Struct gauss_quad::hermite::GaussHermite

pub struct GaussHermite {
    pub nodes: Vec<f64>,
    pub weights: Vec<f64>,
}

A Gauss-Hermite quadrature scheme.

These rules can integrate integrands of the form e^(-x^2) * f(x) over the domain (-∞, ∞).

Example

Integrate e^(-x^2) * cos(x)

// initialize a Gauss-Hermite rule with 20 nodes
let quad = GaussHermite::init(20);

// numerically integrate a function over (-∞, ∞) using the Gauss-Hermite rule
let integral = quad.integrate(|x| x.cos());

assert_abs_diff_eq!(integral, PI.sqrt() / E.powf(0.25), epsilon = 1e-14);

Fields

nodes: Vec<f64>

weights: Vec<f64>

Implementations

impl GaussHermite

pub fn init(deg: usize) -> GaussHermite

Initializes Gauss-Hermite quadrature rule of the given degree by computing the needed nodes and weights.

pub fn nodes_and_weights(deg: usize) -> (Vec<f64>, Vec<f64>)

Apply Golub-Welsch algorithm to determine Gauss-Hermite nodes & weights construct companion matrix A for the Hermite Polynomial using the relation: 1/2 H_{n+1} + n H_{n-1} = x H_n A similar matrix that is symmetrized is constructed by D A D^{-1} Resulting in a symmetric tridiagonal matrix with 0 on the diagonal & sqrt(n/2) on the off-diagonal root & weight finding are equivalent to eigenvalue problem see Gil, Segura, Temme - Numerical Methods for Special Functions

pub fn integrate<F>(&self, integrand: F) -> f64where F: Fn(f64) -> f64,

Perform quadrature of e^(-x^2) * integrand over the domain (-∞, ∞).

Trait Implementations

impl Clone for GaussHermite

fn clone(&self) -> GaussHermite

Returns a copy of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for GaussHermite

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl PartialEq for GaussHermite

fn eq(&self, other: &GaussHermite) -> bool

This method tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl StructuralPartialEq for GaussHermite