num-dual 0.13.7

Generalized (hyper) dual numbers for the calculation of exact (partial) derivatives
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189

use crate::{Derivative, DualNum, DualNumFloat, DualStruct};
use nalgebra::allocator::Allocator;
use nalgebra::*;
use num_traits::{Float, FloatConst, FromPrimitive, Inv, Num, One, Signed, Zero};
use std::fmt;
use std::iter::{Product, Sum};
use std::marker::PhantomData;
use std::ops::{
    Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, RemAssign, Sub, SubAssign,
};

/// A vector second order dual number for the calculation of Hessians.
#[derive(Clone, Debug)]
pub struct Dual2Vec<T: DualNum<F>, F, D: Dim>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    /// Real part of the second order dual number
    pub re: T,
    /// Gradient part of the second order dual number
    pub v1: Derivative<T, F, U1, D>,
    /// Hessian part of the second order dual number
    pub v2: Derivative<T, F, D, D>,
    f: PhantomData<F>,
}

impl<T: DualNum<F> + Copy, F: Copy, const N: usize> Copy for Dual2Vec<T, F, Const<N>> {}

#[cfg(feature = "ndarray")]
impl<T: DualNum<F>, F: DualNumFloat, D: Dim> ndarray::ScalarOperand for Dual2Vec<T, F, D> where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>
{
}

pub type Dual2SVec<T, F, const N: usize> = Dual2Vec<T, F, Const<N>>;
pub type Dual2DVec<T, F> = Dual2Vec<T, F, Dyn>;
pub type Dual2Vec32<D> = Dual2Vec<f32, f32, D>;
pub type Dual2Vec64<D> = Dual2Vec<f64, f64, D>;
pub type Dual2SVec32<const N: usize> = Dual2Vec<f32, f32, Const<N>>;
pub type Dual2SVec64<const N: usize> = Dual2Vec<f64, f64, Const<N>>;
pub type Dual2DVec32 = Dual2Vec<f32, f32, Dyn>;
pub type Dual2DVec64 = Dual2Vec<f64, f64, Dyn>;

impl<T: DualNum<F>, F, D: Dim> Dual2Vec<T, F, D>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    /// Create a new second order dual number from its fields.
    #[inline]
    pub fn new(re: T, v1: Derivative<T, F, U1, D>, v2: Derivative<T, F, D, D>) -> Self {
        Self {
            re,
            v1,
            v2,
            f: PhantomData,
        }
    }
}

impl<T: DualNum<F>, F, const N: usize> Dual2SVec<T, F, N> {
    /// Set the derivative part of variable `index` to 1.
    ///
    /// For most cases, the [`hessian`](crate::hessian) function provides a convenient
    /// interface to calculate derivatives. This function exists for the more edge cases
    /// where more control over the variables is required.
    /// ```
    /// # use num_dual::Dual2SVec64;
    /// # use nalgebra::{U1, U2, matrix};
    /// let x: Dual2SVec64<2> = Dual2SVec64::from_re(5.0).derivative(0);
    /// let y: Dual2SVec64<2> = Dual2SVec64::from_re(3.0).derivative(1);
    /// let z = x * x * y;
    /// assert_eq!(z.re, 75.0);                                                 // x²y
    /// assert_eq!(z.v1.unwrap_generic(U1, U2), matrix![30.0, 25.0]);           // [2xy, x²]
    /// assert_eq!(z.v2.unwrap_generic(U2, U2), matrix![6.0, 10.0; 10.0, 0.0]); // [2y, 2x; 2x, 0]
    /// ```
    #[inline]
    pub fn derivative(mut self, index: usize) -> Self {
        self.v1 = Derivative::derivative_generic(U1, Const::<N>, index);
        self
    }
}

impl<T: DualNum<F>, F> Dual2DVec<T, F> {
    /// Set the derivative part of variable `index` to 1.
    ///
    /// For most cases, the [`hessian`](crate::hessian) function provides a convenient interface
    /// to calculate derivatives. This function exists for the more edge cases
    /// where more control over the variables is required.
    /// ```
    /// # use num_dual::Dual2DVec64;
    /// # use nalgebra::{Dyn, U1, dmatrix};
    /// let x: Dual2DVec64 = Dual2DVec64::from_re(5.0).derivative(2, 0);
    /// let y: Dual2DVec64 = Dual2DVec64::from_re(3.0).derivative(2, 1);
    /// let z = &x * &x * y;
    /// assert_eq!(z.re, 75.0);                                                          // x²y
    /// assert_eq!(z.v1.unwrap_generic(U1, Dyn(2)), dmatrix![30.0, 25.0]);               // [2xy, x²]
    /// assert_eq!(z.v2.unwrap_generic(Dyn(2), Dyn(2)), dmatrix![6.0, 10.0; 10.0, 0.0]); // [2y, 2x; 2x, 0]
    /// ```
    #[inline]
    pub fn derivative(mut self, variables: usize, index: usize) -> Self {
        self.v1 = Derivative::derivative_generic(U1, Dyn(variables), index);
        self
    }
}

impl<T: DualNum<F>, F, D: Dim> Dual2Vec<T, F, D>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    /// Create a new second order dual number from the real part.
    #[inline]
    pub fn from_re(re: T) -> Self {
        Self::new(re, Derivative::none(), Derivative::none())
    }
}

/* chain rule */
impl<T: DualNum<F>, F: Float, D: Dim> Dual2Vec<T, F, D>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    #[inline]
    fn chain_rule(&self, f0: T, f1: T, f2: T) -> Self {
        Self::new(
            f0,
            &self.v1 * f1.clone(),
            &self.v2 * f1 + self.v1.tr_mul(&self.v1) * f2,
        )
    }
}

/* product rule */
impl<T: DualNum<F>, F: Float, D: Dim> Mul<&Dual2Vec<T, F, D>> for &Dual2Vec<T, F, D>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    type Output = Dual2Vec<T, F, D>;
    #[inline]
    fn mul(self, other: &Dual2Vec<T, F, D>) -> Dual2Vec<T, F, D> {
        Dual2Vec::new(
            self.re.clone() * other.re.clone(),
            &other.v1 * self.re.clone() + &self.v1 * other.re.clone(),
            &other.v2 * self.re.clone()
                + self.v1.tr_mul(&other.v1)
                + other.v1.tr_mul(&self.v1)
                + &self.v2 * other.re.clone(),
        )
    }
}

/* quotient rule */
impl<T: DualNum<F>, F: Float, D: Dim> Div<&Dual2Vec<T, F, D>> for &Dual2Vec<T, F, D>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    type Output = Dual2Vec<T, F, D>;
    #[inline]
    fn div(self, other: &Dual2Vec<T, F, D>) -> Dual2Vec<T, F, D> {
        let inv = other.re.recip();
        let inv2 = inv.clone() * inv.clone();
        Dual2Vec::new(
            self.re.clone() * inv.clone(),
            (&self.v1 * other.re.clone() - &other.v1 * self.re.clone()) * inv2.clone(),
            &self.v2 * inv.clone()
                - (&other.v2 * self.re.clone()
                    + self.v1.tr_mul(&other.v1)
                    + other.v1.tr_mul(&self.v1))
                    * inv2.clone()
                + other.v1.tr_mul(&other.v1)
                    * ((T::one() + T::one()) * self.re.clone() * inv2 * inv),
        )
    }
}

/* string conversions */
impl<T: DualNum<F>, F: fmt::Display, D: Dim> fmt::Display for Dual2Vec<T, F, D>
where
    DefaultAllocator: Allocator<U1, D> + Allocator<D, D>,
{
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "{}", self.re)?;
        self.v1.fmt(f, "ε1")?;
        self.v2.fmt(f, "ε1²")
    }
}

impl_second_derivatives!(Dual2Vec, [v1, v2], [D], [U1, D], [D, D]);
impl_dual!(Dual2Vec, [v1, v2], [D], [U1, D], [D, D]);
impl_nalgebra!(Dual2Vec, [v1, v2], [D], [U1, D], [D, D]);