RustQuant_math/

integration.rs

// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// RustQuant: A Rust library for quantitative finance tools.
// Copyright (C) 2023 https://github.com/avhz
// Dual licensed under Apache 2.0 and MIT.
// See:
//      - LICENSE-APACHE.md
//      - LICENSE-MIT.md
// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

//! This module contains functions for numerical integration.
//!
//! The Tanh-Sinh quadrature is used for the integration.
//! This method uses the hyperbolic trig functions to transform
//! the integral over $[-1, +1]$ to an integral over $\mathbb{R} = (-\infty, +\infty)$.
//!
//! We have the approximation:
//!
//! $$
//! \int_{-1}^{+1} f(x) dx \approx \sum_{\mathbb{R}} w_k f(x_k)
//! $$
//!
//! The abscissae and weights are calculated as follows:
//!
//! $$
//! x_k = \tanh \left( \frac{1}{2} \pi \sinh(kh) \right)
//! $$
//!
//! $$
//! w_k = \frac{1}{2} h \pi \cosh(kh) \cosh^{-2} \left( \frac{1}{2} \pi \sinh(kh) \right)
//! $$

// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// FUNCTIONS
// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

/// Integrates a function from `a` to `b`.
/// Uses the Tanh-Sinh quadrature over [-1, +1]
/// and then transforms to an integral over [a, b].
pub fn integrate<F>(f: F, a: f64, b: f64) -> f64
where
    F: Fn(f64) -> f64,
{
    // Apply a linear change of variables:
    //
    // x = c * t + d
    //
    // where:
    //      c = 0.5 * (b - a)
    //      d = 0.5 * (a + b)

    let c = 0.5 * (b - a);
    let d = 0.5 * (a + b);

    c * tanhsinh(|t| {
        let out = f(c * t + d);
        if out.is_finite() {
            out
        } else {
            0.0
        }
    })
}

// This method integrates the function f(c * x + d).
fn tanhsinh<F>(f: F) -> f64
where
    F: Fn(f64) -> f64,
{
    let mut integral = 0.0;

    for i in 0..100 {
        integral += WEIGHTS[i] * f(ABSCISSAE[i]);
    }

    integral
}

// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// ABSCISSAE & WEIGHTS
// These are for the tanh-sinh quadrature.
// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

// Nodes and weights from: https://keisan.casio.com/exec/system/1285151216

/// Abscissae: the nodes for the sum evaluation.
pub const ABSCISSAE: [f64; 100] = [
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -1.0,
    -0.999_999_999_999_999_999_999_999_92,
    -0.999_999_999_999_999_999_999_999_95,
    -0.999_999_999_999_999_999_999_999_98,
    -0.999_999_999_999_999_999_999_999_91,
    -0.999_999_999_999_999_999_999_999_95,
    -0.999_999_999_999_999_999_999_998_77,
    -0.999_999_999_999_999_999_999_103_26,
    -0.999_999_999_999_999_999_703_932_4,
    -0.999_999_999_999_999_950_531_716_2,
    -0.999_999_999_999_995_4,
    -0.999_999_999_999_755_4,
    -0.999_999_999_991_728_4,
    -0.999_999_999_814_670_3,
    -0.999_999_997_111_343_6,
    -0.999_999_967_297_845_9,
    -0.999_999_720_654_543_6,
    -0.999_998_137_805_321,
    -0.999_990_019_143_856_7,
    -0.999_955_840_978_693_4,
    -0.999_834_913_162_265,
    -0.999_467_635_803_999,
    -0.998_491_938_731_602_9,
    -0.996_186_771_585_243_7,
    -0.991_272_560_236_550_9,
    -0.981_701_565_973_876_9,
    -0.964_495_379_931_111_4,
    -0.935_708_688_740_139_8,
    -0.890_613_153_970_636_9,
    -0.824_190_972_468_412_2,
    -0.731_982_954_977_568_5,
    -0.611_228_933_047_550_6,
    -0.462_072_342_445_637_1,
    -0.288_435_163_890_398_6,
    -0.098_120_697_049_078_77,
    0.098_120_697_049_078_75,
    0.288_435_163_890_398_64,
    0.462_072_342_445_637_19,
    0.611_228_933_047_550_7,
    0.731_982_954_977_568_5,
    0.824_190_972_468_412_2,
    0.890_613_153_970_636_9,
    0.935_708_688_740_139_9,
    0.964_495_379_931_111_5,
    0.981_701_565_973_876_9,
    0.991_272_560_236_551,
    0.996_186_771_585_243_7,
    0.998_491_938_731_603,
    0.999_467_635_803_999,
    0.999_834_913_162_265,
    0.999_955_840_978_693_5,
    0.999_990_019_143_856_8,
    0.999_998_137_805_321_1,
    0.999_999_720_654_543_7,
    0.999_999_967_297_846,
    0.999_999_997_111_343_7,
    0.999_999_999_814_670_2,
    0.999_999_999_991_728_4,
    0.999_999_999_999_755_4,
    0.999_999_999_999_995_4,
    0.999_999_999_999_999_950_531_74,
    0.999_999_999_999_999_999_703_94,
    0.999_999_999_999_999_999_999_156,
    0.999_999_999_999_999_999_999_987,
    0.999_999_999_999_999_999_999_985,
    0.999_999_999_999_999_999_999_951,
    0.999_999_999_999_999_999_999_958,
    0.999_999_999_999_999_999_999_955,
    0.999_999_999_999_999_999_999_912,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
    1.0,
];

/// Weights for the sum evaluation.
pub const WEIGHTS: [f64; 100] = [
    0.0,
    4.575_617_930_178_338e-295,
    3.316_994_462_570_346e-260,
    1.865_212_622_495_173_5e-229,
    2.479_121_672_052_092e-202,
    2.081_537_448_295_626e-178,
    2.628_202_011_356_882_4e-157,
    1.072_629_885_654_983_3e-138,
    2.779_482_305_261_637e-122,
    8.296_418_844_663_515e-108,
    4.824_667_051_120_648e-95,
    8.690_880_649_537_159e-84,
    7.300_384_623_604_137e-74,
    4.102_662_749_806_205e-65,
    2.120_927_715_027_845e-57,
    1.335_824_993_406_033_2e-50,
    1.313_406_878_320_599_1e-44,
    2.508_807_816_395_078_6e-39,
    1.129_193_867_059_432_8e-34,
    1.419_894_975_881_043_8e-30,
    5.796_754_421_896_154_4e-27,
    8.772_732_913_927_396e-24,
    5.532_447_125_135_289e-21,
    1.612_026_230_404_078_1e-18,
    2.377_241_245_744_043_6e-16,
    1.922_871_877_455_573e-14,
    9.158_786_378_811_089e-13,
    2.735_019_685_607_782_3e-11,
    5.412_036_348_700_474e-10,
    7.452_095_495_199_113e-9,
    7.455_643_353_281_056e-8,
    5.630_935_258_210_419e-7,
    3.320_892_221_887_424e-6,
    1.575_869_255_697_420_2e-5,
    6.178_940_879_197_28e-5,
    2.049_612_222_935_924_2e-4,
    5.873_088_850_232_362e-4,
    0.001_480_856_522_480_776_8,
    0.003_339_108_906_155_36,
    0.006_827_704_416_155_456,
    0.012_809_851_589_261_179,
    0.022_262_908_367_071_66,
    0.036_106_623_280_003_48,
    0.054_935_001_719_810_43,
    0.078_672_104_392_875_16,
    0.106_225_018_644_775_54,
    0.135_274_854_478_869_08,
    0.162_387_107_725_986_52,
    0.183_570_841_955_285,
    0.195_234_233_228_838_65,
    0.195_234_233_228_838_65,
    0.183_570_841_955_285,
    0.162_387_107_725_986_52,
    0.135_274_854_478_869_08,
    0.106_225_018_644_775_54,
    0.078_672_104_392_875_16,
    0.054_935_001_719_810_43,
    0.036_106_623_280_003_48,
    0.022_262_908_367_071_66,
    0.012_809_851_589_261_179,
    0.006_827_704_416_155_456,
    0.003_339_108_906_155_36,
    0.001_480_856_522_480_776_8,
    5.873_088_850_232_362e-4,
    2.049_612_222_935_924_2e-4,
    6.178_940_879_197_28e-5,
    1.575_869_255_697_420_2e-5,
    3.320_892_221_887_424e-6,
    5.630_935_258_210_419e-7,
    7.455_643_353_281_056e-8,
    7.452_095_495_199_113e-9,
    5.412_036_348_700_474e-10,
    2.735_019_685_607_782_3e-11,
    9.158_786_378_811_089e-13,
    1.922_871_877_455_573e-14,
    2.377_241_245_744_043_6e-16,
    1.612_026_230_404_078_1e-18,
    5.532_447_125_135_289e-21,
    8.772_732_913_927_396e-24,
    5.796_754_421_896_154_4e-27,
    1.419_894_975_881_043_8e-30,
    1.129_193_867_059_432_8e-34,
    2.508_807_816_395_078_6e-39,
    1.313_406_878_320_599_1e-44,
    1.335_824_993_406_033_2e-50,
    2.120_927_715_027_845e-57,
    4.102_662_749_806_205e-65,
    7.300_384_623_604_137e-74,
    8.690_880_649_537_159e-84,
    4.824_667_051_120_648e-95,
    8.296_418_844_663_515e-108,
    2.779_482_305_261_637e-122,
    1.072_629_885_654_983_3e-138,
    2.628_202_011_356_882_4e-157,
    2.081_537_448_295_626e-178,
    2.479_121_672_052_092e-202,
    1.865_212_622_495_173_5e-229,
    3.316_994_462_570_346e-260,
    4.575_617_930_178_338e-295,
    0.0,
];

// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// TESTS
// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#[cfg(test)]
mod tests_integration {
    use super::*;

    use RustQuant_utils::{assert_approx_equal, RUSTQUANT_EPSILON as EPS};

    #[test]
    fn test_quadrature() {
        fn f(x: f64) -> f64 {
            (x.sin()).exp()
        }

        let integral = integrate(f, 0.0, 5.0);

        assert_approx_equal!(integral, 7.189_119_252_343_784, EPS);
    }
}