Skip to content

Loading…

Docs API Reference Changelog

Tensorlucid.Tensor87

Tensor Creation30

Tensor Operations158

Autogradlucid.autograd20

Compilelucid.compile11

Distributionslucid.distributions57

Einopslucid.einops4

Functional Transformslucid.func9

Linear Algebralucid.linalg39

Mixed Precisionlucid.amp2

Model Zoolucid.models46

Neural Networkslucid.nn170

Optimizerslucid.optim30

Profilerlucid.profiler6

Quantizationlucid.quantization31

Serializationlucid.serialization6

Signallucid.signal12

Special Functionslucid.special38

erfcxf i0ef i1f i1ef ndtrf log_ndtrf ndtrif xlog1pyf entrf multigammalnf polygammaf spherical_bessel_j0f chebyshev_polynomial_tf chebyshev_polynomial_uf chebyshev_polynomial_vf chebyshev_polynomial_wf shifted_chebyshev_polynomial_tf shifted_chebyshev_polynomial_uf shifted_chebyshev_polynomial_vf shifted_chebyshev_polynomial_wf hermite_polynomial_hf hermite_polynomial_hef legendre_polynomial_pf laguerre_polynomial_lf bessel_j0f bessel_j1f bessel_y0f bessel_y1f modified_bessel_k0f scaled_modified_bessel_k0f modified_bessel_k1f scaled_modified_bessel_k1f zetaf gammalnf psif expitf modified_bessel_i0f modified_bessel_i1f

Utilslucid.utils1

Weightslucid.weights9

module

special

38 members

lucid.special

`lucid.special` — special mathematical functions.

This sub-package adds functions that complement the elementary math already available at the top level (erf / sinc / lgamma / digamma / i0 / log1p / logit / xlogy / erfc / erfinv / expm1 …). Per H8 there is exactly one canonical path per op — those stay accessible only at the top level, and the extra entries below stay accessible only via lucid.special.<name>.

Implemented (all pure-Python composites over engine ops — no C++ work):

erfcx — scaled complementary error: exp(x²) · erfc(x)
i0e — exponentially-scaled modified Bessel I₀: exp(-|x|) · i0(x)
i1 — modified Bessel of order 1 (Abramowitz polynomial)
i1e — exponentially-scaled i1
ndtr — normal CDF Φ(x) = ½·(1 + erf(x/√2))
ndtri — inverse normal CDF (Beasley-Springer-Moro)
log_ndtr — numerically-stable log(Φ(x))
xlog1py — x · log1p(y) with 0·log1p(0) = 0
entr — entropy element -x · log(x), 0 at x=0
multigammaln — log of the multivariate gamma function
polygamma — derivatives of digamma; n=0 falls through to lucid.digamma, n=1 (trigamma) uses the standard series; n ≥ 2 raises NotImplementedError
spherical_bessel_j0 — sin(x)/x with continuous extension at 0

Deferred (would need oscillatory / Hurwitz-zeta approximations or arbitrary-order Bessel machinery — enable in a follow-up if anyone actually asks): iv, ive, bessel_j0/j1/y0/y1, modified_bessel_k0/k1 (and scaled variants), zeta, the orthogonal-polynomial family (Chebyshev / Legendre / Hermite / Laguerre / Jacobi).

Functions

fnerfcx→ Tensor

Scaled complementary error function.

fni0e→ Tensor

Exponentially scaled modified Bessel function of order 0.

fni1→ Tensor

Modified Bessel function of the first kind, order 1.

fni1e→ Tensor

Exponentially scaled modified Bessel function of order 1.

fnndtr→ Tensor

Standard normal cumulative distribution function.

fnlog_ndtr→ Tensor

Numerically stable logarithm of the standard normal CDF.

fnndtri→ Tensor

Inverse standard normal CDF (probit / quantile function).

fnxlog1py→ Tensor

Safe product $x \log(1 + y)$ with limit-convention zero handling.

fnentr→ Tensor

Element-wise entropy kernel $-x \log x$ .

fnmultigammaln→ Tensor

Logarithm of the multivariate gamma function $\Gamma_p(a)$ .

fnpolygamma→ Tensor

Polygamma function $\psi^{(n)}(x)$ .

fnspherical_bessel_j0→ Tensor

Spherical Bessel function of the first kind, order 0.

fnchebyshev_polynomial_t→ Tensor

Chebyshev polynomial of the first kind, $T_n(x)$ .

fnchebyshev_polynomial_u→ Tensor

Chebyshev polynomial of the second kind, $U_n(x)$ .

fnchebyshev_polynomial_v→ Tensor

Chebyshev polynomial of the third kind, $V_n(x)$ .

fnchebyshev_polynomial_w→ Tensor

Chebyshev polynomial of the fourth kind, $W_n(x)$ .

fnshifted_chebyshev_polynomial_t→ Tensor

Shifted Chebyshev T polynomial $T^*_n(x)$ on $[0, 1]$ .

fnshifted_chebyshev_polynomial_u→ Tensor

Shifted Chebyshev U polynomial $U^*_n(x)$ on $[0, 1]$ .

fnshifted_chebyshev_polynomial_v→ Tensor

Shifted Chebyshev V polynomial $V^*_n(x)$ on $[0, 1]$ .

fnshifted_chebyshev_polynomial_w→ Tensor

Shifted Chebyshev W polynomial $W^*_n(x)$ on $[0, 1]$ .

fnhermite_polynomial_h→ Tensor

Physicist's Hermite polynomial $H_n(x)$ .

fnhermite_polynomial_he→ Tensor

Probabilist's Hermite polynomial $\mathit{He}_n(x)$ .

fnlegendre_polynomial_p→ Tensor

Legendre polynomial $P_n(x)$ .

fnlaguerre_polynomial_l→ Tensor

Laguerre polynomial $L_n(x)$ .

fnbessel_j0→ Tensor

Bessel function of the first kind, order 0.

fnbessel_j1→ Tensor

Bessel function of the first kind, order 1.

fnbessel_y0→ Tensor

Bessel function of the second kind, order 0.

fnbessel_y1→ Tensor

Bessel function of the second kind, order 1.

fnmodified_bessel_k0→ Tensor

Modified Bessel function of the second kind, order 0.

fnscaled_modified_bessel_k0→ Tensor

Exponentially scaled modified Bessel $K_0$ : $e^x K_0(x)$ .

fnmodified_bessel_k1→ Tensor

Modified Bessel function of the second kind, order 1.

fnscaled_modified_bessel_k1→ Tensor

Exponentially scaled modified Bessel $K_1$ : $e^x K_1(x)$ .

fnzeta→ Tensor

Hurwitz zeta function $\zeta(x, q)$ .

fngammaln→ Tensor

Natural logarithm of the absolute value of the gamma function.

fnpsi→ Tensor

Digamma function $\psi(x) = \frac{d}{dx} \log \Gamma(x)$ .

fnexpit→ Tensor

Logistic sigmoid (expit) function $\sigma(x)$ .

fnmodified_bessel_i0→ Tensor

Modified Bessel function of the first kind, order 0.

fnmodified_bessel_i1→ Tensor

Modified Bessel function of the first kind, order 1.

Production-grade ML framework for Apple Silicon. MLX + Accelerate native backend.

Documentation

Quickstart
Installation
Autograd
Metal Device

API Reference

lucid.Tensor
lucid.nn
lucid.optim
lucid.autograd

Resources

GitHub(opens in new tab)
Changelog
BibTeX(opens in new tab)

© 2026 Lucid. Built for Apple Silicon.

Python 3.14+ · macOS arm64

lucid 3.6.0built from 4b61531 (main)on 2026-07-06