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integral.rs
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integral.rs
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use crate::structure::polynomial::{lagrange_polynomial, Calculus, Polynomial};
use crate::traits::fp::FPVector;
use crate::util::non_macro::seq;
#[derive(Debug, Copy, Clone, PartialEq)]
pub enum Integral {
GaussLegendre(usize),
NewtonCotes(usize),
G7K15(f64, u32),
G10K21(f64, u32),
G15K31(f64, u32),
G20K41(f64, u32),
G25K51(f64, u32),
G30K61(f64, u32),
G7K15R(f64, u32),
G10K21R(f64, u32),
G15K31R(f64, u32),
G20K41R(f64, u32),
G25K51R(f64, u32),
G30K61R(f64, u32),
}
impl Integral {
pub fn get_num_node(&self) -> usize {
match self {
Integral::GaussLegendre(n) => *n,
Integral::NewtonCotes(n) => *n,
_ => panic!("This method does not have a fixed number of nodes."),
}
}
pub fn get_tol(&self) -> f64 {
match self {
Integral::G7K15(tol, _) => *tol,
Integral::G10K21(tol, _) => *tol,
Integral::G15K31(tol, _) => *tol,
Integral::G20K41(tol, _) => *tol,
Integral::G25K51(tol, _) => *tol,
Integral::G30K61(tol, _) => *tol,
Integral::G7K15R(tol, _) => *tol,
Integral::G10K21R(tol, _) => *tol,
Integral::G15K31R(tol, _) => *tol,
Integral::G20K41R(tol, _) => *tol,
Integral::G25K51R(tol, _) => *tol,
Integral::G30K61R(tol, _) => *tol,
_ => panic!("This method does not have a tolerance."),
}
}
pub fn get_max_iter(&self) -> u32 {
match self {
Integral::G7K15(_, max_iter) => *max_iter,
Integral::G10K21(_, max_iter) => *max_iter,
Integral::G15K31(_, max_iter) => *max_iter,
Integral::G20K41(_, max_iter) => *max_iter,
Integral::G25K51(_, max_iter) => *max_iter,
Integral::G30K61(_, max_iter) => *max_iter,
Integral::G7K15R(_, max_iter) => *max_iter,
Integral::G10K21R(_, max_iter) => *max_iter,
Integral::G15K31R(_, max_iter) => *max_iter,
Integral::G20K41R(_, max_iter) => *max_iter,
Integral::G25K51R(_, max_iter) => *max_iter,
Integral::G30K61R(_, max_iter) => *max_iter,
_ => panic!("This method does not have a maximum number of iterations."),
}
}
pub fn get_gauss_kronrod_order(&self) -> (u8, u8) {
match self {
Integral::G7K15(_, _) => (7, 15),
Integral::G10K21(_, _) => (10, 21),
Integral::G15K31(_, _) => (15, 31),
Integral::G20K41(_, _) => (20, 41),
Integral::G25K51(_, _) => (25, 51),
Integral::G30K61(_, _) => (30, 61),
Integral::G7K15R(_, _) => (7, 15),
Integral::G10K21R(_, _) => (10, 21),
Integral::G15K31R(_, _) => (15, 31),
Integral::G20K41R(_, _) => (20, 41),
Integral::G25K51R(_, _) => (25, 51),
Integral::G30K61R(_, _) => (30, 61),
_ => panic!("This method does not have a Gauss-Kronrod order."),
}
}
pub fn is_relative(&self) -> bool {
match self {
Integral::G7K15R(_, _) => true,
Integral::G10K21R(_, _) => true,
Integral::G15K31R(_, _) => true,
Integral::G20K41R(_, _) => true,
Integral::G25K51R(_, _) => true,
Integral::G30K61R(_, _) => true,
_ => false,
}
}
pub fn change_tol(&self, tol: f64) -> Self {
match self {
Integral::G7K15(_, max_iter) => Integral::G7K15(tol, *max_iter),
Integral::G10K21(_, max_iter) => Integral::G10K21(tol, *max_iter),
Integral::G15K31(_, max_iter) => Integral::G15K31(tol, *max_iter),
Integral::G20K41(_, max_iter) => Integral::G20K41(tol, *max_iter),
Integral::G25K51(_, max_iter) => Integral::G25K51(tol, *max_iter),
Integral::G30K61(_, max_iter) => Integral::G30K61(tol, *max_iter),
Integral::G7K15R(_, max_iter) => Integral::G7K15R(tol, *max_iter),
Integral::G10K21R(_, max_iter) => Integral::G10K21R(tol, *max_iter),
Integral::G15K31R(_, max_iter) => Integral::G15K31R(tol, *max_iter),
Integral::G20K41R(_, max_iter) => Integral::G20K41R(tol, *max_iter),
Integral::G25K51R(_, max_iter) => Integral::G25K51R(tol, *max_iter),
Integral::G30K61R(_, max_iter) => Integral::G30K61R(tol, *max_iter),
_ => panic!("This method does not have a tolerance."),
}
}
pub fn change_max_iter(&self, max_iter: u32) -> Self {
match self {
Integral::G7K15(tol, _) => Integral::G7K15(*tol, max_iter),
Integral::G10K21(tol, _) => Integral::G10K21(*tol, max_iter),
Integral::G15K31(tol, _) => Integral::G15K31(*tol, max_iter),
Integral::G20K41(tol, _) => Integral::G20K41(*tol, max_iter),
Integral::G25K51(tol, _) => Integral::G25K51(*tol, max_iter),
Integral::G30K61(tol, _) => Integral::G30K61(*tol, max_iter),
Integral::G7K15R(tol, _) => Integral::G7K15R(*tol, max_iter),
Integral::G10K21R(tol, _) => Integral::G10K21R(*tol, max_iter),
Integral::G15K31R(tol, _) => Integral::G15K31R(*tol, max_iter),
Integral::G20K41R(tol, _) => Integral::G20K41R(*tol, max_iter),
Integral::G25K51R(tol, _) => Integral::G25K51R(*tol, max_iter),
Integral::G30K61R(tol, _) => Integral::G30K61R(*tol, max_iter),
_ => panic!("This method does not have a maximum number of iterations."),
}
}
}
/// Type that can be integrated using Newton Cotes Quadrature
pub trait NCIntegrable: std::ops::Sub<Self, Output = Self> + Sized {
type NodeY;
type NCPolynomial;
/// Returns the image of `node_x` under function `f`, in a representation
/// (`Self::NodeY`) suitable for separately computing one Lagrange polynomial
/// for each of the degrees of freedom of `Self` (e.g. a `Vec` with as many
/// entries as the number of degrees of freedom).
fn compute_node_y<F>(f: F, node_x: &[f64]) -> Self::NodeY
where
F: Fn(f64) -> Self;
/// Computes one [`Lagrange polynomial`](lagrange_polynomial) for each one
/// of the degrees of freedom of `Self`, returning them in a representation
/// (`Self::NCPolynomial`) suitable to separately performing operation on
/// them (e.g. [`Vec<Polynomial>`](Polynomial)).
fn compute_polynomial(node_x: &[f64], node_y: &Self::NodeY) -> Self::NCPolynomial;
/// Separately integrates each of the polynomials obtained using
/// [`compute_polynomial`](NCIntegrable::compute_polynomial).
fn integrate_polynomial(p: &Self::NCPolynomial) -> Self::NCPolynomial;
/// Separately evaluates each of the polynomial integrated using
/// [`integrate_polynomial`](NCIntegrable::integrate_polynomial) at `x`,
/// then recombines the result into a `Self`.
fn evaluate_polynomial(p: &Self::NCPolynomial, x: f64) -> Self;
}
impl NCIntegrable for f64 {
type NodeY = Vec<f64>;
type NCPolynomial = Polynomial;
fn compute_node_y<F>(f: F, node_x: &[f64]) -> Vec<f64>
where
F: Fn(f64) -> Self,
{
node_x.to_vec().fmap(|x| f(x))
}
fn compute_polynomial(node_x: &[f64], node_y: &Vec<f64>) -> Polynomial {
lagrange_polynomial(node_x.to_vec(), node_y.to_vec())
}
fn integrate_polynomial(p: &Polynomial) -> Polynomial {
p.integral()
}
fn evaluate_polynomial(p: &Polynomial, x: f64) -> Self {
p.eval(x)
}
}
/// Type that can be integrated using Gauss Legendre or Kronrod Quadrature
pub trait GLKIntegrable:
std::ops::Add<Self, Output = Self> + std::ops::Mul<f64, Output = Self> + Sized
{
/// The neutral element of addition.
const ZERO: Self;
}
impl GLKIntegrable for f64 {
const ZERO: Self = 0f64;
}
/// Type that can be integrated using Gauss Kronrod Quadrature
pub trait GKIntegrable: GLKIntegrable + std::ops::Sub<Self, Output = Self> + Sized + Clone {
/// Returns `true` if this object is neither infinite nor NaN.
fn is_finite(&self) -> bool;
/// Returns the value of the norm used by Gauss Kronrod Quadrature to
/// compute relative tolerances.
fn gk_norm(&self) -> f64;
/// Returns true if this object is less than the tolerance passed as
/// the `tol` argument. By default, returns true if its [`gk_norm`](GKIntegrable::gk_norm)
/// is less than `tol`.
fn is_within_tol(&self, tol: f64) -> bool {
self.gk_norm() < tol
}
}
impl GKIntegrable for f64 {
fn is_finite(&self) -> bool {
f64::is_finite(*self)
}
fn gk_norm(&self) -> f64 {
self.abs()
}
}
/// Numerical Integration
///
/// # Description
/// `fn integrate(f, (a,b), method) -> Y`
///
/// `Y` must implement [`GKIntegrable`] and [`NCIntegrable`], like `f64` or
/// `C64` (`complex` feature only).
///
/// * `f`: Target function (`Fn(f64) -> Y`)
/// * `(a,b)` : Target interval
/// * `method` : Numerical integration method
///
/// # Method
///
/// * Gauss-Legendre Quadrature (up to order 30) : `GaussLegendre(usize)`
/// * Newton-Cotes Quadrature: `NewtonCotes(usize)`
/// * Gauss-Kronrod Quadrature
/// * `G7K15(tol, max_iter)`
/// * `G10K21`
/// * `G15K31`
/// * `G20K41`
/// * `G25K51`
/// * `G30K61`
/// * Gauss-Kronrod Quarature with Relative Error
/// * `G7K15R(rtol, max_iter)`
/// * `G10K21R`
/// * `G15K31R`
/// * `G20K41R`
/// * `G25K51R`
/// * `G30K61R`
pub fn integrate<F, Y>(f: F, (a, b): (f64, f64), method: Integral) -> Y
where
F: Fn(f64) -> Y + Copy,
Y: GKIntegrable + NCIntegrable,
{
match method {
Integral::GaussLegendre(n) => gauss_legendre_quadrature(f, n, (a, b)),
Integral::NewtonCotes(n) => newton_cotes_quadrature(f, n, (a, b)),
method => gauss_kronrod_quadrature(f, (a, b), method),
}
}
/// Newton Cotes Quadrature
pub fn newton_cotes_quadrature<F, Y>(f: F, n: usize, (a, b): (f64, f64)) -> Y
where
F: Fn(f64) -> Y,
Y: NCIntegrable,
{
let h = (b - a) / (n as f64);
let node_x = seq(a, b, h);
let node_y = Y::compute_node_y(f, &node_x);
let p = Y::compute_polynomial(&node_x, &node_y);
let q = Y::integrate_polynomial(&p);
Y::evaluate_polynomial(&q, b) - Y::evaluate_polynomial(&q, a)
}
/// Gauss Legendre Quadrature
///
/// # Type
/// * `f, n, (a,b) -> Y`
/// * `Y` implements [`GLKIntegrable`], like `f64` or `C64` (`complex` feature
/// only)
/// * `f`: Numerical function (`Fn(f64) -> Y`)
/// * `n`: Order of Legendre polynomial (up to 16)
/// * `(a,b)`: Interval of integration
///
/// # Reference
/// * A. N. Lowan et al. (1942)
pub fn gauss_legendre_quadrature<F, Y>(f: F, n: usize, (a, b): (f64, f64)) -> Y
where
F: Fn(f64) -> Y,
Y: GLKIntegrable,
{
unit_gauss_legendre_quadrature(|x| f(x * (b - a) / 2f64 + (a + b) / 2f64), n) * ((b - a) / 2f64)
}
/// Gauss Kronrod Quadrature
///
/// # Type
/// * `f, (a,b), method -> Y`
/// * `Y` implements [`GKIntegrable`], like `f64` or `C64` (`complex` feature
/// only)
/// * `f`: Numerical function (`Fn(f64) -> Y + Copy`)
/// * `(a,b)`: Interval of integration
/// * `method`: Integration method
/// * `G7K15(tol)`
///
/// # Reference
/// * arXiv: [1003.4629](https://arxiv.org/abs/1003.4629)
#[allow(non_snake_case)]
pub fn gauss_kronrod_quadrature<F, T, S, Y>(f: F, (a, b): (T, S), method: Integral) -> Y
where
F: Fn(f64) -> Y + Copy,
T: Into<f64>,
S: Into<f64>,
Y: GKIntegrable,
{
let (g, k) = method.get_gauss_kronrod_order();
let tol = method.get_tol();
let max_iter = method.get_max_iter();
let mut I = Y::ZERO;
let mut S: Vec<(f64, f64, f64, u32)> = vec![];
S.push((a.into(), b.into(), tol, max_iter));
loop {
match S.pop() {
Some((a, b, tol, max_iter)) => {
let G = gauss_legendre_quadrature(f, g as usize, (a, b));
let K = kronrod_quadrature(f, k as usize, (a, b));
let c = (a + b) / 2f64;
let tol_curr = if method.is_relative() {
tol * G.gk_norm()
} else {
tol
};
if (G.clone() - K).is_within_tol(tol_curr) || a == b || max_iter == 0 {
if !G.is_finite() {
return G;
}
I = I + G;
} else {
S.push((a, c, tol / 2f64, max_iter - 1));
S.push((c, b, tol / 2f64, max_iter - 1));
}
}
None => break,
}
}
I
}
pub fn kronrod_quadrature<F, Y>(f: F, n: usize, (a, b): (f64, f64)) -> Y
where
F: Fn(f64) -> Y,
Y: GLKIntegrable,
{
unit_kronrod_quadrature(|x| f(x * (b - a) / 2f64 + (a + b) / 2f64), n) * ((b - a) / 2f64)
}
// =============================================================================
// Gauss Legendre Backends
// =============================================================================
fn unit_gauss_legendre_quadrature<F, Y>(f: F, n: usize) -> Y
where
F: Fn(f64) -> Y,
Y: GLKIntegrable,
{
let (a, x) = gauss_legendre_table(n);
let mut s = Y::ZERO;
for i in 0..a.len() {
s = s + f(x[i]) * a[i];
}
s
}
fn gauss_legendre_table(n: usize) -> (Vec<f64>, Vec<f64>) {
let mut result_root = vec![0f64; n];
let mut result_weight = vec![0f64; n];
let ref_root: &[f64] = match n {
2 => &LEGENDRE_ROOT_2[..],
3 => &LEGENDRE_ROOT_3[..],
4 => &LEGENDRE_ROOT_4[..],
5 => &LEGENDRE_ROOT_5[..],
6 => &LEGENDRE_ROOT_6[..],
7 => &LEGENDRE_ROOT_7[..],
8 => &LEGENDRE_ROOT_8[..],
9 => &LEGENDRE_ROOT_9[..],
10 => &LEGENDRE_ROOT_10[..],
11 => &LEGENDRE_ROOT_11[..],
12 => &LEGENDRE_ROOT_12[..],
13 => &LEGENDRE_ROOT_13[..],
14 => &LEGENDRE_ROOT_14[..],
15 => &LEGENDRE_ROOT_15[..],
16 => &LEGENDRE_ROOT_16[..],
17 => &LEGENDRE_ROOT_17[..],
18 => &LEGENDRE_ROOT_18[..],
19 => &LEGENDRE_ROOT_19[..],
20 => &LEGENDRE_ROOT_20[..],
21 => &LEGENDRE_ROOT_21[..],
22 => &LEGENDRE_ROOT_22[..],
23 => &LEGENDRE_ROOT_23[..],
24 => &LEGENDRE_ROOT_24[..],
25 => &LEGENDRE_ROOT_25[..],
26 => &LEGENDRE_ROOT_26[..],
27 => &LEGENDRE_ROOT_27[..],
28 => &LEGENDRE_ROOT_28[..],
29 => &LEGENDRE_ROOT_29[..],
30 => &LEGENDRE_ROOT_30[..],
_ => panic!("Legendre quadrature is limited up to n = 16"),
};
let ref_weight: &[f64] = match n {
2 => &LEGENDRE_WEIGHT_2[..],
3 => &LEGENDRE_WEIGHT_3[..],
4 => &LEGENDRE_WEIGHT_4[..],
5 => &LEGENDRE_WEIGHT_5[..],
6 => &LEGENDRE_WEIGHT_6[..],
7 => &LEGENDRE_WEIGHT_7[..],
8 => &LEGENDRE_WEIGHT_8[..],
9 => &LEGENDRE_WEIGHT_9[..],
10 => &LEGENDRE_WEIGHT_10[..],
11 => &LEGENDRE_WEIGHT_11[..],
12 => &LEGENDRE_WEIGHT_12[..],
13 => &LEGENDRE_WEIGHT_13[..],
14 => &LEGENDRE_WEIGHT_14[..],
15 => &LEGENDRE_WEIGHT_15[..],
16 => &LEGENDRE_WEIGHT_16[..],
17 => &LEGENDRE_WEIGHT_17[..],
18 => &LEGENDRE_WEIGHT_18[..],
19 => &LEGENDRE_WEIGHT_19[..],
20 => &LEGENDRE_WEIGHT_20[..],
21 => &LEGENDRE_WEIGHT_21[..],
22 => &LEGENDRE_WEIGHT_22[..],
23 => &LEGENDRE_WEIGHT_23[..],
24 => &LEGENDRE_WEIGHT_24[..],
25 => &LEGENDRE_WEIGHT_25[..],
26 => &LEGENDRE_WEIGHT_26[..],
27 => &LEGENDRE_WEIGHT_27[..],
28 => &LEGENDRE_WEIGHT_28[..],
29 => &LEGENDRE_WEIGHT_29[..],
30 => &LEGENDRE_WEIGHT_30[..],
_ => panic!("Legendre quadrature is limited up to n = 16"),
};
match n % 2 {
0 => {
for i in 0..ref_root.len() {
result_root[i] = ref_root[i];
result_weight[i] = ref_weight[i];
}
for i in ref_root.len()..n {
result_root[i] = -ref_root[n - i - 1];
result_weight[i] = ref_weight[n - i - 1];
}
}
1 => {
for i in 0..ref_root.len() {
result_root[i] = ref_root[i];
result_weight[i] = ref_weight[i];
}
for i in ref_root.len()..n {
result_root[i] = -ref_root[n - i];
result_weight[i] = ref_weight[n - i];
}
}
_ => unreachable!(),
}
(result_weight, result_root)
}
// =============================================================================
// Gauss Kronrod Backends
// =============================================================================
fn unit_kronrod_quadrature<F, Y>(f: F, n: usize) -> Y
where
F: Fn(f64) -> Y,
Y: GLKIntegrable,
{
let (a, x) = kronrod_table(n);
let mut s = Y::ZERO;
for i in 0..a.len() {
s = s + f(x[i]) * a[i];
}
s
}
fn kronrod_table(n: usize) -> (Vec<f64>, Vec<f64>) {
let mut result_node = vec![0f64; n];
let mut result_weight = vec![0f64; n];
let ref_node: &[f64] = match n {
15 => &KRONROD_NODES_15[..],
21 => &KRONROD_NODES_21[..],
31 => &KRONROD_NODES_31[..],
41 => &KRONROD_NODES_41[..],
51 => &KRONROD_NODES_51[..],
61 => &KRONROD_NODES_61[..],
_ => panic!("Not yet implemented"),
};
let ref_weight: &[f64] = match n {
15 => &KRONROD_WEIGHTS_15[..],
21 => &KRONROD_WEIGHTS_21[..],
31 => &KRONROD_WEIGHTS_31[..],
41 => &KRONROD_WEIGHTS_41[..],
51 => &KRONROD_WEIGHTS_51[..],
61 => &KRONROD_WEIGHTS_61[..],
_ => panic!("Not yet implemented"),
};
match n % 2 {
0 => {
for i in 0..ref_node.len() {
result_node[i] = ref_node[i];
result_weight[i] = ref_weight[i];
}
for i in ref_node.len()..n {
result_node[i] = -ref_node[n - i - 1];
result_weight[i] = ref_weight[n - i - 1];
}
}
1 => {
for i in 0..ref_node.len() {
result_node[i] = ref_node[i];
result_weight[i] = ref_weight[i];
}
for i in ref_node.len()..n {
result_node[i] = -ref_node[n - i];
result_weight[i] = ref_weight[n - i];
}
}
_ => unreachable!(),
}
(result_weight, result_node)
}
// =============================================================================
// Table for Gauss-Legendre Quadrature (ref. A. N. Lowan et al. (1942))
// =============================================================================
const LEGENDRE_ROOT_2: [f64; 1] = [0.577350269189626];
const LEGENDRE_ROOT_3: [f64; 2] = [0f64, 0.774596669241483];
const LEGENDRE_ROOT_4: [f64; 2] = [0.339981043584856, 0.861136311594053];
const LEGENDRE_ROOT_5: [f64; 3] = [0f64, 0.538469310105683, 0.906179845938664];
const LEGENDRE_ROOT_6: [f64; 3] = [0.238619186083197, 0.661209386466265, 0.932469514203152];
const LEGENDRE_ROOT_7: [f64; 4] = [
0f64,
0.405845151377397,
0.741531185599394,
0.949107912342759,
];
const LEGENDRE_ROOT_8: [f64; 4] = [
0.183434642495650,
0.525532409916329,
0.796666477413627,
0.960289856497536,
];
const LEGENDRE_ROOT_9: [f64; 5] = [
0f64,
0.324253423403809,
0.613371432700590,
0.836031107326636,
0.968160239507626,
];
const LEGENDRE_ROOT_10: [f64; 5] = [
0.148874338981631,
0.433395394129247,
0.679409568299024,
0.865063366688985,
0.973906528517172,
];
const LEGENDRE_ROOT_11: [f64; 6] = [
0f64,
0.269543155952345,
0.519096129110681,
0.730152005574049,
0.887062599768095,
0.978228658146057,
];
const LEGENDRE_ROOT_12: [f64; 6] = [
0.125333408511469,
0.367831498918180,
0.587317954286617,
0.769902674194305,
0.904117256370475,
0.981560634246719,
];
const LEGENDRE_ROOT_13: [f64; 7] = [
0f64,
0.230458315955135,
0.448492751036447,
0.642349339440340,
0.801578090733310,
0.917598399222978,
0.984183054718588,
];
const LEGENDRE_ROOT_14: [f64; 7] = [
0.108054948707344,
0.319112368927890,
0.515248636358154,
0.687292904811685,
0.827201315069765,
0.928434883663574,
0.986283808696812,
];
const LEGENDRE_ROOT_15: [f64; 8] = [
0f64,
0.201194093997435,
0.394151347077563,
0.570972172608539,
0.724417731360170,
0.848206583410427,
0.937273392400706,
0.987992518020485,
];
const LEGENDRE_ROOT_16: [f64; 8] = [
0.095012509837637,
0.281603550779259,
0.458016777657227,
0.617876244402644,
0.755404408355003,
0.865631202387832,
0.944575023073233,
0.989400934991650,
];
const LEGENDRE_ROOT_17: [f64; 9] = [
0f64,
0.178484181495847856,
0.351231763453876315,
0.512690537086476968,
0.657671159216690766,
0.781514003896801407,
0.880239153726985902,
0.950675521768767761,
0.99057547531441734,
];
const LEGENDRE_ROOT_18: [f64; 9] = [
0.084775013041735301,
0.25188622569150551,
0.411751161462842646,
0.559770831073947535,
0.691687043060353208,
0.803704958972523116,
0.892602466497555739,
0.955823949571397755,
0.991565168420930947,
];
const LEGENDRE_ROOT_19: [f64; 10] = [
0f64,
0.160358645640225376,
0.316564099963629832,
0.464570741375960946,
0.600545304661681024,
0.720966177335229379,
0.822714656537142825,
0.903155903614817902,
0.960208152134830031,
0.992406843843584403,
];
const LEGENDRE_ROOT_20: [f64; 10] = [
0.076526521133497334,
0.227785851141645078,
0.373706088715419561,
0.510867001950827098,
0.636053680726515026,
0.746331906460150793,
0.839116971822218823,
0.912234428251325906,
0.963971927277913791,
0.993128599185094925,
];
const LEGENDRE_ROOT_21: [f64; 11] = [
0f64,
0.145561854160895091,
0.288021316802401097,
0.424342120207438784,
0.551618835887219807,
0.667138804197412319,
0.768439963475677909,
0.853363364583317284,
0.920099334150400829,
0.967226838566306294,
0.9937521706203895,
];
const LEGENDRE_ROOT_22: [f64; 11] = [
0.069739273319722221,
0.207860426688221285,
0.341935820892084225,
0.469355837986757026,
0.58764040350691159,
0.69448726318668278,
0.787816805979208162,
0.865812577720300137,
0.926956772187174001,
0.970060497835428727,
0.994294585482399292,
];
const LEGENDRE_ROOT_23: [f64; 12] = [
0f64,
0.133256824298466111,
0.26413568097034493,
0.390301038030290831,
0.50950147784600755,
0.619609875763646156,
0.718661363131950195,
0.804888401618839892,
0.876752358270441667,
0.932971086826016102,
0.972542471218115232,
0.994769334997552124,
];
const LEGENDRE_ROOT_24: [f64; 12] = [
0.064056892862605626,
0.191118867473616309,
0.315042679696163374,
0.433793507626045138,
0.545421471388839536,
0.648093651936975569,
0.740124191578554364,
0.820001985973902922,
0.886415527004401034,
0.938274552002732759,
0.974728555971309498,
0.99518721999702136,
];
const LEGENDRE_ROOT_25: [f64; 13] = [
0f64,
0.122864692610710396,
0.243866883720988432,
0.361172305809387838,
0.47300273144571496,
0.577662930241222968,
0.673566368473468365,
0.759259263037357631,
0.833442628760834001,
0.894991997878275369,
0.942974571228974339,
0.976663921459517512,
0.995556969790498098,
];
const LEGENDRE_ROOT_26: [f64; 13] = [
0.059230093429313207,
0.176858820356890184,
0.292004839485956895,
0.40305175512348631,
0.508440714824505718,
0.606692293017618063,
0.696427260419957265,
0.776385948820678856,
0.845445942788498019,
0.902637861984307074,
0.94715906666171425,
0.97838544595647099,
0.99588570114561693,
];
const LEGENDRE_ROOT_27: [f64; 14] = [
0f64,
0.113972585609529967,
0.226459365439536859,
0.3359939036385089,
0.441148251750026881,
0.54055156457945689,
0.632907971946495141,
0.717013473739423699,
0.791771639070508227,
0.85620790801829449,
0.909482320677491104,
0.950900557814705007,
0.979923475961501223,
0.996179262888988567,
];
const LEGENDRE_ROOT_28: [f64; 14] = [
0.0550792898840342704,
0.164569282133380771,
0.272061627635178078,
0.37625151608907871,
0.475874224955118261,
0.569720471811401719,
0.656651094038864961,
0.735610878013631772,
0.805641370917179171,
0.865892522574395049,
0.915633026392132074,
0.954259280628938197,
0.981303165370872754,
0.99644249757395445,
];
const LEGENDRE_ROOT_29: [f64; 15] = [
0f64,
0.10627823013267923,
0.211352286166001075,
0.314031637867639935,
0.413152888174008664,
0.507592955124227642,
0.59628179713822782,
0.678214537602686515,
0.752462851734477134,
0.81818548761525245,
0.87463780492010279,
0.921180232953058785,
0.957285595778087726,
0.982545505261413175,
0.996679442260596586,
];
const LEGENDRE_ROOT_30: [f64; 15] = [
0.051471842555317696,
0.153869913608583547,
0.254636926167889846,
0.352704725530878113,
0.447033769538089177,
0.536624148142019899,
0.620526182989242861,
0.6978504947933158,
0.767777432104826195,
0.829565762382768397,
0.882560535792052682,
0.926200047429274326,
0.960021864968307512,
0.98366812327974721,
0.99689348407464954,
];
const LEGENDRE_WEIGHT_2: [f64; 1] = [1f64];
const LEGENDRE_WEIGHT_3: [f64; 2] = [0.888888888888889, 0.555555555555556];
const LEGENDRE_WEIGHT_4: [f64; 2] = [0.652145154862546, 0.347854845137454];
const LEGENDRE_WEIGHT_5: [f64; 3] = [0.568888888888889, 0.478628670499366, 0.236926885056189];
const LEGENDRE_WEIGHT_6: [f64; 3] = [0.467913934572691, 0.360761573048139, 0.171324492379170];
const LEGENDRE_WEIGHT_7: [f64; 4] = [
0.417959183673469,
0.381830050505119,
0.279705391489277,
0.129484966168870,
];
const LEGENDRE_WEIGHT_8: [f64; 4] = [
0.362683783378362,
0.313706645877887,
0.222381034453374,
0.101228536290376,
];
const LEGENDRE_WEIGHT_9: [f64; 5] = [
0.330239355001260,
0.312347077040003,
0.260610696402935,
0.180648160694857,
0.081274388361574,
];
const LEGENDRE_WEIGHT_10: [f64; 5] = [
0.295524224714753,
0.269266719309996,
0.219086362515982,
0.149451349150581,
0.066671344308688,
];
const LEGENDRE_WEIGHT_11: [f64; 6] = [
0.272925086777901,
0.262804544510247,
0.233193764591990,
0.186290210927734,
0.125580369464905,
0.055668567116174,
];
const LEGENDRE_WEIGHT_12: [f64; 6] = [
0.249147045813403,
0.233492536538355,
0.203167426723066,
0.160078328543346,
0.106939325995318,
0.047175336386512,
];
const LEGENDRE_WEIGHT_13: [f64; 7] = [
0.232551553230874,
0.226283180262897,
0.207816047536889,
0.178145980761946,
0.138873510219787,
0.092121499837728,
0.040484004765316,
];
const LEGENDRE_WEIGHT_14: [f64; 7] = [
0.215263853463158,
0.205198463721290,
0.185538397477938,
0.157203167158194,
0.121518570687903,
0.080158087159760,
0.035119460331752,
];
const LEGENDRE_WEIGHT_15: [f64; 8] = [
0.202578241925561,
0.198431485327111,
0.186161000015562,
0.166269205816994,
0.139570677926154,
0.107159220467172,
0.070366047488108,
0.030753241996117,
];
const LEGENDRE_WEIGHT_16: [f64; 8] = [
0.189450610455069,
0.182603415044924,
0.169156519395003,
0.149595988816577,
0.124628971255534,
0.095158511682493,
0.062253523938648,
0.027152459411754,
];
const LEGENDRE_WEIGHT_17: [f64; 9] = [
0.179446470356206526,
0.17656270536699265,
0.168004102156450045,
0.154045761076810288,
0.13513636846852547,
0.111883847193403971,
0.085036148317179181,
0.055459529373987201,
0.024148302868547932,
];
const LEGENDRE_WEIGHT_18: [f64; 9] = [
0.169142382963143592,
0.16427648374583272,
0.15468467512626524,
0.140642914670650651,
0.12255520671147846,
0.100942044106287166,
0.07642573025488906,
0.049714548894969796,
0.02161601352648331,
];
const LEGENDRE_WEIGHT_19: [f64; 10] = [
0.161054449848783696,
0.158968843393954348,
0.152766042065859667,
0.14260670217360661,
0.128753962539336228,
0.111566645547333995,
0.091490021622449999,
0.06904454273764123,
0.0448142267656996,
0.019461788229726477,
];
const LEGENDRE_WEIGHT_20: [f64; 10] = [
0.15275338713072585,
0.149172986472603747,
0.142096109318382051,
0.13168863844917663,
0.118194531961518417,
0.10193011981724044,
0.083276741576704749,
0.062672048334109064,
0.040601429800386941,
0.0176140071391521183,
];
const LEGENDRE_WEIGHT_21: [f64; 11] = [
0.146081133649690427,
0.144524403989970059,
0.139887394791073155,
0.13226893863333746,
0.12183141605372853,
0.108797299167148378,
0.093444423456033862,
0.0761001136283793,
0.05713442542685721,
0.036953789770852494,
0.0160172282577743333,
];
const LEGENDRE_WEIGHT_22: [f64; 11] = [
0.139251872855631993,
0.136541498346015171,
0.131173504787062371,
0.123252376810512424,
0.112932296080539218,
0.100414144442880965,
0.085941606217067727,
0.06979646842452049,
0.052293335152683286,
0.033774901584814155,
0.014627995298272201,
];
const LEGENDRE_WEIGHT_23: [f64; 12] = [
0.13365457218610618,
0.13246203940469662,
0.12890572218808215,
0.12304908430672953,
0.114996640222411365,
0.10489209146454141,
0.09291576606003515,
0.079281411776718955,
0.064232421408525852,
0.048037671731084669,
0.030988005856979444,
0.013411859487141772,