multicalc 0.7.0

Rust scientific computing for single and multi-variable calculus
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
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211

use multicalc::numerical_derivative::autodiff::{AutoDiffMulti, AutoDiffSingle};
use multicalc::numerical_derivative::derivator::*;
use multicalc::numerical_derivative::finite_difference::*;
use multicalc::numerical_derivative::hessian::Hessian;
use multicalc::numerical_derivative::jacobian::Jacobian;
use multicalc::numerical_derivative::mode::*;
use multicalc::scalar::{Numeric, VectorFn, c};
use multicalc::utils::error_codes::*;
use multicalc::{scalar_fn, scalar_fn_vec};

// ----- autodiff (the default backend): exact derivatives -----

#[test]
fn ad_single_derivative() {
    // f(x) = x^3 -> f' = 3x^2, f'' = 6x, f''' = 6
    let func = scalar_fn!(|x| x * x * x);
    let d = AutoDiffSingle::default();

    assert!(f64::abs(d.get(1, &func, 2.0).unwrap() - 12.0) < 1e-12);
    assert!(f64::abs(d.get(2, &func, 2.0).unwrap() - 12.0) < 1e-12);
    assert!(f64::abs(d.get(3, &func, 2.0).unwrap() - 6.0) < 1e-12);
}

#[test]
fn ad_first_partials() {
    // f(x, y) = 3x^2 + 2xy -> df/dx = 6x + 2y, df/dy = 2x
    let func = scalar_fn!(|v: &[f64; 2]| c(3.0) * v[0] * v[0] + c(2.0) * v[0] * v[1]);
    let d = AutoDiffMulti::default();
    let point = [1.0, 3.0];

    assert!(f64::abs(d.get_single_partial(&func, 0, &point).unwrap() - 12.0) < 1e-12);
    assert!(f64::abs(d.get_single_partial(&func, 1, &point).unwrap() - 2.0) < 1e-12);
}

#[test]
fn ad_first_partials_transcendental() {
    // f(x, y, z) = y*sin(x) + x*cos(y) + x*y*e^z
    let func = scalar_fn!(|v: &[f64; 3]| {
        v[1] * v[0].sin() + v[0] * v[1].cos() + v[0] * v[1] * v[2].exp()
    });
    let d = AutoDiffMulti::default();
    let point = [1.0, 2.0, 3.0];

    // df/dx = y*cos(x) + cos(y) + y*e^z
    let dx = 2.0 * f64::cos(1.0) + f64::cos(2.0) + 2.0 * f64::exp(3.0);
    assert!(f64::abs(d.get_single_partial(&func, 0, &point).unwrap() - dx) < 1e-12);

    // df/dz = x*y*e^z
    let dz = 1.0 * 2.0 * f64::exp(3.0);
    assert!(f64::abs(d.get_single_partial(&func, 2, &point).unwrap() - dz) < 1e-12);
}

#[test]
fn ad_second_partials() {
    // f(x, y, z) = y*sin(x) + x*cos(y) + x*y*e^z
    let func = scalar_fn!(|v: &[f64; 3]| {
        v[1] * v[0].sin() + v[0] * v[1].cos() + v[0] * v[1] * v[2].exp()
    });
    let d = AutoDiffMulti::default();
    let point = [1.0, 2.0, 3.0];

    // d2f/dx2 = -y*sin(x)
    let dxx = -2.0 * f64::sin(1.0);
    assert!(f64::abs(d.get_double_partial(&func, &[0, 0], &point).unwrap() - dxx) < 1e-12);

    // mixed d2f/dx dy = cos(x) - sin(y) + e^z
    let dxy = f64::cos(1.0) - f64::sin(2.0) + f64::exp(3.0);
    assert!(f64::abs(d.get_double_partial(&func, &[0, 1], &point).unwrap() - dxy) < 1e-12);
}

#[test]
fn ad_third_partials() {
    // f = x^3 y^3 z^3:  d3/dx dy dz = 27 x^2 y^2 z^2 = 972;  d3/dx2 dy = 18 x y^2 z^3 = 1944
    let func = scalar_fn!(|v: &[f64; 3]| v[0].powi(3) * v[1].powi(3) * v[2].powi(3));
    let d = AutoDiffMulti::default();
    let point = [1.0, 2.0, 3.0];

    assert!(f64::abs(d.get(&func, &[0, 1, 2], &point).unwrap() - 972.0) < 1e-9);
    assert!(f64::abs(d.get(&func, &[0, 0, 1], &point).unwrap() - 1944.0) < 1e-9);
}

#[test]
fn ad_jacobian() {
    // (x*y*z, x^2 + y^2)
    let f = scalar_fn_vec!(|v: &[f64; 3]| [v[0] * v[1] * v[2], v[0] * v[0] + v[1] * v[1]]);
    let jacobian: Jacobian = Jacobian::default();
    let result = jacobian.get(&f, &[1.0, 2.0, 3.0]).unwrap();

    let expected = [[6.0, 3.0, 2.0], [2.0, 4.0, 0.0]];
    for i in 0..2 {
        for j in 0..3 {
            assert!(f64::abs(result[i][j] - expected[i][j]) < 1e-12);
        }
    }
}

#[test]
#[cfg(feature = "alloc")]
fn ad_jacobian_on_heap() {
    let f = scalar_fn_vec!(|v: &[f64; 3]| [v[0] * v[1] * v[2], v[0] * v[0] + v[1] * v[1]]);
    let jacobian: Jacobian = Jacobian::default();
    let result = jacobian.get_on_heap(&f, &[1.0, 2.0, 3.0]).unwrap();

    let expected = [[6.0, 3.0, 2.0], [2.0, 4.0, 0.0]];
    for i in 0..2 {
        for j in 0..3 {
            assert!(f64::abs(result[i][j] - expected[i][j]) < 1e-12);
        }
    }
}

#[test]
fn ad_hessian() {
    // f(x, y) = y*sin(x) + 2*x*e^y
    let func = scalar_fn!(|v: &[f64; 2]| v[1] * v[0].sin() + c(2.0) * v[0] * v[1].exp());
    let hessian: Hessian = Hessian::default();
    let result = hessian.get(&func, &[1.0, 2.0]).unwrap();

    let expected = [
        [-2.0 * f64::sin(1.0), f64::cos(1.0) + 2.0 * f64::exp(2.0)],
        [f64::cos(1.0) + 2.0 * f64::exp(2.0), 2.0 * f64::exp(2.0)],
    ];
    for i in 0..2 {
        for j in 0..2 {
            assert!(f64::abs(result[i][j] - expected[i][j]) < 1e-12);
        }
    }
}

#[test]
fn ad_f32() {
    // x*x at 0.5; first derivative 2x = 1.0, exact under autodiff
    let func = scalar_fn!(|x| x * x);
    let d = AutoDiffSingle::<f32>::default();
    assert!(f32::abs(d.get(1, &func, 0.5_f32).unwrap() - 1.0) < 1e-6);
}

// ----- autodiff error handling -----

#[test]
fn ad_error_index_out_of_range() {
    let func = scalar_fn!(|v: &[f64; 3]| v[0] + v[1] + v[2]);
    let d = AutoDiffMulti::default();
    let result = d.get_single_partial(&func, 5, &[1.0, 2.0, 3.0]);
    assert_eq!(result.unwrap_err(), CalcError::IndexOutOfRange);
}

#[test]
fn ad_error_order_zero() {
    let func = scalar_fn!(|v: &[f64; 3]| v[0] + v[1] + v[2]);
    let d = AutoDiffMulti::default();
    let idx: [usize; 0] = [];
    assert_eq!(
        d.get(&func, &idx, &[1.0, 2.0, 3.0]).unwrap_err(),
        CalcError::DerivativeOrderZero
    );
}

#[test]
fn ad_error_order_unsupported() {
    // autodiff multi caps at third order; a fourth-order partial is rejected
    let func = scalar_fn!(|v: &[f64; 3]| v[0] + v[1] + v[2]);
    let d = AutoDiffMulti::default();
    assert_eq!(
        d.get(&func, &[0, 1, 2, 0], &[1.0, 2.0, 3.0]).unwrap_err(),
        CalcError::DerivativeOrderUnsupported
    );
}

#[test]
fn ad_jacobian_empty_error() {
    // a function with no outputs is an empty function set
    struct EmptyVectorFn;
    impl VectorFn<3, 0> for EmptyVectorFn {
        fn eval<S: Numeric>(&self, _point: &[S; 3]) -> [S; 0] {
            []
        }
    }

    let jacobian: Jacobian = Jacobian::default();
    let result = jacobian.get(&EmptyVectorFn, &[1.0, 2.0, 3.0]);
    assert_eq!(result.unwrap_err(), CalcError::EmptyFunctionSet);
}

// ----- finite differences: kept as a sparse fallback for the engine and the cases autodiff
//       does not cover (high-order mixed partials, zero-step-size error) -----

#[test]
fn fd_single_derivative_modes() {
    // x^2/2, derivative x; check all three finite-difference modes still work
    let func = scalar_fn!(|x| c(0.5) * x * x);
    for mode in [
        FiniteDifferenceMode::Forward,
        FiniteDifferenceMode::Backward,
        FiniteDifferenceMode::Central,
    ] {
        let mut d = FiniteDifferenceSingle::default();
        d.config.method = mode;
        assert!(f64::abs(d.get(1, &func, 2.0).unwrap() - 2.0) < 0.001);
    }
}

#[test]
fn fd_step_size_zero_error() {
    let func = scalar_fn!(|v: &[f64; 3]| v[1] * v[0].sin());
    let d = FiniteDifferenceMulti::from_parameters(0.0, FiniteDifferenceMode::Central, 1.0);
    assert_eq!(
        d.get(&func, &[0], &[1.0, 2.0, 3.0]).unwrap_err(),
        CalcError::StepSizeZero
    );
}