-15] + [+/- 1.86e-15]*I)]
>>> L*A*R
[[([2.00000000000 +/- 1.10e-12] + [+/- 1.08e-12]*I), ([+/- 1.44e-12] + [+/- 1.42e-12]*I)],
[([+/- 9.76e-13] + [+/- 9.63e-13]*I), ([3.00000000000 +/- 1.27e-12] + [+/- 1.25e-12]*I)]]
>>> L*R
[[([1.00000000000 +/- 3.26e-13] + [+/- 3.20e-13]*I), ([+/- 3.72e-13] + [+/- 3.67e-13]*I)],
[([+/- 2.77e-13] + [+/- 2.73e-13]*I), ([1.00000000000 +/- 3.17e-13] + [+/- 3.13e-13]*I)]]

>>> A = Mat(CF)([[1,2],[-1,4]])
>>> D, L, R = A.diagonalization()
>>> D
[2.000000000000000, 3.000000000000000]
>>> L*A*R
[[2.000000000000000, -8.275113827716402e-16],
[0, 3.000000000000000]]
>>> L*R
[[1.000000000000000, -8.275113803054639e-17],
[0, 1.000000000000000]]

class flint_ctypes.Vec(element_domain, n=None)

Parent class for vector domains.

__init__(element_domain, n=None)

class flint_ctypes.gr_vec(*args, **kwargs)

__init__(*args, **kwargs)

>>> VecZZ(range(3, 20, 3))
[3, 6, 9, 12, 15, 18]

sum()

Sum of the elements in this vector.

>>> VecZZ(list(range(1,101))).sum()
5050
>>> VecZZ([]).sum()
0
>>> Vec(ZZmod(100))(list(range(1,101))).sum()
50

product()

Product of the elements in this vector.

>>> VecZZ(list(range(1,11))).product()
3628800
>>> VecZZ([]).product()
1
>>> Vec(ZZmod(103))(list(range(1,101))).product()
51

flint_ctypes.PolynomialRing

alias of flint_ctypes.PolynomialRing_gr_poly

flint_ctypes.PowerSeriesRing

alias of flint_ctypes.PowerSeriesRing_gr_series

flint_ctypes.PowerSeriesModRing

alias of flint_ctypes.PowerSeriesModRing_gr_series

flint_ctypes.ZZmod(n)

flint_ctypes.ModularGroup

alias of flint_ctypes.ModularGroup_psl2z

flint_ctypes.DirichletGroup

alias of flint_ctypes.DirichletGroup_dirichlet_char

flint_ctypes.SymmetricGroup

alias of flint_ctypes.SymmetricGroup_perm

flint_ctypes.timing(f, *args, **kwargs)

flint_ctypes.raises(f, exception)

flint_ctypes.test_perm()

flint_ctypes.test_psl2z()

flint_ctypes.test_matrix()

flint_ctypes.test_fq()

flint_ctypes.test_floor_ceil_trunc_nint()

flint_ctypes.test_zz()

flint_ctypes.test_qq()

flint_ctypes.test_qqbar()

flint_ctypes.test_arb()

flint_ctypes.test_vec()

flint_ctypes.test_all()

flint_ctypes.test_float()

flint_ctypes.test_special()