Struct Rule 

pub struct Rule { /* private fields */ }

A one-dimensional Gauss-Kronrod integration rule.

A Gaussian numerical integration rule approximates an integral of a function by performing a weighted sum of the function evaluated at defined points/abscissae. The order of an integration rule, $n$, denotes the number of abscissae, $x_{i}$, at which the function is evaluated and the number of weights $w_{i}$ for the weighted sum, such that the approximation is, $$ I = \int_{b}^{a} f(x) dx \approx \sum_{i = 1}^{n} W_{i} f(X_{i}) = I_{n} $$ where the $X_{i}$ and $W_{i}$ are the rescaled abscissae and weights, $$ X_{i} = \frac{b + a + (a - b) x_{i}}{2} ~~~~~~~~ W_{i} = \frac{(a - b) w_{i}}{2} $$ A Gauss-Kronrod integration rule combines two rules of different order for efficient estimation of the numerical error. The rules for an $n$-point Gauss-Kronrod rule contain $m = (n - 1) / 2$ abscissae shared by the Gaussian and Kronrod rules and an extended set of $n - m$ Kronrod abscissae. The weighted sum of the full set of $n$ Kronrod function evaluations are used to approximate the result of the integration, while the weighted sum of the lower order set of $m$ Gaussian points are used to calculate the numerical error in the routine, $$ E = |I_{n} - I_{m}| $$ This approach is efficient, as only $n$ total function evaluations are required to obtain the result approximation and error estimate.

The Rule struct defines a Gauss-Kronrod quadrature rule for use in the one-dimensional numerical integration routines Basic, Adaptive, and AdaptiveSingularity. Rules of varying order $n$ are generated through dedicated constructor functions Rule::gk*.

Implementations

impl Rule

Constructors

pub const fn gk15() -> Self

15-point Gauss-Kronrod rule

Generates the evaluation points/abscissae $x_{i}$ and weights $w_{i}$ for a 15-point Gauss-Kronrod integration rule.

pub const fn gk21() -> Self

21-point Gauss-Kronrod rule

Generates the evaluation points/abscissae $x_{i}$ and weights $w_{i}$ for a 21-point Gauss-Kronrod integration rule.

pub const fn gk31() -> Self

31-point Gauss-Kronrod rule

Generates the evaluation points/abscissae $x_{i}$ and weights $w_{i}$ for a 31-point Gauss-Kronrod integration rule.

pub const fn gk41() -> Self

41-point Gauss-Kronrod rule

Generates the evaluation points/abscissae $x_{i}$ and weights $w_{i}$ for a 41-point Gauss-Kronrod integration rule.

pub const fn gk51() -> Self

51-point Gauss-Kronrod rule

Generates the evaluation points/abscissae $x_{i}$ and weights $w_{i}$ for a 51-point Gauss-Kronrod integration rule.

pub const fn gk61() -> Self

61-point Gauss-Kronrod rule

Generates the evaluation points/abscissae $x_{i}$ and weights $w_{i}$ for a 61-point Gauss-Kronrod integration rule.

Trait Implementations

impl Clone for Rule

fn clone(&self) -> Rule

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for Rule

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

Formats the value using the given formatter.

impl PartialEq for Rule

fn eq(&self, other: &Rule) -> 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 PartialOrd for Rule

fn partial_cmp(&self, other: &Rule) -> Option<Ordering>

This method returns an ordering between self and other values if one exists.

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

Tests less than (for self and other) and is used by the < operator.

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

Tests less than or equal to (for self and other) and is used by the <= operator.

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

Tests greater than (for self and other) and is used by the > operator.

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

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl Copy for Rule

impl StructuralPartialEq for Rule