rint 0.1.1

A pure Rust library for the numerical integration of real or complex valued functions of real variables in multiple dimensions.
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

use std::cmp::Ordering;

use crate::multi::Integrator;
use crate::multi::Rule;
use crate::Integrand;
use crate::Limits;
use crate::ScalarF64;

/// The estimated integral value of a calculated region.
///
/// This is the output of the core [`Integrator`], used internally by the different numerical
/// integration routines in [`multi`]. It contains an estimate of the numerical integration
/// `result`, the estimated `error`, and then further information about the integration which is
/// used in the [`Adaptive`] routines for error estimation and analysis. A key feature of the
/// adaptive routines is the use of a [`BinaryHeap`] for storing the various [`Region`]s which have
/// been integrated. As such, the [`Region`] type implements the [`Ord`] and [`PartialOrd`] traits
/// and order against the `error` (primary) and `limits` (secondary) fields. Thus, when pushed on
/// to the [`BinaryHeap`], the [`Region`] with the largest `error` estimate _or_ if all have equal
/// `error` the region with the smallest [`Limits::length`] will be obtained first upon using
/// [`BinaryHeap::pop`]. This offloads the ordering of the [`Region`]s to the standard libary
/// implementation of the [`BinaryHeap`].
///
/// [`Integrator`]: crate::quadrature::Integrator
/// [`Adaptive`]: crate::quadrature::Adaptive
#[derive(Debug, Clone)]
pub(crate) struct Region<T: ScalarF64, const N: usize> {
    pub(crate) error: f64,
    pub(crate) result: T,
    pub(crate) limits: [Limits; N],
    pub(crate) bisection_axis: usize,
    pub(crate) volume: f64,
}

impl<T: ScalarF64, const N: usize> PartialEq for Region<T, N> {
    fn eq(&self, other: &Self) -> bool {
        (self.result == other.result)
            && (self.error == other.error)
            && (self.bisection_axis == other.bisection_axis)
            && (self.limits == other.limits)
            && (self.volume == other.volume)
    }
}

impl<T: ScalarF64, const N: usize> Eq for Region<T, N> {}

impl<T: ScalarF64, const N: usize> PartialOrd for Region<T, N> {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        Some(self.cmp(other))
    }
}

impl<T: ScalarF64, const N: usize> Ord for Region<T, N> {
    fn cmp(&self, other: &Self) -> Ordering {
        let mut ordering = self.total_cmp_error(other);
        if let Ordering::Equal = ordering {
            ordering = self.total_cmp_interval_length(other);
            if let Ordering::Equal = ordering {
                ordering = self.total_cmp_volume(other);
            }
        }
        ordering
    }
}

impl<T: ScalarF64, const N: usize> Region<T, N> {
    pub(crate) fn unevaluated() -> Self {
        let zero = T::zero();
        Self {
            error: 0.0,
            result: zero,
            limits: [Limits::new_unchecked(0.0, 0.0); N],
            bisection_axis: 0,
            volume: 0.0,
        }
    }

    pub(crate) fn with_error(mut self, error: f64) -> Self {
        self.error = error;
        self
    }

    pub(crate) fn with_result(mut self, result: T) -> Self {
        self.result = result;
        self
    }

    pub(crate) fn with_limits(mut self, limits: [Limits; N]) -> Self {
        self.limits = limits;
        self
    }

    pub(crate) fn with_bisection_axis(mut self, bisection_axis: usize) -> Self {
        self.bisection_axis = bisection_axis;
        self
    }

    pub(crate) fn with_volume(mut self, volume: f64) -> Self {
        self.volume = volume;
        self
    }

    pub(crate) fn error(&self) -> f64 {
        self.error
    }

    pub(crate) fn result(&self) -> T {
        self.result
    }

    pub(crate) fn limits(&self) -> &[Limits; N] {
        &self.limits
    }

    pub(crate) fn bisection_axis(&self) -> usize {
        self.bisection_axis
    }

    /// Primary comparison function for the [`Region`].
    ///
    /// Uses [`f64::total_cmp`] to determine the ordering of two [`Region`]s based on the `error`
    /// estimates.
    pub(crate) fn total_cmp_error(&self, other: &Self) -> Ordering {
        self.error.total_cmp(&other.error)
    }

    /// Secondary comparison function for the [`Region`].
    ///
    /// In the event that two regions have the same `error` estimate we use the _inverse_ length of
    /// the integration region to determine the ordering. The rational here is that if the `error`
    /// due to each region is the same then the smaller region is likely to be more problematic and
    /// should be tackled first.
    pub(crate) fn total_cmp_interval_length(&self, other: &Self) -> Ordering {
        let width = self.limits[self.bisection_axis].width().abs();
        let other_width = other.limits[other.bisection_axis].width().abs();
        let inverse_length = 1.0 / width;
        let other_inverse_length = 1.0 / other_width;
        inverse_length.total_cmp(&other_inverse_length)
    }

    pub(crate) fn total_cmp_volume(&self, other: &Self) -> Ordering {
        let inverse_volume = 1.0 / self.volume.abs();
        let other_inverse_volume = 1.0 / other.volume.abs();
        inverse_volume.total_cmp(&other_inverse_volume)
    }

    #[allow(clippy::needless_borrow)]
    pub(crate) fn bisect<
        I: Integrand<Point = [f64; N], Scalar = T>,
        const J: usize,
        const K: usize,
    >(
        &self,
        function: &I,
        rule: &Rule<N, J, K>,
    ) -> [Region<T, N>; 2] {
        let axis_to_bisect = self.bisection_axis;
        let previous_limits = self.limits();
        let previous_result = self.result();

        let [lower, upper] = previous_limits[axis_to_bisect].bisect();

        let mut upper_limits = *previous_limits;
        upper_limits[axis_to_bisect] = upper;

        let mut lower_limits = *previous_limits;
        lower_limits[axis_to_bisect] = lower;

        let mut upper_integral = Integrator::new(&function, &rule, upper_limits).integrate();
        let mut lower_integral = Integrator::new(&function, &rule, lower_limits).integrate();

        // adjust error estimates according to Bernsten & Espelid 1991
        {
            let new_result = upper_integral.result() + lower_integral.result();

            let mut upper_error = upper_integral.error();
            let mut lower_error = lower_integral.error();

            let est1 = (previous_result - new_result).abs();
            let est2 = lower_error + upper_error;

            let error_coeff = rule.adaptive_error_coeff();

            if est2 > 0.0 {
                lower_error *= 1.0 + error_coeff.c5() * est1 / est2;
                upper_error *= 1.0 + error_coeff.c5() * est1 / est2;
            }

            upper_error += error_coeff.c6() * est1;
            lower_error += error_coeff.c6() * est1;

            upper_integral.error = upper_error;
            lower_integral.error = lower_error;
        }

        [lower_integral, upper_integral]
    }
}