Struct mathru::optimization::Newton

pub struct Newton<T> { /* fields omitted */ }

Newton's method

f \colon \mathbb{R}^n \to \mathbb{R}

is a twice differentiable function.

Newton's method solves the minimization problem

 f(x) \to min

input: $f \colon \mathbb{R}^{n} \to \mathbb{R}$ with initial approximation $x_{0} \in \mathbb{R}^{n}$

output: $x_{k}$

1. Initialization: Choose $\sigma \in (0, 1)$

   $\rho > 0, k := 0$

2. Solve de equation system $\nabla^{2} f(x_{k})d_{k} = -\nabla f(x_{k})$ 3. If the euqation is not solvable, or the condition $\nabla f(x_{k})^{T}d_{k} \leq -\rho \lvert \lvert \nabla f(x_k) \rvert \rvert_{2}^{2}$ is not fullfilled Than $d_{k} := \nabla f(x_{k})$

3. $\alpha_{k} := 1$

4. while $f(x_{k} + \alpha_{k}d_{k}) > f(x_{k}) + \sigma \alpha_{k} \nabla f(x_{k})^{T}d_{k}$ set $\alpha_{k}$ 6. $x_{k + 1} := x_{k} + d_{k}$ 7. $k := k + 1$ go to 2.

Implementations

impl<T> Newton<T>

pub fn new(iters: u64, sigma: T, rho: T) -> Newton<T>

Creates an instance of newtons method

Arguments

• 'iters': Number of iterations

impl<T> Newton<T> where
    T: Real, 

pub fn minimize<F>(&self, func: &F, x_0: &Vector<T>) -> OptimResult<Vector<T>> where
    F: Optim<T>, 

Minimize function func

Arguments

• 'func0': Function to be minimized
• 'x_0': Initial guess for the minimum

Return

local minimum