rlst 0.8.0

//! Functions for the evaluation and conversion of Chebychev series
//!
//! This module contains functions to
//! - Evaluate a Chebychev series
//! - Convert between data at Chebychev points and series coefficients
//! - Take the derivative of Chebychev expansions
//!
//! All routines are available for 1, 2, or 3-dimensional Chebychev expansions.
//! In 2 and 3 dimensions the Chebychev expansions are assumed to be defined as tensors
//! of direct products of 1-dimensional expansions.

use std::ops::{DivAssign, MulAssign};

use itertools::izip;
use num::PrimInt;

use crate::dense::base_array::BaseArray;
use crate::dense::data_container::ArrayContainer;
use crate::fftw::{FftPlanInplace, R2RPlanKind};
use crate::{Array, fftw::R2RPlan};
use crate::{DynArray, MutableArrayImpl, RlstScalar, ValueArrayImpl};
use crate::{c32, c64};

/// Convert between tensor-grid data and Chebychev coefficients.
///
/// Implemented for one-, two-, and three-dimensional arrays. Every axis uses
/// second-kind points and its array extent determines the number of points.
pub trait ChebychevCoefficientsToData<Item, const NDIM: usize> {
    /// Compute Chebychev coefficients from values at Chebychev points
    ///
    /// Given a 1, 2, or 3-dimensional array `arr`. This function interprets
    /// the contents of the array as data at Chebychev points of the second kind
    /// and computes the associated series expansion coefficients.
    fn chebychev_coeffs_from_data_second_kind(&self) -> DynArray<Item, NDIM>;
    /// Compute data at Chebychev points from series coefficients
    ///
    /// Given a 1, 2, or 3-dimensional array `arr`. This function interprets
    /// the contents of the array as Chebychev expansion coefficients and
    /// computes the associated values at Chebychev points of the second kind.
    fn chebychev_data_from_coeffs_second_kind(&self) -> DynArray<Item, NDIM>;
}

/// Evaluation of Chebychev Coefficients in one dimension
pub trait EvaluateChebychevCoefficients1d<Item>
where
    Item: RlstScalar,
{
    /// Evaluate a 1d array assuming that its entries are Chebychev coefficients
    ///
    /// The implementation uses the [Clenshaw algorithm](https://en.wikipedia.org/wiki/Clenshaw_algorithm).
    fn evaluate_chebychev_series_second_kind<ArrayImplPoints, ArrayImplResult>(
        &self,
        eval_points: &Array<ArrayImplPoints, 1>,
        result: &mut Array<ArrayImplResult, 1>,
    ) where
        ArrayImplPoints: ValueArrayImpl<Item::Real, 1>,
        ArrayImplResult: MutableArrayImpl<Item, 1>;
}

/// Evaluation of Chebychev Coefficients in two dimensions
pub trait EvaluateChebychevCoefficients2d<Item>
where
    Item: RlstScalar,
{
    /// Evaluate a 2d Chebychev expansion at an array of points
    ///
    /// The points are stored as columns of the `(2, npts)` array `eval_points`.
    /// The output array is a one-dimensional array of length `npts`.
    fn evaluate_chebychev_series_second_kind<ArrayImplPoints, ArrayImplResult>(
        &self,
        eval_points: &Array<ArrayImplPoints, 2>,
        result: &mut Array<ArrayImplResult, 1>,
    ) where
        ArrayImplPoints: ValueArrayImpl<Item::Real, 2>,
        ArrayImplResult: MutableArrayImpl<Item, 1>;
    /// Evaluate a 2d Chebychev expansion at the two-dimensional tensor product of 1d coordinates
    ///
    /// The evaluation points are assumed to be direct products of the points `eval_points_x` and
    /// `eval_points_y` in the x and y directions.
    /// The implementation uses the [Clenshaw algorithm](https://en.wikipedia.org/wiki/Clenshaw_algorithm).
    fn tensor_evaluate_chebychev_series_second_kind<
        ArrayImplPointsX,
        ArrayImplPointsY,
        ArrayImplResult,
    >(
        &self,
        eval_points_x: &Array<ArrayImplPointsX, 1>,
        eval_points_y: &Array<ArrayImplPointsY, 1>,
        result: &mut Array<ArrayImplResult, 2>,
    ) where
        ArrayImplPointsX: ValueArrayImpl<Item::Real, 1>,
        ArrayImplPointsY: ValueArrayImpl<Item::Real, 1>,
        ArrayImplResult: MutableArrayImpl<Item, 2>;
}

/// Evaluation of Chebychev coefficients in three dimensions
pub trait EvaluateChebychevCoefficients3d<Item>
where
    Item: RlstScalar,
{
    /// Evaluate a 3d Chebychev expansion at an array of points
    ///
    /// The points are stored as columns of the `(3, npts)` array `eval_points`.
    /// The output array is a one-dimensional array of length `npts`.
    fn evaluate_chebychev_series_second_kind<ArrayImplPoints, ArrayImplResult>(
        &self,
        eval_points: &Array<ArrayImplPoints, 2>,
        result: &mut Array<ArrayImplResult, 1>,
    ) where
        ArrayImplPoints: ValueArrayImpl<Item::Real, 2>,
        ArrayImplResult: MutableArrayImpl<Item, 1>;

    /// Evaluate a 3d Chebychev expansion at the three-dimensional tensor product of 1d coordinates
    ///
    /// The implementation uses the [Clenshaw algorithm](https://en.wikipedia.org/wiki/Clenshaw_algorithm).
    fn tensor_evaluate_chebychev_series_second_kind<
        ArrayImplPointsX,
        ArrayImplPointsY,
        ArrayImplPointsZ,
        ArrayImplResult,
    >(
        &self,
        eval_points_x: &Array<ArrayImplPointsX, 1>,
        eval_points_y: &Array<ArrayImplPointsY, 1>,
        eval_points_z: &Array<ArrayImplPointsZ, 1>,
        result: &mut Array<ArrayImplResult, 3>,
    ) where
        ArrayImplPointsX: ValueArrayImpl<Item::Real, 1>,
        ArrayImplPointsY: ValueArrayImpl<Item::Real, 1>,
        ArrayImplPointsZ: ValueArrayImpl<Item::Real, 1>,
        ArrayImplResult: MutableArrayImpl<Item, 3>;
}

/// Compute the series coefficients of the derivative of a Chebychev series
pub trait DerivativeOfChebychevSeries1d<Item> {
    /// Compute the series coefficients of the derivative of a chebychev series of the second kind.
    ///
    /// The coefficients of the derivative overwrite the original array without changing its shape.
    /// Coefficients above the differentiated polynomial's degree become zero.
    /// The parameter `order` determines how many derivatives are taken. `order=1` computes
    /// the first derivative and `order=0` leaves the coefficients unchanged.
    fn chebychev_derivative_second_kind(&mut self, order: usize);
}

/// Compute the series coefficients of the derivative of a two-dimensional Chebychev series
pub trait DerivativeOfChebychevSeries2d<Item> {
    /// Compute the series coefficients of the derivative of a two dimensional Chebychev series of the second kind.
    ///
    /// The coefficients of the derivative overwrite the original array.
    /// The parameter `order` determines how many derivatives are taken. `order=[0, 2]` means
    /// that the output has no derivative along the x-axis and two derivatives
    /// along the y-axis.
    fn chebychev_derivative_second_kind(&mut self, order: [usize; 2]);
}

/// Compute the series coefficients of the derivative of a three-dimensional Chebychev series
pub trait DerivativeOfChebychevSeries3d<Item> {
    /// Compute the series coefficients of the derivative of a three-dimensional Chebychev series of the second kind.
    ///
    /// The coefficients of the derivative overwrite the original array.
    /// The parameter `order` determines how many derivatives are taken. `order=[0, 2, 1]` means
    /// that the output has no derivative along the x-axis, two derivatives
    /// along the y-axis and one derivative along the z axis.
    fn chebychev_derivative_second_kind(&mut self, order: [usize; 3]);
}

impl<Item, ArrayImpl> DerivativeOfChebychevSeries1d<Item> for Array<ArrayImpl, 1>
where
    Item: RlstScalar,
    ArrayImpl: MutableArrayImpl<Item, 1>,
{
    #[inline(always)]
    fn chebychev_derivative_second_kind(&mut self, order: usize) {
        let n = self.len();

        // If the order is zero there is nothing to do
        if order == 0 {
            return;
        }

        // We repeat the derivative loop `order` times

        for _ in 0..order {
            let mut bkp1 = Item::default(); // The series element k + 1;
            let mut dp1 = Item::default(); // The cofficient d_{k+1} of the derivative series.
            let mut dp2 = Item::default(); // The coefficient d_{k+2} of the derivative series.
            let two = Item::from_f64(2.0).unwrap();
            let half = two.recip();

            for (index, d) in self
                .r_mut()
                .reverse_axis(0)
                .iter_mut()
                .take(n - 1)
                .enumerate()
            {
                // The coefficients up to 1
                let kp1 = Item::from_usize(n - index).unwrap();
                let tmp = *d;
                *d = dp2 + two * kp1 * bkp1;
                bkp1 = tmp;
                (dp1, dp2) = (*d, dp1); // We need this in the next iteration
            }

            // Finally, the zero coefficient.

            self[[0]] = bkp1 + half * dp2;
        }
    }
}

impl<Item, ArrayImpl> DerivativeOfChebychevSeries2d<Item> for Array<ArrayImpl, 2>
where
    Item: RlstScalar,
    ArrayImpl: MutableArrayImpl<Item, 2>,
{
    fn chebychev_derivative_second_kind(&mut self, order: [usize; 2]) {
        // We first compute the derivatives in the x-direction.
        // The axes parameter `multislice` is 1 for this since we evaluate for each fixed y-value the derivative over all x-values

        for (_, mut slice) in self.multislice_iter_mut::<1, 1>([1]) {
            slice.chebychev_derivative_second_kind(order[0]);
        }

        // And now the derivatives in the y-direction
        for (_, mut slice) in self.multislice_iter_mut::<1, 1>([0]) {
            slice.chebychev_derivative_second_kind(order[1]);
        }
    }
}

impl<Item, ArrayImpl> DerivativeOfChebychevSeries3d<Item> for Array<ArrayImpl, 3>
where
    Item: RlstScalar,
    ArrayImpl: MutableArrayImpl<Item, 3>,
{
    fn chebychev_derivative_second_kind(&mut self, order: [usize; 3]) {
        // We first compute the derivatives in the x-direction.
        // The axes parameter `multislice` is 1 for this since we evaluate for each fixed y-value the derivative over all x-values

        for (_, mut slice) in self.multislice_iter_mut::<2, 1>([1, 2]) {
            slice.chebychev_derivative_second_kind(order[0]);
        }

        // And now the derivatives in the y-direction
        for (_, mut slice) in self.multislice_iter_mut::<2, 1>([0, 2]) {
            slice.chebychev_derivative_second_kind(order[1]);
        }

        // And finally in the z direction
        for (_, mut slice) in self.multislice_iter_mut::<2, 1>([0, 1]) {
            slice.chebychev_derivative_second_kind(order[2]);
        }
    }
}

impl<Item, ArrayImpl> EvaluateChebychevCoefficients1d<Item> for Array<ArrayImpl, 1>
where
    Item: RlstScalar,
    ArrayImpl: ValueArrayImpl<Item, 1>,
{
    fn evaluate_chebychev_series_second_kind<ArrayImplPoints, ArrayImplResult>(
        &self,
        eval_points: &Array<ArrayImplPoints, 1>,
        result: &mut Array<ArrayImplResult, 1>,
    ) where
        ArrayImplPoints: ValueArrayImpl<Item::Real, 1>,
        ArrayImplResult: MutableArrayImpl<Item, 1>,
    {
        // Make sure that the arrays make sense
        assert!(!self.is_empty());
        assert_eq!(eval_points.len(), result.len());

        // The degree of the Chebychev expansion
        let n = self.len() - 1;
        let mut bkp1;
        let mut bkp2;
        let a0 = self.get_value([0]).unwrap();

        let two = Item::from_f64(2.0).unwrap();

        for (x, r) in izip!(eval_points.iter_value(), result.iter_mut()) {
            bkp1 = Item::default();
            bkp2 = Item::default();
            // We only do the coefficients a_n up to a_1.
            // The coefficient a_0 is treated separately
            let x = Item::from_real(x);
            for cheb_coeff in self.r().reverse_axis(0).iter_value().take(n) {
                (bkp1, bkp2) = (cheb_coeff + two * x * bkp1 - bkp2, bkp1);
            }
            *r = a0 + x * bkp1 - bkp2;
        }
    }
}

impl<Item, ArrayImpl> EvaluateChebychevCoefficients2d<Item> for Array<ArrayImpl, 2>
where
    Item: RlstScalar,
    ArrayImpl: ValueArrayImpl<Item, 2>,
{
    fn evaluate_chebychev_series_second_kind<ArrayImplPoints, ArrayImplResult>(
        &self,
        eval_points: &Array<ArrayImplPoints, 2>,
        result: &mut Array<ArrayImplResult, 1>,
    ) where
        ArrayImplPoints: ValueArrayImpl<Item::Real, 2>,
        ArrayImplResult: MutableArrayImpl<Item, 1>,
    {
        // We check that the dimensions of `eval_points` and `result` are compatible.

        let points_shape = eval_points.shape();
        let npoints = points_shape[1];
        let points_dim = points_shape[0];

        assert_eq!(npoints, result.len());
        assert_eq!(points_dim, 2);

        // Now evaluate the points successively along each dimension

        let mut p0 =
            Array::<_, 1>::new(BaseArray::new(ArrayContainer::<Item::Real, 1>::new(), [1]));
        let mut p1 =
            Array::<_, 1>::new(BaseArray::new(ArrayContainer::<Item::Real, 1>::new(), [1]));
        let mut res = Array::<_, 2>::new(BaseArray::new(ArrayContainer::<Item, 1>::new(), [1, 1]));

        // For each point we evaluate along all dimensions

        for (point, r) in izip!(eval_points.col_iter(), result.iter_mut()) {
            p0[[0]] = point.get_value([0]).unwrap();
            p1[[0]] = point.get_value([1]).unwrap();
            self.tensor_evaluate_chebychev_series_second_kind(&p0, &p1, &mut res);

            *r = res[[0, 0]];
        }
    }

    fn tensor_evaluate_chebychev_series_second_kind<
        ArrayImplPointsX,
        ArrayImplPointsY,
        ArrayImplResult,
    >(
        &self,
        eval_points_x: &Array<ArrayImplPointsX, 1>,
        eval_points_y: &Array<ArrayImplPointsY, 1>,
        result: &mut Array<ArrayImplResult, 2>,
    ) where
        ArrayImplPointsX: ValueArrayImpl<Item::Real, 1>,
        ArrayImplPointsY: ValueArrayImpl<Item::Real, 1>,
        ArrayImplResult: MutableArrayImpl<Item, 2>,
    {
        // Check that the dimensions are correct

        let [mx, my] = result.shape();

        let [_nx, ny] = self.shape();

        assert_eq!(mx, eval_points_x.len());
        assert_eq!(my, eval_points_y.len());

        // We need to create a temporary matrix that holds the intermediate result
        // of transforming along one direction.

        let mut tmp = DynArray::<Item, 2>::from_shape([mx, ny]);

        // Now iterate through the columns of the input matrix and evaluate each column into tmp

        for (col_in, mut col_tmp) in izip!(self.col_iter(), tmp.col_iter_mut()) {
            col_in.evaluate_chebychev_series_second_kind(eval_points_x, &mut col_tmp);
        }

        // And now we iterate through the rows and evaluate along y to get the result for each row.
        // This is stored in the output array.

        for (row_tmp, mut row_res) in izip!(tmp.row_iter(), result.row_iter_mut()) {
            row_tmp.evaluate_chebychev_series_second_kind(eval_points_y, &mut row_res);
        }
    }
}

impl<Item, ArrayImpl> EvaluateChebychevCoefficients3d<Item> for Array<ArrayImpl, 3>
where
    Item: RlstScalar,
    ArrayImpl: ValueArrayImpl<Item, 3>,
{
    fn tensor_evaluate_chebychev_series_second_kind<
        ArrayImplPointsX,
        ArrayImplPointsY,
        ArrayImplPointsZ,
        ArrayImplResult,
    >(
        &self,
        eval_points_x: &Array<ArrayImplPointsX, 1>,
        eval_points_y: &Array<ArrayImplPointsY, 1>,
        eval_points_z: &Array<ArrayImplPointsZ, 1>,
        result: &mut Array<ArrayImplResult, 3>,
    ) where
        ArrayImplPointsX: ValueArrayImpl<Item::Real, 1>,
        ArrayImplPointsY: ValueArrayImpl<Item::Real, 1>,
        ArrayImplPointsZ: ValueArrayImpl<Item::Real, 1>,
        ArrayImplResult: MutableArrayImpl<Item, 3>,
    {
        // Check that the dimensions are correct

        let [mx, my, mz] = result.shape();

        let [_nx, ny, nz] = self.shape();

        assert_eq!(mx, eval_points_x.len());
        assert_eq!(my, eval_points_y.len());
        assert_eq!(mz, eval_points_z.len());

        // We first evaluate along the x direction.

        let mut tmp_x = DynArray::<Item, 3>::from_shape([mx, ny, nz]);

        for (indices, slice) in self.r().multislice_iter::<_, 1>([1, 2]) {
            slice.evaluate_chebychev_series_second_kind(
                eval_points_x,
                &mut tmp_x.r_mut().multislice([1, 2], indices),
            );
        }

        // Now we evaluate along the y direction

        let mut tmp_y = DynArray::<Item, 3>::from_shape([mx, my, nz]);

        for (indices, slice) in tmp_x.r().multislice_iter::<_, 1>([0, 2]) {
            slice.evaluate_chebychev_series_second_kind(
                eval_points_y,
                &mut tmp_y.r_mut().multislice([0, 2], indices),
            );
        }

        // And now along the z direction

        for (indices, slice) in tmp_y.r().multislice_iter::<_, 1>([0, 1]) {
            slice.evaluate_chebychev_series_second_kind(
                eval_points_z,
                &mut result.r_mut().multislice([0, 1], indices),
            );
        }
    }

    fn evaluate_chebychev_series_second_kind<ArrayImplPoints, ArrayImplResult>(
        &self,
        eval_points: &Array<ArrayImplPoints, 2>,
        result: &mut Array<ArrayImplResult, 1>,
    ) where
        ArrayImplPoints: ValueArrayImpl<<Item as RlstScalar>::Real, 2>,
        ArrayImplResult: MutableArrayImpl<Item, 1>,
    {
        // We check that the dimensions of `eval_points` and `result` are compatible.

        let points_shape = eval_points.shape();
        let npoints = points_shape[1];
        let points_dim = points_shape[0];

        assert_eq!(npoints, result.len());
        assert_eq!(points_dim, 3);

        // Now evaluate the points successively along each dimension

        let mut p0 =
            Array::<_, 1>::new(BaseArray::new(ArrayContainer::<Item::Real, 1>::new(), [1]));
        let mut p1 =
            Array::<_, 1>::new(BaseArray::new(ArrayContainer::<Item::Real, 1>::new(), [1]));
        let mut p2 =
            Array::<_, 1>::new(BaseArray::new(ArrayContainer::<Item::Real, 1>::new(), [1]));
        let mut res =
            Array::<_, 3>::new(BaseArray::new(ArrayContainer::<Item, 1>::new(), [1, 1, 1]));

        // For each point we evaluate along all dimensions

        for (point, r) in izip!(eval_points.col_iter(), result.iter_mut()) {
            p0[[0]] = point.get_value([0]).unwrap();
            p1[[0]] = point.get_value([1]).unwrap();
            p2[[0]] = point.get_value([2]).unwrap();
            self.tensor_evaluate_chebychev_series_second_kind(&p0, &p1, &p2, &mut res);

            *r = res[[0, 0, 0]];
        }
    }
}

macro_rules! impl_cheb_coeffs_complex {
    ($dtype:ty) => {
        impl<ArrayImpl, const NDIM: usize> ChebychevCoefficientsToData<$dtype, NDIM>
            for Array<ArrayImpl, NDIM>
        where
            ArrayImpl: ValueArrayImpl<$dtype, NDIM>,
        {
            fn chebychev_coeffs_from_data_second_kind(&self) -> DynArray<$dtype, NDIM> {
                let shape = self.shape();
                let mut re = DynArray::<<$dtype as RlstScalar>::Real, NDIM>::from_shape(shape);
                let mut im = DynArray::<<$dtype as RlstScalar>::Real, NDIM>::from_shape(shape);

                re.fill_from(&self.r().unary_op(|elem: $dtype| elem.re()));
                im.fill_from(&self.r().unary_op(|elem: $dtype| elem.im()));

                let re_coeffs = re.chebychev_coeffs_from_data_second_kind();
                let im_coeffs = im.chebychev_coeffs_from_data_second_kind();

                let mut out = DynArray::<$dtype, NDIM>::from_shape(shape);
                for (o, r, i) in izip!(
                    out.iter_mut(),
                    re_coeffs.iter_value(),
                    im_coeffs.iter_value()
                ) {
                    *o = <$dtype>::new(r, i);
                }
                out
            }

            fn chebychev_data_from_coeffs_second_kind(&self) -> DynArray<$dtype, NDIM> {
                let shape = self.shape();
                let mut re = DynArray::<<$dtype as RlstScalar>::Real, NDIM>::from_shape(shape);
                let mut im = DynArray::<<$dtype as RlstScalar>::Real, NDIM>::from_shape(shape);

                re.fill_from(&self.r().unary_op(|elem: $dtype| elem.re()));
                im.fill_from(&self.r().unary_op(|elem: $dtype| elem.im()));

                let re_data = re.chebychev_data_from_coeffs_second_kind();
                let im_data = im.chebychev_data_from_coeffs_second_kind();

                let mut out = DynArray::<$dtype, NDIM>::from_shape(shape);
                for (o, r, i) in izip!(out.iter_mut(), re_data.iter_value(), im_data.iter_value()) {
                    *o = <$dtype>::new(r, i);
                }
                out
            }
        }
    };
}

impl_cheb_coeffs_complex!(c64);
impl_cheb_coeffs_complex!(c32);

macro_rules! impl_cheb_coeffs {
    ($dtype:ty) => {
        impl<ArrayImpl, const NDIM: usize> ChebychevCoefficientsToData<$dtype, NDIM>
            for Array<ArrayImpl, NDIM>
        where
            ArrayImpl: ValueArrayImpl<$dtype, NDIM>,
        {
            fn chebychev_coeffs_from_data_second_kind(&self) -> DynArray<$dtype, NDIM> {
                if NDIM > 3 {
                    unimplemented!();
                }

                let mut result = DynArray::<$dtype, NDIM>::from_shape(self.shape());
                result.fill_from(self);

                let shape = result.shape();

                // Compute the discrete cosine transform of the data

                let kind = vec![R2RPlanKind::FFTW_REDFT00; NDIM];

                result
                    .r_mut()
                    .into_fft()
                    .r2r_fft(crate::fftw::FftwPlanFlags::Estimate, &kind)
                    .unwrap()
                    .execute();

                // We scale all values.
                let n_recip =
                    (shape.iter().map(|elem| elem - 1).product::<usize>() as $dtype).recip();
                result *= n_recip;

                // We now have to deal with the boundary data

                // For NDIM = 1 we just need to adjust the single boundary points
                // For NDIM = 2 and NDIM = 3 we need to adjust the boundary values of the hypercube
                // formed by the values.

                if NDIM == 1 {
                    *result.data_mut().unwrap().first_mut().unwrap() /= 2.0;
                    *result.data_mut().unwrap().last_mut().unwrap() /= 2.0;
                } else if NDIM == 2 {
                    for (axis, dim) in shape.iter().enumerate() {
                        for index in [0, dim - 1] {
                            result
                                .r_mut()
                                .coerce_dim::<2>()
                                .unwrap()
                                .slice::<1>(axis, index)
                                .div_assign(2.0);
                        }
                    }
                } else if NDIM == 3 {
                    for (axis, dim) in shape.iter().enumerate() {
                        for index in [0, dim - 1] {
                            result
                                .r_mut()
                                .coerce_dim::<3>()
                                .unwrap()
                                .slice::<2>(axis, index)
                                .div_assign(2.0);
                        }
                    }
                }

                result
            }

            fn chebychev_data_from_coeffs_second_kind(&self) -> DynArray<$dtype, NDIM> {
                if NDIM > 3 {
                    unimplemented!();
                }

                let mut result = DynArray::<$dtype, NDIM>::from_shape(self.shape());
                result.fill_from(self);

                let shape = result.shape();

                // Rescale the values.

                result *= (2.pow(NDIM as u32) as $dtype).recip();

                if NDIM == 1 {
                    *result.data_mut().unwrap().first_mut().unwrap() *= 2.0;
                    *result.data_mut().unwrap().last_mut().unwrap() *= 2.0;
                } else if NDIM == 2 {
                    for (axis, dim) in shape.iter().enumerate() {
                        for index in [0, dim - 1] {
                            result
                                .r_mut()
                                .coerce_dim::<2>()
                                .unwrap()
                                .slice::<1>(axis, index)
                                .mul_assign(2.0);
                        }
                    }
                } else if NDIM == 3 {
                    for (axis, dim) in shape.iter().enumerate() {
                        for index in [0, dim - 1] {
                            result
                                .r_mut()
                                .coerce_dim::<3>()
                                .unwrap()
                                .slice::<2>(axis, index)
                                .mul_assign(2.0);
                        }
                    }
                }

                // Compute the discrete cosine transform of the data

                let kind = vec![R2RPlanKind::FFTW_REDFT00; NDIM];

                result
                    .r_mut()
                    .into_fft()
                    .r2r_fft(crate::fftw::FftwPlanFlags::Estimate, &kind)
                    .unwrap()
                    .execute();

                // This neutralises the boundary scaling and leaves only the interior values divided by two.
                // This is easier than trying to scale only interior values.

                result
            }
        }
    };
}

impl_cheb_coeffs!(f64);
impl_cheb_coeffs!(f32);

#[cfg(test)]
mod test {
    use num::Complex;

    use crate::{
        EvaluateArray, Linspace, assert_array_relative_eq,
        dense::tools::PrettyPrint,
        interpolation::{
            Kind, barycentric_chebychev_weights, barycentric_evaluate_2d, barycentric_evaluate_3d,
            chebychev_points, tensor_barycentric_evaluate_2d, tensor_barycentric_evaluate_3d,
        },
    };

    use super::*;

    #[test]
    pub fn test_data_to_coeffs_1d() {
        let n = 3;

        let points = chebychev_points::<f64>(Kind::Second, n);
        let mut one = DynArray::<f64, _>::from_shape([n]);
        one.fill_with_value(1.0);

        let data = 2.0_f64 * points.r() * points.r() + points.r();

        let coeffs = data.chebychev_coeffs_from_data_second_kind();

        let data_out = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_relative_eq!(coeffs, one, 1E-10);
        crate::assert_array_relative_eq!(data, data_out, 1E-10);
    }

    #[test]
    pub fn test_data_to_coeffs_1d_f32() {
        let n = 3;

        let points = chebychev_points::<f32>(Kind::Second, n);
        let mut one = DynArray::<f32, _>::from_shape([n]);
        one.fill_with_value(1.0);

        let data = 2.0_f32 * points.r() * points.r() + points.r();

        let coeffs = data.chebychev_coeffs_from_data_second_kind();

        let data_out = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_relative_eq!(coeffs, one, 1E-5);
        crate::assert_array_relative_eq!(data, data_out, 1E-5);
    }

    #[test]
    pub fn test_data_to_coeffs_2d() {
        let n = 3;

        let points = chebychev_points::<f64>(Kind::Second, n);
        let mut data = DynArray::<f64, 2>::from_shape([n, n]);

        for (i, pi) in points.iter_value().enumerate() {
            for (j, pj) in points.iter_value().enumerate() {
                data[[i, j]] = pi * pj + pi + pj + 2.0 * pi * pi + 2.0 * pj * pj - 1.0;
            }
        }

        let mut coeffs_expected = DynArray::<f64, 2>::from_shape([3, 3]);
        // All values are one except the very last entry
        coeffs_expected[[0, 0]] = 1.0;
        coeffs_expected[[1, 0]] = 1.0;
        coeffs_expected[[2, 0]] = 1.0;
        coeffs_expected[[0, 1]] = 1.0;
        coeffs_expected[[1, 1]] = 1.0;
        coeffs_expected[[0, 2]] = 1.0;

        let coeffs = data.chebychev_coeffs_from_data_second_kind();

        coeffs.pretty_print();

        crate::assert_array_relative_eq!(coeffs_expected, coeffs, 1E-10);

        let data_out = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_abs_diff_eq!(data, data_out, 1E-10);
    }

    #[test]
    pub fn test_data_to_coeffs_3d() {
        let n = 3;

        let points = chebychev_points::<f64>(Kind::Second, n);
        let mut data = DynArray::<f64, 3>::from_shape([n, n, n]);

        for (i, pi) in points.iter_value().enumerate() {
            for (j, pj) in points.iter_value().enumerate() {
                for (k, pk) in points.iter_value().enumerate() {
                    data[[i, j, k]] =
                        pi * pj + pi + pj + 2.0 * pi * pi + 2.0 * pj * pj - 1.0 + 3.0 * pk;
                }
            }
        }

        let mut coeffs_expected = DynArray::<f64, 3>::from_shape([3, 3, 3]);
        // All values are one except the very last entry
        coeffs_expected[[0, 0, 0]] = 1.0;
        coeffs_expected[[1, 0, 0]] = 1.0;
        coeffs_expected[[2, 0, 0]] = 1.0;
        coeffs_expected[[0, 1, 0]] = 1.0;
        coeffs_expected[[1, 1, 0]] = 1.0;
        coeffs_expected[[0, 2, 0]] = 1.0;
        coeffs_expected[[0, 0, 1]] = 3.0;

        let coeffs = data.chebychev_coeffs_from_data_second_kind();

        crate::assert_array_relative_eq!(coeffs_expected, coeffs, 1E-10);

        let data_out = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_abs_diff_eq!(data, data_out, 1E-10);
    }

    #[test]
    pub fn test_random_data_to_coefficients_and_back_1d() {
        let n = 11;

        let mut expected = DynArray::<f64, _>::from_shape([n]);
        expected.fill_from_seed_normally_distributed(0);

        let coeffs = expected.chebychev_coeffs_from_data_second_kind();

        let actual = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_relative_eq!(actual, expected, 1E-10);
    }

    #[test]
    pub fn test_complex_random_data_to_coefficients_and_back_1d() {
        let n = 11;

        let mut expected = DynArray::<c64, _>::from_shape([n]);
        expected.fill_from_seed_normally_distributed(0);

        let coeffs = expected.chebychev_coeffs_from_data_second_kind();

        let actual = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_relative_eq!(actual, expected, 1E-10);
    }

    #[test]
    pub fn test_random_data_to_coefficients_and_back_2d() {
        let n = 5;
        let m = 7;

        let mut expected = DynArray::<f64, _>::from_shape([m, n]);
        expected.fill_from_seed_normally_distributed(0);

        let coeffs = expected.chebychev_coeffs_from_data_second_kind();

        let actual = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_relative_eq!(actual, expected, 1E-10);
    }

    #[test]
    pub fn test_random_data_to_coefficients_and_back_3d() {
        let m = 7;
        let n = 5;
        let k = 8;

        let mut expected = DynArray::<f64, _>::from_shape([m, n, k]);
        expected.fill_from_seed_normally_distributed(0);

        let coeffs = expected.chebychev_coeffs_from_data_second_kind();

        let actual = coeffs.chebychev_data_from_coeffs_second_kind();

        crate::assert_array_relative_eq!(actual, expected, 1E-10);
    }

    #[test]
    pub fn test_eval_cheb_series_1d_second_kind() {
        let n = 3;
        let m = 11;
        let mut eval_points = DynArray::<f64, 1>::from_shape([m]);
        eval_points.linspace(-1.0, 1.0);

        let mut coeffs = DynArray::<f64, 1>::from_shape([n]);

        // This defines the polynomial p(x) = 5.0x^2 + 2x - 1.5

        coeffs[[0]] = 1.0;
        coeffs[[1]] = 2.0;
        coeffs[[2]] = 2.5;

        let mut result = DynArray::<f64, 1>::from_shape([m]);
        coeffs.evaluate_chebychev_series_second_kind(&eval_points, &mut result);

        let mut expected = eval_points.eval();
        expected
            .iter_mut()
            .for_each(|x| *x = 5.0 * *x * *x + 2.0 * *x - 1.5);

        crate::assert_array_relative_eq!(result, expected, 1E-13);
    }

    #[test]
    pub fn test_tensor_eval_cheb_series_2d_second_kind() {
        let nx = 5;
        let ny = 10;
        let mx = 60;
        let my = 70;

        let mut eval_points_x = DynArray::<f64, 1>::from_shape([mx]);
        let mut eval_points_y = DynArray::<f64, 1>::from_shape([my]);
        eval_points_x.linspace(-1.0, 1.0);
        eval_points_y.linspace(-1.0, 1.0);

        let mut coeffs = DynArray::<f64, 2>::from_shape([nx, ny]);
        coeffs.fill_from_seed_equally_distributed(0);

        // we fill the coefficients with random values. we then compare the direct evaluation with
        // the evaluation obtained by a dct to physical space, followed by barycentric evaluation.

        let mut actual = DynArray::<f64, 2>::from_shape([mx, my]);
        coeffs.tensor_evaluate_chebychev_series_second_kind(
            &eval_points_x,
            &eval_points_y,
            &mut actual,
        );

        // we now compute the same by doing a dft followed by barycentric evaluation.

        let data = coeffs.chebychev_data_from_coeffs_second_kind();

        let mut expected = DynArray::<f64, 2>::from_shape([mx, my]);

        let cheb_points_x = chebychev_points(Kind::Second, nx);
        let cheb_points_y = chebychev_points(Kind::Second, ny);
        let weights_x = barycentric_chebychev_weights::<f64>(Kind::Second, nx);
        let weights_y = barycentric_chebychev_weights::<f64>(Kind::Second, ny);

        tensor_barycentric_evaluate_2d(
            &eval_points_x,
            &eval_points_y,
            &cheb_points_x,
            &cheb_points_y,
            &weights_x,
            &weights_y,
            &data,
            &mut expected,
        );

        assert_array_relative_eq!(actual, expected, 1e-10);
    }

    #[test]
    pub fn test_eval_cheb_series_2d_second_kind() {
        let nx = 5;
        let ny = 10;
        let m = 60;

        let mut eval_points = DynArray::<f64, 2>::from_shape([2, m]);
        eval_points.fill_from_seed_equally_distributed(0);
        for value in eval_points.iter_mut() {
            // Shift random values to [-1, 1]
            *value = 2.0 * *value - 1.0
        }

        let mut coeffs = DynArray::<f64, 2>::from_shape([nx, ny]);
        coeffs.fill_from_seed_normally_distributed(0);

        // we fill the coefficients with random values. we then compare the direct evaluation with
        // the evaluation obtained by a dct to physical space, followed by barycentric evaluation.

        let mut actual = DynArray::<f64, 1>::from_shape([m]);
        coeffs.evaluate_chebychev_series_second_kind(&eval_points, &mut actual);
        let data = coeffs.chebychev_data_from_coeffs_second_kind();

        let mut expected = DynArray::<f64, 1>::from_shape([m]);

        let cheb_points_x = chebychev_points(Kind::Second, nx);
        let cheb_points_y = chebychev_points(Kind::Second, ny);
        let weights_x = barycentric_chebychev_weights::<f64>(Kind::Second, nx);
        let weights_y = barycentric_chebychev_weights::<f64>(Kind::Second, ny);

        barycentric_evaluate_2d(
            &eval_points,
            &cheb_points_x,
            &cheb_points_y,
            &weights_x,
            &weights_y,
            &data,
            &mut expected,
        );

        assert_array_relative_eq!(actual, expected, 1e-10);
    }

    #[test]
    pub fn test_tensor_eval_cheb_series_3d_second_kind() {
        let nx = 5;
        let ny = 10;
        let nz = 13;
        let mx = 18;
        let my = 19;
        let mz = 14;

        let mut eval_points_x = DynArray::<f64, 1>::from_shape([mx]);
        let mut eval_points_y = DynArray::<f64, 1>::from_shape([my]);
        let mut eval_points_z = DynArray::<f64, 1>::from_shape([mz]);
        eval_points_x.linspace(-1.0, 1.0);
        eval_points_y.linspace(-1.0, 1.0);
        eval_points_z.linspace(-1.0, 1.0);

        let mut coeffs = DynArray::<f64, 3>::from_shape([nx, ny, nz]);
        coeffs.fill_from_seed_equally_distributed(0);

        // we fill the coefficients with random values. we then compare the direct evaluation with
        // the evaluation obtained by a dct to physical space, followed by barycentric evaluation.

        let mut actual = DynArray::<f64, 3>::from_shape([mx, my, mz]);
        coeffs.tensor_evaluate_chebychev_series_second_kind(
            &eval_points_x,
            &eval_points_y,
            &eval_points_z,
            &mut actual,
        );

        // We now compute the same by doing a DFT followed by barycentric evaluation.

        let data = coeffs.chebychev_data_from_coeffs_second_kind();

        let mut expected = DynArray::<f64, 3>::from_shape([mx, my, mz]);

        let cheb_points_x = chebychev_points(Kind::Second, nx);
        let cheb_points_y = chebychev_points(Kind::Second, ny);
        let cheb_points_z = chebychev_points(Kind::Second, nz);
        let weights_x = barycentric_chebychev_weights::<f64>(Kind::Second, nx);
        let weights_y = barycentric_chebychev_weights::<f64>(Kind::Second, ny);
        let weights_z = barycentric_chebychev_weights::<f64>(Kind::Second, nz);

        tensor_barycentric_evaluate_3d(
            &eval_points_x,
            &eval_points_y,
            &eval_points_z,
            &cheb_points_x,
            &cheb_points_y,
            &cheb_points_z,
            &weights_x,
            &weights_y,
            &weights_z,
            &data,
            &mut expected,
        );

        assert_array_relative_eq!(actual, expected, 1E-10);
    }

    #[test]
    pub fn test_eval_cheb_series_3d_second_kind() {
        let nx = 5;
        let ny = 10;
        let nz = 13;
        let m = 60;

        let mut eval_points = DynArray::<f64, 2>::from_shape([3, m]);
        eval_points.fill_from_seed_equally_distributed(0);
        for value in eval_points.iter_mut() {
            // Shift random values to [-1, 1]
            *value = 2.0 * *value - 1.0
        }

        let mut coeffs = DynArray::<f64, 3>::from_shape([nx, ny, nz]);
        coeffs.fill_from_seed_normally_distributed(0);

        // we fill the coefficients with random values. we then compare the direct evaluation with
        // the evaluation obtained by a dct to physical space, followed by barycentric evaluation.

        let mut actual = DynArray::<f64, 1>::from_shape([m]);
        coeffs.evaluate_chebychev_series_second_kind(&eval_points, &mut actual);
        let data = coeffs.chebychev_data_from_coeffs_second_kind();

        let mut expected = DynArray::<f64, 1>::from_shape([m]);

        let cheb_points_x = chebychev_points(Kind::Second, nx);
        let cheb_points_y = chebychev_points(Kind::Second, ny);
        let cheb_points_z = chebychev_points(Kind::Second, nz);
        let weights_x = barycentric_chebychev_weights::<f64>(Kind::Second, nx);
        let weights_y = barycentric_chebychev_weights::<f64>(Kind::Second, ny);
        let weights_z = barycentric_chebychev_weights::<f64>(Kind::Second, nz);

        barycentric_evaluate_3d(
            &eval_points,
            &cheb_points_x,
            &cheb_points_y,
            &cheb_points_z,
            &weights_x,
            &weights_y,
            &weights_z,
            &data,
            &mut expected,
        );

        assert_array_relative_eq!(actual, expected, 1e-10);
    }

    #[test]
    fn test_derivative_1d() {
        // We sample the sine function from -1 to 1, compute the coefficients, take the derivative
        // and transform back. The result is compared with the cosine function.

        let n = 20; // Order of the Chebychev expansion

        let points = chebychev_points::<f64>(Kind::Second, n);

        // We fill the data array
        let mut values = points.eval();
        values.iter_mut().for_each(|elem| *elem = elem.sin());

        // and the array of expected values.
        let mut expected = points.eval();
        expected.iter_mut().for_each(|elem| *elem = elem.cos());

        // We compute the coefficients of the expansion
        let coeffs = values.chebychev_coeffs_from_data_second_kind();

        // Now compute the derivative of the coefficients
        let mut deriv = coeffs.eval();
        deriv.chebychev_derivative_second_kind(1);

        // We transform back
        let values_deriv = deriv.chebychev_data_from_coeffs_second_kind();

        // We compare with the sine of the original values.

        crate::assert_array_relative_eq!(values_deriv, expected, 1E-10);

        // Now check the second derivative
        let mut deriv_2 = coeffs.eval();
        deriv_2.chebychev_derivative_second_kind(2);

        let values_deriv_2 = deriv_2.chebychev_data_from_coeffs_second_kind();

        // The result should be close to -sin(x)
        let mut expected = points.eval();
        expected.iter_mut().for_each(|elem| *elem = -elem.sin());

        crate::assert_array_relative_eq!(values_deriv_2, expected, 1E-10);
    }

    #[test]
    fn test_derivative_complex_1d() {
        // We sample the sine function from -1 to 1, compute the coefficients, take the derivative
        // and transform back. The result is compared with the cosine function.

        let n = 30; // Order of the Chebychev expansion

        let points = chebychev_points::<f64>(Kind::Second, n);

        // We fill the data array
        let mut values = points.r().cast::<c64>().eval();
        values
            .iter_mut()
            .for_each(|elem| *elem = Complex::new(elem.re().sin(), (5.0_f64 * elem.re()).cos()));

        // and the array of expected values.
        let mut expected = points.r().cast::<c64>().eval();
        expected.iter_mut().for_each(|elem| {
            *elem = Complex::new(elem.re().cos(), -5.0_f64 * (5.0_f64 * elem.re()).sin())
        });

        // We compute the coefficients of the expansion
        let coeffs = values.chebychev_coeffs_from_data_second_kind();

        // Now compute the derivative of the coefficients
        let mut deriv = coeffs.eval();
        deriv.chebychev_derivative_second_kind(1);

        // We transform back
        let values_deriv = deriv.chebychev_data_from_coeffs_second_kind();

        // We compare with the sine of the original values.

        crate::assert_array_relative_eq!(values_deriv, expected, 1E-10);
    }

    #[test]
    fn test_derivative_2d() {
        // We sample sin(x * y) in the unit cube, compute the coefficients, take two derivatives
        // with respect to y and one derivative with respect to x
        // and transform back. The result is compared with the analytical derivative.

        let nx = 20; // Order of the Chebychev expansion
        let ny = 30; // Order of the Chebychev expansion

        let points_x = chebychev_points::<f64>(Kind::Second, nx);
        let points_y = chebychev_points::<f64>(Kind::Second, ny);

        // We fill the data array
        let mut values = DynArray::from_shape([nx, ny]);
        for (indx, px) in points_x.iter_value().enumerate() {
            for (indy, py) in points_y.iter_value().enumerate() {
                values[[indx, indy]] = f64::sin(px * py);
            }
        }

        // and the array of expected values.
        let mut expected = DynArray::from_shape([nx, ny]);
        for (indx, px) in points_x.iter_value().enumerate() {
            for (indy, py) in points_y.iter_value().enumerate() {
                expected[[indx, indy]] =
                    // 2 derivatives with respect to y and one derivative with respect to x
                    -px * px * py * f64::cos(px * py) - 2.0 * px * f64::sin(px * py);
            }
        }

        // We compute the coefficients of the expansion
        let coeffs = values.chebychev_coeffs_from_data_second_kind();

        // Now compute the derivative of the coefficients
        let mut deriv = coeffs.eval();
        deriv.chebychev_derivative_second_kind([1, 2]);

        // We transform back
        let values_deriv = deriv.chebychev_data_from_coeffs_second_kind();

        // We compare with the sine of the original values.

        crate::assert_array_relative_eq!(values_deriv, expected, 1E-8);
    }

    #[test]
    fn test_derivative_3d() {
        // We sample sin(x * y * z) in the unit cube, compute the coefficients, take two derivatives
        // with respect to y, one derivative with respect to x, and 3 derivatives with
        // respect to z and transform back. The result is compared with the analytical derivative.

        let nx = 14; // Order of the Chebychev expansion
        let ny = 15; // Order of the Chebychev expansion
        let nz = 16; // Order of the Chebychev expansion

        let points_x = chebychev_points::<f64>(Kind::Second, nx);
        let points_y = chebychev_points::<f64>(Kind::Second, ny);
        let points_z = chebychev_points::<f64>(Kind::Second, nz);

        // We fill the data array
        let mut values = DynArray::from_shape([nx, ny, nz]);
        for (indx, px) in points_x.iter_value().enumerate() {
            for (indy, py) in points_y.iter_value().enumerate() {
                for (indz, pz) in points_z.iter_value().enumerate() {
                    values[[indx, indy, indz]] = f64::sin(px * py * pz);
                }
            }
        }

        // and the array of expected values.
        let mut expected = DynArray::from_shape([nx, ny, nz]);
        for (indx, px) in points_x.iter_value().enumerate() {
            for (indy, py) in points_y.iter_value().enumerate() {
                for (indz, pz) in points_z.iter_value().enumerate() {
                    expected[[indx, indy, indz]] = -18.0 * px * px * py * f64::cos(px * py * pz)
                        + px * px
                            * px
                            * px
                            * py
                            * py
                            * py
                            * 11.0
                            * pz
                            * pz
                            * f64::cos(px * py * pz)
                        + px * px * px * py * py * 30.0 * pz * f64::sin(px * py * pz)
                        - px * px
                            * px
                            * px
                            * px
                            * py
                            * py
                            * py
                            * py
                            * pz
                            * pz
                            * pz
                            * f64::sin(px * py * pz);
                }
                // 2 derivatives with respect to y and one derivative with respect to x
            }
        }

        // We compute the coefficients of the expansion
        let coeffs = values.chebychev_coeffs_from_data_second_kind();

        // Now compute the derivative of the coefficients
        let mut deriv = coeffs.eval();
        deriv.chebychev_derivative_second_kind([1, 2, 3]);

        // We transform back
        let values_deriv = deriv.chebychev_data_from_coeffs_second_kind();

        // We compare with the sine of the original values.

        crate::assert_array_abs_diff_eq!(values_deriv, expected, 1E-4);
    }
}