# abax
A lightweight Rust library providing high-precision mathematical constants and special functions.
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## Features
- **Mathematical Constants**:
- Bernoulli numbers (<math><msub><mi>B</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>) up to <math><mi>n</mi><mo>=</mo><mn>16</mn></math>.
- Riemann Zeta function values for integers <math><mn>2</mn></math> through <math><mn>16</mn></math>.
- Euler-Mascheroni constant.
- Stirling series coefficients for asymptotic expansions.
- **Special Functions**:
- **Gamma and Polygamma Functions**:
- `gamma(x)`: Gamma function <math><mi>Γ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> computed with the Lanczos approximation and reflection formula.
- `gammaln(x)`: Natural logarithm of the Gamma function <math><mi>ln</mi><mo>(</mo><mi>Γ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> using Stirling's approximation.
- `digamma(x)`, `trigamma(x)`, `tetragamma(x)`: Polygamma functions <math><msup><mi>ψ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math> implemented with reflection and recurrence shifting.
- `psi(k, x)`: Polygamma interface for arbitrary order $k$ with stable handling of poles and infinities.
- `gammainc(x, a, lower, scaled)`: Regularized incomplete gamma function with scaling for improved conditioning.
- `gammaincinv(y, a, lower)`: Robust inverse incomplete gamma function using a combined Halley-Newton-Bisection method.
- **Beta Functions**:
- `beta(z, w)`, `betaln(z, w)`: Beta function and its natural logarithm.
- `betainc(x, z, w, lower)`: Regularized incomplete beta function <math><mrow><msub><mi>I</mi><mi>x</mi></msub><mo>(</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></math>.
- `betaincinv(y, z, w, lower)`: Inverse of the regularized incomplete beta function.
- **Error Functions**:
- `erf(x)`, `erfc(x)`, `erfcx(x)`: Error function, complementary error function, and scaled variant.
- `erfinv(x)`, `erfcinv(x)`: Inverse error functions implemented with piecewise rational approximations.
- **Bessel Functions**:
- `besseli(nu, x, scale)`: Modified Bessel function of the first kind <math><msub><mi>I</mi><mi>ν</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> with optional exponential scaling to prevent overflow.
- **Probability Distributions**:
| Family | PDF | CDF | Inverse (Quantile) |
|:---|:---|:---|:---|
| **Normal** | `normpdf(x, μ, σ)` | `normcdf(x, μ, σ, upper)` | `norminv(p, μ, σ)` |
| **Lognormal** | `lognpdf(x, μ, σ)` | `logncdf(x, μ, σ, upper)` | `logninv(p, μ, σ)` |
| **Student's T** | `tpdf(x, v)` | `tcdf(x, v, upper)` | `tinv(p, v)` |
| **Noncentral T**| `nctpdf(x, ν, δ)`| `nctcdf(x, ν, δ, upper)` | `nctinv(p, ν, δ)` |
| **F** | `fpdf(x, v1, v2)` | `fcdf(x, v1, v2, upper)` | `finv(p, v1, v2)` |
| **Gamma** | `gampdf(x, a, b)` | `gamcdf(x, a, b, upper)` | `gaminv(p, a, b)` |
| **Beta** | `betapdf(x, a, b)` | `betacdf(x, a, b, upper)` | `betainv(p, a, b)` |
| **Exponential** | `expdf(x, λ)` | `expcdf(x, λ, upper)` | `expinv(p, λ)` |
| **Chi-squared** | `chi2pdf(x, v)` | `chi2cdf(x, v, upper)` | `chi2inv(p, v)` |
| **Weibull** | `wblpdf(x, a, b)` | `wblcdf(x, a, b, upper)` | `wblinv(p, a, b)` |
| **Extreme Value** | `evpdf(x, μ, σ)` | `evcdf(x, μ, σ, upper)` | `evinv(p, μ, σ)` |
| **Generalized Extreme Value** | `gevpdf(x, k, σ, μ)` | `gevcdf(x, k, sigma, mu, upper)` | `gevinv(p, k, sigma, mu)` |
| **Poisson** | `poisspdf(x, λ)` | `poisscdf(x, λ, upper)` | `poissinv(p, λ)` |
| **Binomial** | `binopdf(x, n, p)` | `binocdf(x, n, p, upper)` | `binoinv(y, n, p)` |
## Usage
Add this to your `Cargo.toml`:
```toml
[dependencies]
abax = "*"
```
## License
This project is licensed under the MIT License - see the LICENSE file for details.
## Repository
Source code is available on GitHub.