Struct GaussJacobi

pub struct GaussJacobi { /* private fields */ }

A Gauss-Jacobi quadrature scheme.

This rule can integrate expressions of the form (1 - x)^alpha * (1 + x)^beta * f(x), where f(x) is a smooth function on a finite domain, alpha > -1 and beta > -1, and where f(x) is transformed from the domain [a, b] to the domain [-1, 1]. This enables the approximation of integrals with singularities at the end points of the domain.

Examples

// initialize the quadrature rule.
let quad = GaussJacobi::new(10, -0.5, 0.0)?;

let integral = quad.integrate(0.0, 2.0, |x| (-x).exp());

assert_abs_diff_eq!(integral, 0.9050798148074449, epsilon = 1e-14);

Implementations

impl GaussJacobi

pub fn new(deg: usize, alpha: f64, beta: f64) -> Result<Self, GaussJacobiError>

Initializes Gauss-Jacobi quadrature rule of the given degree by computing the nodes and weights needed for the given parameters. alpha is the exponent of the (1 - x) factor and beta is the exponent of the (1 + x) factor.

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

Applies the Golub-Welsch algorithm to determine Gauss-Jacobi nodes & weights. See Gil, Segura, Temme - Numerical Methods for Special Functions

Errors

Returns an error if deg is smaller than 2, and/or if alpha and/or beta are smaller than or equal to -1.

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

where F: Fn(f64) -> f64,

Perform quadrature of integrand from a to b. This will integrate
(1 - x)^alpha * (1 + x)^beta * integrand(x)
where alpha and beta were given in the call to new, and the integrand is transformed from the domain [a, b] to the domain [-1, 1].

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

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

Available on crate feature rayon only.

Same as integrate but runs in parallel.

pub const fn alpha(&self) -> f64

Returns the value of the alpha parameter.

pub const fn beta(&self) -> f64

Returns the value of the beta parameter.

impl GaussJacobi

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

Returns an iterator over the nodes of the quadrature rule.

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

Returns an iterator over the weights of the quadrature rule.

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

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 GaussJacobi

fn clone(&self) -> GaussJacobi

Returns a copy of the value.

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

Performs copy-assignment from source.

impl Debug for GaussJacobi

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

Formats the value using the given formatter.

impl<'de> Deserialize<'de> for GaussJacobi

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

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl From<GaussChebyshevFirstKind> for GaussJacobi

Gauss-Chebyshev quadrature of the first kind is equivalent to Gauss-Jacobi quadrature with alpha = beta = -0.5.

fn from(value: GaussChebyshevFirstKind) -> Self

Converts to this type from the input type.

impl From<GaussChebyshevSecondKind> for GaussJacobi

Gauss-Chebyshev quadrature of the second kind is equivalent to Gauss-Jacobi quadrature with alpha = beta = 0.5.

fn from(value: GaussChebyshevSecondKind) -> Self

Converts to this type from the input type.

impl From<GaussLegendre> for GaussJacobi

Gauss-Legendre quadrature is equivalent to Gauss-Jacobi quadrature with alpha = beta = 0.

fn from(value: GaussLegendre) -> Self

Converts to this type from the input type.

impl<'a> IntoIterator for &'a GaussJacobi

type IntoIter = GaussJacobiIter<'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 GaussJacobi

type IntoIter = GaussJacobiIntoIter

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 GaussJacobi

fn eq(&self, other: &GaussJacobi) -> 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 GaussJacobi

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 GaussJacobi