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 4(5)
    - Automatic step size control with starting step size

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

- Statistics
    - probability distribution
        - Bernoulli
        - Beta
        - Binomial
        - Chisquared
        - Exponential
        - Gamma
        - Chi-squared
        - Multinomial
        - 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

```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 ODE with initial condition:

```rust use mathru::*; use mathru::algebra::linear::{Vector}; use mathru::analysis::ode::{DormandPrince54}; use mathru::analysis::ode::ExplicitODE;

// Define the ODE // x' = (5t^2 - x) / e^(t + x) `$ // x(0) = 1 pub struct ExplicitODE1 { timespan: (f64, f64), initcond: Vector }

impl Default for ExplicitODE1 { fn default() -> ExplicitODE1 { ExplicitODE1 { timespan: (0.0, 10.0), initcond: vector![1.0], } } }

impl ExplicitODE for ExplicitODE1 { fn func(self: &Self, t: &f64, x: &Vector) -> Vector { return vector!((5.0 * t * t - *x.get(0)) / (t + *x.get(0)).exp()); }

// Returns the time span
fn time_span(self: &Self) -> (f64, f64)
{
    return self.time_span;
}

// Initial value at x(0)
fn init_cond(self: &Self) -> Vector<f64>
{
    return self.init_cond.clone();
}

}

fn main() { let h0: f64 = 0.001; let nmax: u32 = 300; let abs_tol: f64 = 10e-7;

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

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

} ```

Fitting with Levenberg-Marquardt

```rust ///y = 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 Jacobian for Example { 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();
    }

    return (residual.transpose() * jacobian_f * -2.0).into();
}

}

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

let noise: Normal = 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 y_vec: Vec<f64> = Vec::with_capacity(num_samples);

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

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

    //Add some noise
    y_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 y: Vector<f64> = Vector::new_column(num_samples, y_vec.clone());

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

// Optimise fitting parameters with Levenberg-Marquard algorithm
let optim: LevenbergMarquardt<f64> = LevenbergMarquardt::new(100, 0.3, 0.95);
let beta_opt: Vector<f64> = optim.minimize(&example_function, &vector![-1.5; 1.0; -2.0]).arg();

println!("{}", beta_opt);

} ```

Contributions

Any contribution is welcome!