Struct GaussHermite

pub struct GaussHermite { /* private fields */ }

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::new(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);

Implementations

impl GaussHermite

pub fn new(deg: usize) -> Result<Self, GaussHermiteError>

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

A rule of degree n can integrate polynomials of degree 2n-1 exactly.

Applies the Golub-Welsch algorithm to determine Gauss-Hermite nodes & weights. Constructs the 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

Errors

Returns an error if deg is smaller than 2.

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

where F: Fn(f64) -> f64,

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

pub fn par_integrate<F>(&self, integrand: F) -> f64

where F: Fn(f64) -> f64 + Sync,

Available on crate feature rayon only.

Same as integrate but runs in parallel.

impl GaussHermite

pub fn nodes(&self) -> GaussHermiteNodes<'_>

Returns an iterator over the nodes of the quadrature rule.

pub fn weights(&self) -> GaussHermiteWeights<'_>

Returns an iterator over the weights of the quadrature rule.

pub fn iter(&self) -> GaussHermiteIter<'_>

Returns an iterator over the node-weight pairs of the quadrature rule.

pub fn as_node_weight_pairs(&self) -> &[(Node, Weight)]

Returns a slice of all the node-weight pairs of the quadrature rule.

pub fn into_node_weight_pairs(self) -> Vec<(Node, Weight)>

Converts the quadrature rule into a vector of node-weight pairs.

This function just returns the underlying vector without any computation or cloning.

pub fn degree(&self) -> usize

Returns the degree of the quadrature rule.

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<'de> Deserialize<'de> for GaussHermite

fn deserialize<__D>(__deserializer: __D) -> Result<Self, __D::Error>

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl<'a> IntoIterator for &'a GaussHermite

type IntoIter = GaussHermiteIter<'a>

Which kind of iterator are we turning this into?

type Item = &'a (f64, f64)

The type of the elements being iterated over.

fn into_iter(self) -> Self::IntoIter

Creates an iterator from a value.

impl IntoIterator for GaussHermite

type IntoIter = GaussHermiteIntoIter

Which kind of iterator are we turning this into?

type Item = (f64, f64)

The type of the elements being iterated over.

fn into_iter(self) -> Self::IntoIter

Creates an iterator from a value.

impl PartialEq for GaussHermite

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

Tests for self and other values to be equal, and is used by ==.

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

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

impl Serialize for GaussHermite

fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl StructuralPartialEq for GaussHermite