mathru

mathru is a numeric library containing algorithms for linear algebra, analysis and statistics written in pure Rust with BLAS/LAPACK support.

Features

- Linear algebra
    - Vector
    - Matrix
        - Basic matrix operations(+,-,*)
        - Transposition
        - LU decomposition (native/lapack)
        - QR decomposition (native/lapack)
        - Hessenberg decomposition (native/lapack)
        - Singular value decomposition
        - Inverse (native/lapack)
        - Pseudo inverse (native/lapack)
        - Determinant (native/lapack)
        - Trace
        - Eigenvalue (native/lapack)

- Ordinary differential equation (ODE)
    - Explicit methods
        - Heun's method
        - Euler method
        - Midpoint method
        - Ralston's method
        - Kutta 3rd order
        - Runge-Kutta 4th order
        - Runge-Kutta-Felhberg 4(5)
        - Dormand-Prince 4(5)
        - Cash-Karp 4(5)
        - Tsitouras 4(5)
        - Bogacki-Shampine 2(3)
        - Adams-Bashforth
    - Automatic step size control with starting step size
    - Implizit methods
        - Implizit Euler

- Optimization
    - Gauss-Newton algorithm
    - Gradient descent
    - Newton method
    - Levenberg-Marquardt algorithm
    - Conjugate gradient method

- Statistics
    - probability distribution
        - Bernoulli
        - Beta
        - Binomial 
        - Exponential
        - Gamma
        - Chi-squared
        - Normal
        - Poisson
        - Raised cosine
        - Student-t
        - Uniform
    - test
        - Chi-squared 
        - G
        - Student-t

- elementary functions
    - trigonometric functions
    - hyperbolic functions
    - exponential functions

- special functions
    - gamma functions
    - beta functions
    - hypergeometrical functions

Usage

Add this to your Cargo.toml for the native Rust implementation:

toml [dependencies.mathru] version = "^0.6" Add the following lines to 'Cargo.toml' if the blas/lapack backend should be used:

toml [dependencies.mathru] version = "^0.6" default-features = false features = ["blaslapack"]

Then import the modules and it is ready to be used.

Example

Calculate the LU decomposition of a matrix

```rust use mathru::algebra::linear::{Matrix};

// Compute the LU decomposition of a 2x2 matrix let a: Matrix = Matrix::new(2, 2, vec![1.0, -2.0, 3.0, -7.0]); let l_ref: Matrix = Matrix::new(2, 2, vec![1.0, 0.0, 1.0 / 3.0, 1.0]);

let (l, u, p): (Matrix, Matrix, Matrix) = a.dec_lu();

asserteq!(lref, l); ```

Solve an initial value problem with Dormand-Prince method:

```rust use mathru::algebra::linear::{Vector}; use mathru::analysis::ode::{ExplicitODE, DormandPrince54}; use mathru::analysis::ode::problem::Euler; use plotters::prelude::*;

fn main() { // Create an ODE instance let problem: Euler = Euler::default();

let (x_start, x_end) = problem.time_span();

// Create a ODE solver instance
let h_0: f64 = 0.001;
let n_max: u32 = 800;
let abs_tol: f64 = 10e-7;

let solver: DormandPrince54<f64> = DormandPrince54::new(abs_tol, h_0, n_max);

// Solve ODE
let (x, y): (Vec<f64>, Vec<Vector<f64>>) = solver.solve(&problem).unwrap();

//Create chart
let mut graph_x1: Vec<(f64, f64)> = Vec::with_capacity(x.len());
let mut graph_x2: Vec<(f64, f64)> = Vec::with_capacity(x.len());
let mut graph_x3: Vec<(f64, f64)> = Vec::with_capacity(x.len());

for i in 0..x.len()
{
    let x_i = x[i];
    graph_x1.push((x_i, *y[i].get(0)));
    graph_x2.push((x_i, *y[i].get(1)));
    graph_x3.push((x_i, *y[i].get(2)));
}

let root_area = BitMapBackend::new("../figure/ode_explicit.png", (1200, 800))
    .into_drawing_area();
root_area.fill(&WHITE).unwrap();

let mut ctx = ChartBuilder::on(&root_area)
    .margin(20)
    .set_label_area_size(LabelAreaPosition::Left, 40)
    .set_label_area_size(LabelAreaPosition::Bottom, 40)
    .caption("ODE solved with Dormand-Prince", ("Arial", 40))
    .build_ranged(x_start..x_end, -1.0f64..1.5f64)
    .unwrap();

ctx.configure_mesh()
    .x_desc("Time t")
    .axis_desc_style(("sans-serif", 25).into_font())
    .draw().unwrap();


ctx.draw_series(LineSeries::new(graph_x1, &BLACK)).unwrap();

ctx.draw_series(LineSeries::new(graph_x2, &BLACK)).unwrap();

ctx.draw_series(LineSeries::new(graph_x3, &BLACK)).unwrap();

} ``` Example image

Solve an initial value problem with implicit Euler method:

```rust use mathru::algebra::linear::{Vector}; use mathru::analysis::ode::{ImplicitODE, ImplicitEuler}; use mathru::analysis::ode::problem::VanDerPolOsc; use plotters::prelude::*;

fn main() { // Create an ODE instance let problem: VanDerPolOsc = VanDerPolOsc::default();

// Create a ODE solver instance
let solver: ImplicitEuler<f64> = ImplicitEuler::new(0.0005);

// Solve ODE
let (t, x): (Vec<f64>, Vec<Vector<f64>>) = solver.solve(&problem).unwrap();

//Create chart
let mut graph_x1: Vec<(f64, f64)> = Vec::with_capacity(x.len());
let mut graph_x2: Vec<(f64, f64)> = Vec::with_capacity(x.len());

for i in 0..x.len()
{
    let t_i = t[i];
    graph_x1.push((t_i, *x[i].get(0)));
    graph_x2.push((t_i, *x[i].get(1)));
}

let root_area = BitMapBackend::new("../figure/ode_implicit.png", (1200, 800))
    .into_drawing_area();

root_area.fill(&WHITE).unwrap();

let (t_start, t_end) = problem.time_span();

let mut ctx = ChartBuilder::on(&root_area)
    .margin(20)
    .set_label_area_size(LabelAreaPosition::Left, 40)
    .set_label_area_size(LabelAreaPosition::Bottom, 40)
    .caption("ODE solved with implicit Euler", ("Arial", 40))
    .build_ranged(t_start..t_end, -2.0f64..2.0f64)
    .unwrap();

ctx.configure_mesh()
    .x_desc("Time t")
    .y_desc("x, x'")
    .axis_desc_style(("sans-serif", 25).into_font())
    .draw().unwrap();

ctx.draw_series(LineSeries::new(graph_x1, &BLACK)).unwrap();

ctx.draw_series(LineSeries::new(graph_x2, &RED)).unwrap();

} ```

Example image

Fitting with Levenberg-Marquardt

```rust use mathru::; use mathru::algebra::linear::{Vector, Matrix}; use mathru::statistics::distrib::{Distribution, Normal}; use mathru::optimization::{Optim, LevenbergMarquardt}; use plotters::prelude::;

///y(t) = a + b * exp(c * t) = f(t) pub struct Example { x: Vector, y: Vector }

impl Example { pub fn new(x: Vector, y: Vector) -> Example { Example { x: x, y: y } }

pub fn function(x: f64, beta: &Vector<f64>) -> f64
{
    let beta_0: f64 = *beta.get(0);
    let beta_1: f64 = *beta.get(1);
    let beta_2: f64 = *beta.get(2);
    let f_x: f64 = beta_0 + beta_1 * (beta_2 * x).exp();

    return f_x;
}

}

impl Optim for Example { // y(xi) - f(xi) fn eval(self: &Self, beta: &Vector) -> Vector { let fx = self.x.clone().apply(&|x: &f64| Example::function(*x, beta)); let r: Vector = &self.y - &fx; return vector![r.dotp(&r)] }

fn jacobian(self: &Self, beta: &Vector<f64>) -> Matrix<f64>
{
    let (x_m, _x_n) = self.x.dim();
    let (beta_m, _beta_n) = beta.dim();

    let mut jacobian_f: Matrix<f64> = Matrix::zero(x_m, beta_m);

    let f_x = self.x.clone().apply(&|x: &f64| Example::function(*x, beta));

    let residual: Vector<f64> = &self.y - &f_x;

    for i in 0..x_m
    {
        let beta_1: f64 = *beta.get(1);
        let beta_2: f64 = *beta.get(2);

        let x_i: f64 = *self.x.get(i);

        *jacobian_f.get_mut(i, 0) = 1.0;
        *jacobian_f.get_mut(i, 1) = (beta_2 * x_i).exp();
        *jacobian_f.get_mut(i, 2) = beta_1 * x_i * (beta_2 * x_i).exp();

    }

    let jacobian: Matrix<f64> = (residual.transpose() * jacobian_f * -2.0).into();
    return jacobian;
}

}

fn main() { let num_samples: usize = 100;

let noise: Normal<f64> = Normal::new(0.0, 0.05);

let mut t_vec: Vec<f64> = Vec::with_capacity(num_samples);

// Start time
let t_0 = 0.0f64;
// End time
let t_1 = 5.0f64;

let mut x_vec: Vec<f64> = Vec::with_capacity(num_samples);

// True function parameters
let beta: Vector<f64> = vector![0.5; 5.0; -1.0];

for i in 0..num_samples
{
    let t_i: f64 = (t_1 - t_0) / (num_samples as f64) * (i as f64);

    //Add some noise
    x_vec.push(Example::function(t_i, &beta) + noise.random());
    t_vec.push(t_i);
}

let t: Vector<f64> = Vector::new_column(num_samples, t_vec.clone());
let x: Vector<f64> = Vector::new_column(num_samples, x_vec.clone());

let example_function = Example::new(t, x);

let optim: LevenbergMarquardt<f64> = LevenbergMarquardt::new(100, 0.3, 0.95);

// Fit parameter
let beta_0: Vector<f64> = vector![-1.5; 1.0; -2.0];
let beta_opt: Vector<f64> = optim.minimize(&example_function, &beta_0).arg();

//Create chart
let mut graph_x: Vec<(f64, f64)> = Vec::with_capacity(x_vec.len());
let mut graph_x_hat: Vec<(f64, f64)> = Vec::with_capacity(x_vec.len());

for i in 0..x_vec.len()
{
    let t_i = t_vec[i];
    graph_x.push((t_i, x_vec[i]));

    let x_hat = Example::function(t_i, &beta_opt);
    graph_x_hat.push((t_i, x_hat));
}

let root_area = BitMapBackend::new("../figure/fit_lm.png", (1200, 800))
    .into_drawing_area();
root_area.fill(&WHITE).unwrap();

let mut ctx = ChartBuilder::on(&root_area)
    .margin(20)
    .set_label_area_size(LabelAreaPosition::Left, 40)
    .set_label_area_size(LabelAreaPosition::Bottom, 40)
    .caption("Parameter fitting with Levenberg Marquardt", ("Arial", 40))
    .build_ranged(t_0..t_1, -0.5f64..6.0f64)
    .unwrap();

ctx.configure_mesh()
    .x_desc("Time t")
    .axis_desc_style(("sans-serif", 25).into_font())
    .draw().unwrap();


ctx.draw_series(LineSeries::new(graph_x, &BLACK)).unwrap();

ctx.draw_series(LineSeries::new(graph_x_hat, &RED)).unwrap();

} ``` Fitting with Levenberg-Marquardt

License

Licensed under

MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

Contribution

Any contribution is welcome!