Special functions

Nautilus.Special provides the mathematical special functions that underlie probability distributions, physics simulations, and numerical analysis. It replaces scipy.special for Chelis programs.

All functions are pure (no effects), operate on f32 scalars, and are safe to differentiate via grad.

Imports

import Nautilus.Special (erf, erfc, erfinv, gamma, log_gamma, digamma, trigamma, beta, lbeta)
import Nautilus.Special (bessel_j0, bessel_j1, bessel_y0, bessel_y1)
import Nautilus.Special (bessel_i0, bessel_i1, bessel_k0, bessel_k1)
import Nautilus.Special (airy_ai, airy_bi, ellipk, ellipe)

Function families

Error functions

• erf(x): the error function, ~1e-7 relative precision
• erfc(x): complementary error function 1 - erf(x), same precision domain as erf
• erfinv(x): inverse error function for x in (-1, 1), ~1e-8

Gamma-related

• gamma(x): the gamma function via Lanczos + reflection, +inf at non-positive integers
• log_gamma(x): log of the gamma function (Lanczos, g=7), ~1e-9
• digamma(x): psi function (derivative of log_gamma), ~1e-7
• trigamma(x): derivative of digamma, ~1e-6
• beta(a, b): the beta function B(a, b) = Gamma(a)Gamma(b)/Gamma(a+b)
• lbeta(a, b): log of the beta function

Bessel functions

• bessel_j0(x), bessel_j1(x): Bessel functions of the first kind (all reals, J0 even, J1 odd)
• bessel_y0(x), bessel_y1(x): Bessel functions of the second kind (x > 0, -inf at 0)
• bessel_i0(x), bessel_i1(x): modified Bessel, first kind (all reals)
• bessel_k0(x), bessel_k1(x): modified Bessel, second kind (x > 0, +inf at 0)

Airy functions

• airy_ai(x): Airy Ai (power series for |x| <= 5, asymptotic for x > 5)
• airy_bi(x): Airy Bi

Elliptic integrals

• ellipk(m): complete elliptic integral of the first kind, m in [0, 1)
• ellipe(m): complete elliptic integral of the second kind, m in [0, 1]

Both use the arithmetic-geometric mean (AGM) recurrence, which converges quadratically in about 10 iterations.