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//! Generalized, recursive, scalar and vector (hyper) dual numbers for the automatic and exact calculation of (partial) derivatives.
//!
//! ## Example
//! This example defines a generic scalar and a generic vector function that can be called using any (hyper-) dual number and automatically calculates derivatives.
//! ```
//! use num_dual::*;
//! use nalgebra::SVector;
//!
//! fn foo<D: DualNum<f64>>(x: D) -> D {
//! x.powi(3)
//! }
//!
//! fn bar<D: DualNum<f64>, const N: usize>(x: SVector<D, N>) -> D {
//! x.dot(&x).sqrt()
//! }
//!
//! fn main() {
//! // Calculate a simple derivative
//! let (f, df) = first_derivative(foo, 5.0);
//! assert_eq!(f, 125.0);
//! assert_eq!(df, 75.0);
//!
//! // Manually construct the dual number
//! let x = Dual64::new(5.0, 1.0);
//! println!("{}", foo(x)); // 125 + [75]ε
//!
//! // Calculate a gradient
//! let (f, g) = gradient(bar, SVector::from([4.0, 3.0]));
//! assert_eq!(f, 5.0);
//! assert_eq!(g[0], 0.8);
//!
//! // Calculate a Hessian
//! let (f, g, h) = hessian(bar, SVector::from([4.0, 3.0]));
//! println!("{h}"); // [[0.072, -0.096], [-0.096, 0.128]]
//!
//! // for x=cos(t) calculate the third derivative of foo w.r.t. t
//! let (f0, f1, f2, f3) = third_derivative(|t| foo(t.cos()), 1.0);
//! println!("{f3}"); // 1.5836632930100278
//! }
//! ```
#![warn(clippy::all)]
#![allow(clippy::needless_range_loop)]
use num_traits::{Float, FloatConst, FromPrimitive, Inv, NumAssignOps, NumOps, Signed};
use std::fmt;
use std::iter::{Product, Sum};
#[macro_use]
mod macros;
#[macro_use]
mod derivatives;
mod bessel;
mod derivative;
mod dual;
mod dual2;
mod dual2_vec;
mod dual3;
mod dual_vec;
mod hyperdual;
mod hyperdual_vec;
mod hyperhyperdual;
pub use bessel::BesselDual;
pub use derivative::Derivative;
pub use dual::{first_derivative, try_first_derivative, Dual, Dual32, Dual64};
pub use dual2::{second_derivative, try_second_derivative, Dual2, Dual2_32, Dual2_64};
pub use dual2_vec::{
hessian, try_hessian, Dual2DVec32, Dual2DVec64, Dual2SVec32, Dual2SVec64, Dual2Vec, Dual2Vec32,
Dual2Vec64,
};
pub use dual3::{third_derivative, try_third_derivative, Dual3, Dual3_32, Dual3_64};
pub use dual_vec::{
gradient, jacobian, try_gradient, try_jacobian, DualDVec32, DualDVec64, DualSVec32, DualSVec64,
DualVec, DualVec32, DualVec64,
};
pub use hyperdual::{
second_partial_derivative, try_second_partial_derivative, HyperDual, HyperDual32, HyperDual64,
};
pub use hyperdual_vec::{
partial_hessian, try_partial_hessian, HyperDualDVec32, HyperDualDVec64, HyperDualSVec32,
HyperDualSVec64, HyperDualVec, HyperDualVec32, HyperDualVec64,
};
pub use hyperhyperdual::{
third_partial_derivative, third_partial_derivative_vec, try_third_partial_derivative,
try_third_partial_derivative_vec, HyperHyperDual, HyperHyperDual32, HyperHyperDual64,
};
#[cfg(feature = "linalg")]
pub mod linalg;
#[cfg(feature = "python")]
pub mod python;
#[cfg(feature = "python_macro")]
mod python_macro;
/// A generalized (hyper) dual number.
pub trait DualNum<F>:
NumOps
+ for<'r> NumOps<&'r Self>
+ Signed
+ NumOps<F>
+ NumAssignOps
+ NumAssignOps<F>
+ Clone
+ Inv<Output = Self>
+ Sum
+ Product
+ FromPrimitive
+ From<F>
+ fmt::Display
+ PartialEq
+ fmt::Debug
+ 'static
{
/// Highest derivative that can be calculated with this struct
const NDERIV: usize;
/// Real part (0th derivative) of the number
fn re(&self) -> F;
/// Reciprocal (inverse) of a number `1/x`.
fn recip(&self) -> Self;
/// Power with integer exponent `x^n`
fn powi(&self, n: i32) -> Self;
/// Power with real exponent `x^n`
fn powf(&self, n: F) -> Self;
/// Square root
fn sqrt(&self) -> Self;
/// Cubic root
fn cbrt(&self) -> Self;
/// Exponential `e^x`
fn exp(&self) -> Self;
/// Exponential with base 2 `2^x`
fn exp2(&self) -> Self;
/// Exponential minus 1 `e^x-1`
fn exp_m1(&self) -> Self;
/// Natural logarithm
fn ln(&self) -> Self;
/// Logarithm with arbitrary base
fn log(&self, base: F) -> Self;
/// Logarithm with base 2
fn log2(&self) -> Self;
/// Logarithm with base 10
fn log10(&self) -> Self;
/// Logarithm on x plus one `ln(1+x)`
fn ln_1p(&self) -> Self;
/// Sine
fn sin(&self) -> Self;
/// Cosine
fn cos(&self) -> Self;
/// Tangent
fn tan(&self) -> Self;
/// Calculate sine and cosine simultaneously
fn sin_cos(&self) -> (Self, Self);
/// Arcsine
fn asin(&self) -> Self;
/// Arccosine
fn acos(&self) -> Self;
/// Arctangent
fn atan(&self) -> Self;
/// Hyperbolic sine
fn sinh(&self) -> Self;
/// Hyperbolic cosine
fn cosh(&self) -> Self;
/// Hyperbolic tangent
fn tanh(&self) -> Self;
/// Area hyperbolic sine
fn asinh(&self) -> Self;
/// Area hyperbolic cosine
fn acosh(&self) -> Self;
/// Area hyperbolic tangent
fn atanh(&self) -> Self;
/// 0th order spherical Bessel function of the first kind
fn sph_j0(&self) -> Self;
/// 1st order spherical Bessel function of the first kind
fn sph_j1(&self) -> Self;
/// 2nd order spherical Bessel function of the first kind
fn sph_j2(&self) -> Self;
/// Fused multiply-add
#[inline]
fn mul_add(&self, a: Self, b: Self) -> Self {
self.clone() * a + b
}
/// Power with dual exponent `x^n`
#[inline]
fn powd(&self, exp: Self) -> Self {
(self.ln() * exp).exp()
}
}
/// The underlying data type of individual derivatives. Usually f32 or f64.
pub trait DualNumFloat:
Float + FloatConst + FromPrimitive + Signed + fmt::Display + fmt::Debug + Sync + Send + 'static
{
}
impl<T> DualNumFloat for T where
T: Float
+ FloatConst
+ FromPrimitive
+ Signed
+ fmt::Display
+ fmt::Debug
+ Sync
+ Send
+ 'static
{
}
macro_rules! impl_dual_num_float {
($float:ty) => {
impl DualNum<$float> for $float {
const NDERIV: usize = 0;
fn re(&self) -> $float {
*self
}
fn mul_add(&self, a: Self, b: Self) -> Self {
<$float>::mul_add(*self, a, b)
}
fn recip(&self) -> Self {
<$float>::recip(*self)
}
fn powi(&self, n: i32) -> Self {
<$float>::powi(*self, n)
}
fn powf(&self, n: Self) -> Self {
<$float>::powf(*self, n)
}
fn powd(&self, n: Self) -> Self {
<$float>::powf(*self, n)
}
fn sqrt(&self) -> Self {
<$float>::sqrt(*self)
}
fn exp(&self) -> Self {
<$float>::exp(*self)
}
fn exp2(&self) -> Self {
<$float>::exp2(*self)
}
fn ln(&self) -> Self {
<$float>::ln(*self)
}
fn log(&self, base: Self) -> Self {
<$float>::log(*self, base)
}
fn log2(&self) -> Self {
<$float>::log2(*self)
}
fn log10(&self) -> Self {
<$float>::log10(*self)
}
fn cbrt(&self) -> Self {
<$float>::cbrt(*self)
}
fn sin(&self) -> Self {
<$float>::sin(*self)
}
fn cos(&self) -> Self {
<$float>::cos(*self)
}
fn tan(&self) -> Self {
<$float>::tan(*self)
}
fn asin(&self) -> Self {
<$float>::asin(*self)
}
fn acos(&self) -> Self {
<$float>::acos(*self)
}
fn atan(&self) -> Self {
<$float>::atan(*self)
}
fn sin_cos(&self) -> (Self, Self) {
<$float>::sin_cos(*self)
}
fn exp_m1(&self) -> Self {
<$float>::exp_m1(*self)
}
fn ln_1p(&self) -> Self {
<$float>::ln_1p(*self)
}
fn sinh(&self) -> Self {
<$float>::sinh(*self)
}
fn cosh(&self) -> Self {
<$float>::cosh(*self)
}
fn tanh(&self) -> Self {
<$float>::tanh(*self)
}
fn asinh(&self) -> Self {
<$float>::asinh(*self)
}
fn acosh(&self) -> Self {
<$float>::acosh(*self)
}
fn atanh(&self) -> Self {
<$float>::atanh(*self)
}
fn sph_j0(&self) -> Self {
if self.abs() < <$float>::EPSILON {
1.0 - self * self / 6.0
} else {
self.sin() / self
}
}
fn sph_j1(&self) -> Self {
if self.abs() < <$float>::EPSILON {
self / 3.0
} else {
let sc = self.sin_cos();
let rec = self.recip();
(sc.0 * rec - sc.1) * rec
}
}
fn sph_j2(&self) -> Self {
if self.abs() < <$float>::EPSILON {
self * self / 15.0
} else {
let sc = self.sin_cos();
let s2 = self * self;
((3.0 - s2) * sc.0 - 3.0 * self * sc.1) / (self * s2)
}
}
}
};
}
impl_dual_num_float!(f32);
impl_dual_num_float!(f64);