rint 0.1.2

mod fully_symmetric_07;
mod fully_symmetric_09;
mod fully_symmetric_09_2d;
mod fully_symmetric_11_3d;
mod fully_symmetric_13_2d;

use crate::multi::generator::Generator;
use crate::multi::two_pow_n_f64;
use crate::InitialisationError;
use crate::ScalarF64;

/// A multi-dimensional fully-symmetric integration rule.
///
/// # Overview
///
/// This provides fully-symmetric numerical integration rules for approximating $N$-dimensional
/// integrals of the form
/// $$
/// I = \int_{\Sigma_{N}} f(\mathbf{x}) d\mathbf{x}
/// $$
/// where $\mathbf{x} = (x_{1},\dots,x_{N})$ is an $N$-dimensional coordinate and $\Sigma_{N}$ is
/// an $N$-dimensional hypercube. A [`Rule`] of a given order $n$ denotes a set of five
/// fully-symmetric rules. The rule with polynomial order $n = 2m + 1$ is used to approximate the
/// integral $I$ using a weighted sum,
/// $$
/// I = \int_{\Sigma_{N}} f(\mathbf{x}) d\mathbf{x}
/// \approx R\[f\] = \sum_{i = 1}^{L} w_{i} f(\mathbf{x}\_{i})
/// $$
/// where $L$ is the total number of evaluation points $\mathbf{x}\_{i} = (x_{1},\dots,x_{N})$ and
/// $w_{i}$ are the rule weights. The remaining four rules are _null rules_ of order $2m-1$,
/// $2m-1$, $2m-3$, and $2m-5$, used to calculate,
/// $$
/// N_{j}\[f\] = \sum_{i = 1}^{L} w_{i}^{j} f(\mathbf{x}\_{i}) ~~~~~~ (j = 1,2,3,4).
/// $$
/// The null-rules are evaluated with the same set of points $\mathbf{x}\_{i}$, however the weights
/// $w_{i}^{j}$ are such that a null rule of degree $d$ will _integrate to zero_ all monomials of
/// degree $\le d$. These are used in the estimation of the error.
///
/// Each rule is defined by a set of _generators_ and weights. The generators are used to generate
/// all of the evaluation points $\mathbf{x}\_{i}$ from a set of base generator _types_. A rule for
/// the $N$-dimensional hypercube $x_{i} \in [-1,+1]^{N}$ is _fully-symmetric_ if it contains all
/// points that can be generated from the base generator type by permutations and/or sign-changes
/// of the coordinates with the same associated weight. The base generator types are:
///
/// <center>
///
/// | Types | Generators | Number of points |
/// | -------- | -------- | -------- |
/// | $(0,0,0,0,\dots,0)$                               | $(0,0,0,0,\dots,0)$ $i = K_{0}$                                                   | $1$ |
/// | $(\alpha,0,0,0,\dots,0)$                          | $(\alpha_{i},0,0,0,\dots,0)$ $i = K_{1}$                                          | $2N$ |
/// | $(\beta,\beta,0,0,\dots,0)$                       | $(\beta_{i},\beta_{i},0,0,\dots,0)$ $i = K_{2}$                                   | $2N(N-1)$ |
/// | $(\gamma,\delta,0,0,\dots,0)$                     | $(\gamma_{i},\delta_{i},0,0,\dots,0)$ $i = K_{3}$                                 | $4N(N-1)$ |
/// | $(\epsilon,\epsilon,\epsilon,0,\dots,0)$          | $(\epsilon_{i},\epsilon_{i},\epsilon_{i},0,\dots,0)$ $i = K_{4}$                  | $4N(N-1)(N-2)/3$ |
/// | $(\zeta,\zeta,\eta,0,\dots,0)$                    | $(\zeta_{i},\zeta_{i},\eta_{i},0,\dots,0)$ $i = K_{5}$                            | $4N(N-1)(N-2)$ |
/// | $(\lambda,\lambda,\lambda,\lambda,\dots,\lambda)$ | $(\lambda_{i},\lambda_{i},\lambda_{i},\lambda_{i},\dots,\lambda_{i})$ $i = K_{6}$ | $2^{N}$ |
///
/// </center>
///
/// The number of unique generators of a particular type is given by the structure parameter
/// $K_{j}$ (note that $K_{0}$ can only take the values $0$ or $1$). As an example, consider an
/// $N=2$ dimensional integral, which contains the generator of type $\mathbf{x}=(\alpha, 0)$ which
/// is associated with a weight $w$. The fully-symmetric rule will evaluate the function
/// $f(\mathbf{x})$ at each of the points $\mathbf{x}\_{1}=(\alpha,0)$,
/// $\mathbf{x}\_{2}=(-\alpha,0)$, $\mathbf{x}\_{3}=(0,\alpha)$, $\mathbf{x}\_{4}=(0,-\alpha)$,
/// weighting each function value with the same $w$. The null rules use the same sets of generators
/// but different weights so error estimation is very inexpensive. For more details on the error
/// estimation algorithm see:
/// - Bernsten, Espelid, & Genz. 1991. An adaptive algorithm for the approximate calculation of
/// multiple integrals. ACM Trans. Math. Softw. 17, 4 (Dec. 1991), 437–451.
/// <https://doi.org/10.1145/210232.210233>
///
///
/// # Available fully-symmetric integration rules
///
/// A number of fully-symmetric integration rules of different order $n$ are provided. The
/// underlying [`Rule`] type should _not_ be required to call directly. Instead, type alias' are
/// provided of the form `RuleX`, with generators `RuleX::generate` to construct them for use with
/// the integrators. There are three rules provided which are only valid for a specific
/// dimensionality $N$. These are,
/// - [`multi::Rule13`]: A 13-point fully symmetric integration rule for functions of $N=2$
/// dimension.
/// - [`multi::Rule11`]: An 11-point fully symmetric integration rule for functions of $N=3$
/// dimension.
/// - [`multi::Rule09N2`]: A 9-point fully symmetric integration rule for functions of $N=2$
/// dimension.
///
/// A further two rules are provided which are more general, with each being suitable for
/// integration in up to $N=15$ dimensions,
/// - [`multi::Rule09`]: A 9-point fully symmetric integration rule for functions of
/// $3 \le N \le 15$ dimension.
/// - [`multi::Rule07`]: A 7-point fully symmetric integration rule for functions of
/// $2 \le N \le 15$ dimension.
///
/// [`multi::Rule07`]: crate::multi::Rule07
/// [`multi::Rule09`]: crate::multi::Rule09
/// [`multi::Rule09N2`]: crate::multi::Rule09N2
/// [`multi::Rule11`]: crate::multi::Rule11
/// [`multi::Rule13`]: crate::multi::Rule13
#[derive(Debug, Clone, PartialEq, PartialOrd)]
pub struct Rule<const N: usize, const FINAL: usize, const TOTAL: usize> {
    initial_data: [Data<N>; 3],
    final_data: [Data<N>; FINAL],
    scales_norms: [ScalesNorms<TOTAL>; 3],
    basic_error_coeff: BasicErrorCoeff,
    adaptive_error_coeff: AdaptiveErrorCoeff,
    evaluations: usize,
    ratio: f64,
}

impl<const N: usize, const FINAL: usize, const TOTAL: usize> Rule<N, FINAL, TOTAL> {
    pub(crate) const fn initial_data(&self) -> &[Data<N>; 3] {
        &self.initial_data
    }

    pub(crate) const fn final_data(&self) -> &[Data<N>; FINAL] {
        &self.final_data
    }

    pub(crate) const fn scales_norms(&self) -> &[ScalesNorms<TOTAL>] {
        &self.scales_norms
    }

    pub(crate) const fn basic_error_coeff(&self) -> &BasicErrorCoeff {
        &self.basic_error_coeff
    }

    pub(crate) const fn adaptive_error_coeff(&self) -> &AdaptiveErrorCoeff {
        &self.adaptive_error_coeff
    }

    pub(crate) const fn evaluations(&self) -> usize {
        self.evaluations
    }

    pub(crate) const fn ratio(&self) -> f64 {
        self.ratio
    }
}

/// A 7-point fully-symmetric integration rule valid for $2 \le N \le 15$.
///
/// This is a general integration [`Rule`] valid for functions with dimensionality $2 \le N \le
/// 15$. The structure parameters $K_{i}$ for the rule generators are,
///
/// <center>
///
/// | $K_{0}$ | $K_{1}$ | $K_{2}$ | $K_{3}$ | $K_{4}$ | $K_{5}$ | $K_{6}$ |
/// | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: |
/// |   $1$   |   $3$   |   $1$   |   $0$   |   $0$   |   $0$   |   $1$   |
///
/// </center>
///
/// While the total number of evaluation points $L$ depends on the dimensionality $N$ via,
/// $$
/// L = 1 + 6N + 2N(N-1) + 2^{N}
/// $$
/// This is the recommended rule to use when a significant amount of adaptivity is required.
///
/// - Genz & Malik, _An imbedded family of fully symmetric numerical integration rules_, SIAM J.
/// Numer. Anal., 20 (1983)
pub type Rule07<const N: usize> = Rule<N, 3, 6>;

impl<const N: usize> Rule07<N> {
    /// Generate a fully-symmetric 7-point integration rule for $2 \le N \le 15$ dimensional integration.
    ///
    /// # Errors
    /// Will fail if $N < 2$ or $N > 15$.
    pub const fn generate() -> Result<Self, InitialisationError> {
        fully_symmetric_07::generate_rule::<N>()
    }
}

/// A 9-point fully-symmetric integration rule valid for $3 \le N \le 15$.
///
/// This is a general integration [`Rule`] valid for functions with dimensionality $3 \le N \le
/// 15$. The structure parameters $K_{i}$ for the rule generators are,
///
/// <center>
///
/// | $K_{0}$ | $K_{1}$ | $K_{2}$ | $K_{3}$ | $K_{4}$ | $K_{5}$ | $K_{6}$ |
/// | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: |
/// |   $1$   |   $4$   |   $2$   |   $1$   |   $0$   |   $0$   |   $0$   |
///
/// </center>
///
/// While the total number of evaluation points $L$ depends on the dimensionality $N$ via,
/// $$
/// L = 1 + 8N + 6N(N-1) + 4N(N-1)(N-2)/3 + 2^{N}
/// $$
///
/// - Genz & Malik, _An imbedded family of fully symmetric numerical integration rules_, SIAM J.
/// Numer. Anal., 20 (1983)
pub type Rule09<const N: usize> = Rule<N, 6, 9>;

impl<const N: usize> Rule09<N> {
    /// Generate a fully-symmetric 9-point integration rule for $3 \le N \le 15$ dimensional integration.
    ///
    /// # Errors
    /// Will fail if $N < 3$ or $N > 15$.
    pub const fn generate() -> Result<Self, InitialisationError> {
        fully_symmetric_09::generate_rule::<N>()
    }
}

/// A 9-point fully-symmetric integration rule valid for $N = 2$.
///
/// This [`Rule`] is only valid for functions with $N=2$ dimensions. The structure parameters
/// $K_{i}$ and total number of evaluation points $L$ for the rule generators are,
///
/// <center>
///
/// | $K_{0}$ | $K_{1}$ | $K_{2}$ | $K_{3}$ | $K_{4}$ | $K_{5}$ | $K_{6}$ | $L$ |
/// | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-: |
/// |   $1$   |   $4$   |   $2$   |   $1$   |   $0$   |   $0$   |   $0$   | $33$ |
///
/// </center>
///
/// - Genz & Malik, _An imbedded family of fully symmetric numerical integration rules_, SIAM J.
/// Numer. Anal., 20 (1983)
pub type Rule09N2 = Rule<2, 5, 8>;

impl Rule09N2 {
    /// Generate a fully-symmetric 9-point integration rule for $N = 2$ dimensional integration.
    #[must_use]
    pub const fn generate() -> Self {
        fully_symmetric_09_2d::generate_rule()
    }
}

/// A 11-point fully-symmetric integration rule valid for $N = 3$.
///
/// This [`Rule`] is only valid for functions with $N=3$ dimensions. The structure parameters
/// $K_{i}$ and total number of evaluation points $L$ for the rule generators are,
///
/// <center>
///
/// | $K_{0}$ | $K_{1}$ | $K_{2}$ | $K_{3}$ | $K_{4}$ | $K_{5}$ | $K_{6}$ | $L$ |
/// | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-: |
/// |   $1$   |   $5$   |   $2$   |   $0$   |   $3$   |   $2$   |   $0$   | $127$ |
///
/// </center>
///
/// - Bernsten & Espelid, _On the construction of higher degree three dimensional embedded
/// integration rules_, Reports in Informatics 16, Dept. of Informatics. Univ. of Bergen, 1985.
/// - Bernsten & Espelid, _On the construction of higher degree three-dimensional embedded
/// integration rules_, SIAM J. Numer. Anal. 25, 1 (1988)
pub type Rule11 = Rule<3, 10, 13>;

impl Rule11 {
    /// Generate a fully-symmetric 11-point integration rule for $N = 3$ dimensional integration.
    #[must_use]
    pub const fn generate() -> Self {
        fully_symmetric_11_3d::generate_rule()
    }
}

/// A 13-point fully-symmetric integration rule valid for $N = 2$.
///
/// This is the highest polynomial order [`Rule`] available for multi-dimensional integration, and
/// is only valid for functions with $N=2$ dimensions. The structure parameters $K_{i}$ and total
/// number of evaluation points $L$ for the rule generators are,
///
/// <center>
///
/// | $K_{0}$ | $K_{1}$ | $K_{2}$ | $K_{3}$ | $K_{4}$ | $K_{5}$ | $K_{6}$ |   $L$   |
/// | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: | :-----: |
/// |   $1$   |   $5$   |   $5$   |   $3$   |   $0$   |   $0$   |   $0$   |   $65$  |
///
/// </center>
///
/// - Eriksen, _On the development of embedded fully symetric quadrature rules fpr the square_,
/// Thesis for the degree Cand. Scient., Dept. of Informatics, Univ. of Bergen, 1986.
pub type Rule13 = Rule<2, 11, 14>;

impl Rule13 {
    /// Generate a fully-symmetric 13-point integration rule for $N = 2$ dimensional integration.
    #[must_use]
    pub const fn generate() -> Self {
        fully_symmetric_13_2d::generate_rule()
    }
}

/// A generator and its associated weights
///
/// Data structure containing a generator and the five weights associated with it.
#[derive(Debug, Clone, PartialEq, PartialOrd)]
pub(crate) struct Data<const N: usize> {
    generator: Generator<N>,
    weight: f64,
    null1: f64,
    null2: f64,
    null3: f64,
    null4: f64,
}

impl<const N: usize> Data<N> {
    /// Create a new instance of [`Data`]
    pub(crate) const fn new(generator: Generator<N>, weights: [f64; 5]) -> Self {
        let [weight, null1, null2, null3, null4] = weights;
        Self {
            generator,
            weight,
            null1,
            null2,
            null3,
            null4,
        }
    }

    pub(crate) const fn generator(&self) -> &Generator<N> {
        &self.generator
    }

    pub(crate) fn get_first_value(&self) -> f64 {
        // unwrap is fine as generators are never empty
        *self.generator.generator().first().unwrap()
    }

    pub(crate) const fn weight(&self) -> f64 {
        self.weight
    }

    pub(crate) const fn null1(&self) -> f64 {
        self.null1
    }

    pub(crate) const fn null2(&self) -> f64 {
        self.null2
    }

    pub(crate) const fn null3(&self) -> f64 {
        self.null3
    }

    pub(crate) const fn null4(&self) -> f64 {
        self.null4
    }
}

#[derive(Debug, Clone, PartialEq, PartialOrd)]
pub(crate) struct ScalesNorms<const TOTAL: usize> {
    scales: [f64; TOTAL],
    norms: [f64; TOTAL],
}

impl<const TOTAL: usize> ScalesNorms<TOTAL> {
    const fn new(scales: [f64; TOTAL], norms: [f64; TOTAL]) -> Self {
        Self { scales, norms }
    }

    const fn scales(&self) -> &[f64] {
        &self.scales
    }

    const fn norms(&self) -> &[f64] {
        &self.norms
    }

    pub(crate) fn max_consecutive_null<T: ScalarF64>(&self, null1: T, null2: T) -> T {
        let scales = self.scales();
        let norms = self.norms();

        scales
            .iter()
            .zip(norms)
            .map(|(s, n)| (null1 * s + null2) * n)
            .fold(T::zero(), |a, b| if a.abs() < b.abs() { b } else { a })
    }
}

pub(crate) const fn scales_norms<const N: usize, const TOTAL: usize>(
    weights: &[[f64; 5]; TOTAL],
    rule_points: [f64; TOTAL],
) -> [ScalesNorms<TOTAL>; 3] {
    let two_ndim = two_pow_n_f64(N);

    let mut scales = [[0f64; TOTAL]; 3];
    let mut norms = [[0f64; TOTAL]; 3];
    let mut we = [0f64; 14];

    let mut i = 0;
    while i < TOTAL {
        let mut k = 0;
        while k < 3 {
            scales[k][i] = if weights[i][k + 1] != 0.0 {
                -weights[i][k + 2] / weights[i][k + 1]
            } else {
                100.0
            };
            let mut j = 0;
            while j < TOTAL {
                we[j] = weights[j][k + 2] + scales[k][i] * weights[j][k + 1];
                j += 1;
            }
            norms[k][i] = 0.0;
            let mut j = 0;
            while j < TOTAL {
                let weabs = we[j].abs();

                norms[k][i] += rule_points[j] * weabs;
                j += 1;
            }
            norms[k][i] = two_ndim / norms[k][i];

            k += 1;
        }
        i += 1;
    }

    let scales_norms0 = ScalesNorms::new(scales[0], norms[0]);
    let scales_norms1 = ScalesNorms::new(scales[1], norms[1]);
    let scales_norms2 = ScalesNorms::new(scales[2], norms[2]);

    [scales_norms0, scales_norms1, scales_norms2]
}

#[derive(Debug, Clone, Copy, PartialEq, PartialOrd)]
pub(crate) struct BasicErrorCoeff {
    c1: f64,
    c2: f64,
    c3: f64,
    c4: f64,
}

impl BasicErrorCoeff {
    pub(crate) const fn new(c1: f64, c2: f64, c3: f64, c4: f64) -> Self {
        Self { c1, c2, c3, c4 }
    }
    #[inline]
    pub(crate) const fn c1(&self) -> f64 {
        self.c1
    }

    #[inline]
    pub(crate) const fn c2(&self) -> f64 {
        self.c2
    }

    #[inline]
    pub(crate) const fn c3(&self) -> f64 {
        self.c3
    }

    #[inline]
    pub(crate) const fn c4(&self) -> f64 {
        self.c4
    }
}

#[derive(Debug, Clone, Copy, PartialEq, PartialOrd)]
pub(crate) struct AdaptiveErrorCoeff {
    c5: f64,
    c6: f64,
}

impl AdaptiveErrorCoeff {
    pub(crate) const fn new(c5: f64, c6: f64) -> Self {
        Self { c5, c6 }
    }

    #[inline]
    pub(crate) const fn c5(&self) -> f64 {
        self.c5
    }

    #[inline]
    pub(crate) const fn c6(&self) -> f64 {
        self.c6
    }
}

const ADAPTIVE_ERROR_COEFF: AdaptiveErrorCoeff = AdaptiveErrorCoeff::new(0.5, 0.25);

#[cfg(test)]
mod util {
    use super::*;

    pub(crate) fn rel_or_abs_diff(a: f64, b: f64) -> f64 {
        if a == 0.0 {
            (a - b).abs()
        } else {
            (a - b).abs() / a.abs()
        }
    }

    pub(crate) fn assert_check_vec_tol<const WL: usize, const TY: usize>(
        calc: &[[f64; TY]; WL],
        should_be: &[[f64; TY]; WL],
        tol: f64,
    ) {
        for (x, y) in calc.iter().zip(should_be.iter()) {
            for (a, b) in x.iter().zip(y.iter()) {
                let val = rel_or_abs_diff(*a, *b);

                assert!(val < tol);
            }
        }
    }

    pub(crate) fn assert_check_slice_tol(calc: &[f64], should_be: &[f64], tol: f64) {
        for (x, y) in calc.iter().zip(should_be.iter()) {
            let val = rel_or_abs_diff(*x, *y);

            assert!(val < tol);
        }
    }

    #[allow(clippy::similar_names)]
    pub(crate) fn assert_check_vec_data_tol<const WL: usize, const TY: usize>(
        calc: &[Data<TY>; WL],
        should_be: &[Data<TY>; WL],
        tol: f64,
    ) {
        for (x, y) in calc.iter().zip(should_be.iter()) {
            let genx = x.generator().generator();
            let geny = y.generator().generator();
            for (gx, gy) in genx.iter().zip(geny.iter()) {
                let val = rel_or_abs_diff(*gx, *gy);
                assert!(val < tol);
            }

            let wx = x.weight();
            let wy = y.weight();
            let val = rel_or_abs_diff(wx, wy);
            assert!(val < tol);

            let n1x = x.null1();
            let n1y = y.null1();
            let val = rel_or_abs_diff(n1x, n1y);
            assert!(val < tol);

            let n2x = x.null2();
            let n2y = y.null2();
            let val = rel_or_abs_diff(n2x, n2y);
            assert!(val < tol);

            let n3x = x.null3();
            let n3y = y.null3();
            let val = rel_or_abs_diff(n3x, n3y);
            assert!(val < tol);

            let n4x = x.null4();
            let n4y = y.null4();
            let val = rel_or_abs_diff(n4x, n4y);
            assert!(val < tol);
        }
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn error_when_wrong_dimension() {
        {
            const N: usize = 1;
            let err = Rule07::<N>::generate();
            assert!(err.is_err());
        }
        {
            const N: usize = 2;
            let err = Rule07::<N>::generate();
            assert!(err.is_ok());
        }
        {
            const N: usize = 8;
            let err = Rule07::<N>::generate();
            assert!(err.is_ok());
        }
        {
            const N: usize = 15;
            let err = Rule07::<N>::generate();
            assert!(err.is_ok());
        }
        {
            const N: usize = 16;
            let err = Rule07::<N>::generate();
            assert!(err.is_err());
        }

        {
            const N: usize = 1;
            let err = Rule09::<N>::generate();
            assert!(err.is_err());
        }
        {
            const N: usize = 2;
            let err = Rule09::<N>::generate();
            assert!(err.is_err());
        }
        {
            const N: usize = 8;
            let err = Rule09::<N>::generate();
            assert!(err.is_ok());
        }
        {
            const N: usize = 15;
            let err = Rule09::<N>::generate();
            assert!(err.is_ok());
        }
        {
            const N: usize = 16;
            let err = Rule09::<N>::generate();
            assert!(err.is_err());
        }
    }
}