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    #[cfg(feature = "serde-serialize")]
    use serde::{Deserialize, Serialize};
    use num::Zero;

    use alga::general::ComplexField;
    use crate::allocator::{Allocator, Reallocator};
    use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, Unit, VectorN};
    use crate::constraint::{SameNumberOfRows, ShapeConstraint};
    use crate::dimension::{Dim, DimMin, DimMinimum, U1};
    use crate::storage::{Storage, StorageMut};

    use crate::geometry::Reflection;
    use crate::linalg::householder;

    /// The QR decomposition of a general matrix.
    #[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
    #[cfg_attr(
        feature = "serde-serialize",
        serde(bound(
            serialize = "DefaultAllocator: Allocator<N, R, C> +
                               Allocator<N, DimMinimum<R, C>>,
             MatrixMN<N, R, C>: Serialize,
             VectorN<N, DimMinimum<R, C>>: Serialize"
        ))
    )]
    #[cfg_attr(
        feature = "serde-serialize",
        serde(bound(
            deserialize = "DefaultAllocator: Allocator<N, R, C> +
                               Allocator<N, DimMinimum<R, C>>,
             MatrixMN<N, R, C>: Deserialize<'de>,
             VectorN<N, DimMinimum<R, C>>: Deserialize<'de>"
        ))
    )]
    #[derive(Clone, Debug)]
    pub struct QR<N: ComplexField, R: DimMin<C>, C: Dim>
    where DefaultAllocator: Allocator<N, R, C> + Allocator<N, DimMinimum<R, C>>
    {
        qr: MatrixMN<N, R, C>,
        diag: VectorN<N, DimMinimum<R, C>>,
    }

    impl<N: ComplexField, R: DimMin<C>, C: Dim> Copy for QR<N, R, C>
    where
        DefaultAllocator: Allocator<N, R, C> + Allocator<N, DimMinimum<R, C>>,
        MatrixMN<N, R, C>: Copy,
        VectorN<N, DimMinimum<R, C>>: Copy,
    {}

    impl<N: ComplexField, R: DimMin<C>, C: Dim> QR<N, R, C>
    where DefaultAllocator: Allocator<N, R, C> + Allocator<N, R> + Allocator<N, DimMinimum<R, C>>
    {
        /// Computes the QR decomposition using householder reflections.
        pub fn new(mut matrix: MatrixMN<N, R, C>) -> Self {
            let (nrows, ncols) = matrix.data.shape();
            let min_nrows_ncols = nrows.min(ncols);

            let mut diag = unsafe { MatrixMN::new_uninitialized_generic(min_nrows_ncols, U1) };

            if min_nrows_ncols.value() == 0 {
                return QR {
                    qr: matrix,
                    diag: diag,
                };
            }

            for ite in 0..min_nrows_ncols.value() {
                householder::clear_column_unchecked(&mut matrix, &mut diag[ite], ite, 0, None);
            }

            QR {
                qr: matrix,
                diag: diag,
            }
        }

        /// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
        #[inline]
        pub fn r(&self) -> MatrixMN<N, DimMinimum<R, C>, C>
        where
            DefaultAllocator: Allocator<N, DimMinimum<R, C>, C>,
        {
            let (nrows, ncols) = self.qr.data.shape();
            let mut res = self.qr.rows_generic(0, nrows.min(ncols)).upper_triangle();
            res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
            res
        }

        /// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
        ///
        /// This is usually faster than `r` but consumes `self`.
        #[inline]
        pub fn unpack_r(self) -> MatrixMN<N, DimMinimum<R, C>, C>
        where
            DefaultAllocator: Reallocator<N, R, C, DimMinimum<R, C>, C>,
        {
            let (nrows, ncols) = self.qr.data.shape();
            let mut res = self.qr.resize_generic(nrows.min(ncols), ncols, N::zero());
            res.fill_lower_triangle(N::zero(), 1);
            res.set_partial_diagonal(self.diag.iter().map(|e| N::from_real(e.modulus())));
            res
        }

        /// Computes the orthogonal matrix `Q` of this decomposition.
        pub fn q(&self) -> MatrixMN<N, R, DimMinimum<R, C>>
        where DefaultAllocator: Allocator<N, R, DimMinimum<R, C>> {
            let (nrows, ncols) = self.qr.data.shape();

            // NOTE: we could build the identity matrix and call q_mul on it.
            // Instead we don't so that we take in account the matrix sparseness.
            let mut res = Matrix::identity_generic(nrows, nrows.min(ncols));
            let dim = self.diag.len();

            for i in (0..dim).rev() {
                let axis = self.qr.slice_range(i.., i);
                // FIXME: sometimes, the axis might have a zero magnitude.
                let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());

                let mut res_rows = res.slice_range_mut(i.., i..);
                refl.reflect_with_sign(&mut res_rows, self.diag[i].signum());
            }

            res
        }

        /// Unpacks this decomposition into its two matrix factors.
        pub fn unpack(
            self,
        ) -> (
            MatrixMN<N, R, DimMinimum<R, C>>,
            MatrixMN<N, DimMinimum<R, C>, C>,
        )
        where
            DimMinimum<R, C>: DimMin<C, Output = DimMinimum<R, C>>,
            DefaultAllocator:
                Allocator<N, R, DimMinimum<R, C>> + Reallocator<N, R, C, DimMinimum<R, C>, C>,
        {
            (self.q(), self.unpack_r())
        }

        #[doc(hidden)]
        pub fn qr_internal(&self) -> &MatrixMN<N, R, C> {
            &self.qr
        }

        /// Multiplies the provided matrix by the transpose of the `Q` matrix of this decomposition.
        pub fn q_tr_mul<R2: Dim, C2: Dim, S2>(&self, rhs: &mut Matrix<N, R2, C2, S2>)
        // FIXME: do we need a static constraint on the number of rows of rhs?
        where S2: StorageMut<N, R2, C2> {
            let dim = self.diag.len();

            for i in 0..dim {
                let axis = self.qr.slice_range(i.., i);
                let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());

                let mut rhs_rows = rhs.rows_range_mut(i..);
                refl.reflect_with_sign(&mut rhs_rows, self.diag[i].signum().conjugate());
            }
        }
    }

    impl<N: ComplexField, D: DimMin<D, Output = D>> QR<N, D, D>
    where DefaultAllocator: Allocator<N, D, D> + Allocator<N, D>
    {
        /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
        ///
        /// Returns `None` if `self` is not invertible.
        pub fn solve<R2: Dim, C2: Dim, S2>(
            &self,
            b: &Matrix<N, R2, C2, S2>,
        ) -> Option<MatrixMN<N, R2, C2>>
        where
            S2: StorageMut<N, R2, C2>,
            ShapeConstraint: SameNumberOfRows<R2, D>,
            DefaultAllocator: Allocator<N, R2, C2>,
        {
            let mut res = b.clone_owned();

            if self.solve_mut(&mut res) {
                Some(res)
            } else {
                None
            }
        }

        /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
        ///
        /// If the decomposed matrix is not invertible, this returns `false` and its input `b` is
        /// overwritten with garbage.
        pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<N, R2, C2, S2>) -> bool
        where
            S2: StorageMut<N, R2, C2>,
            ShapeConstraint: SameNumberOfRows<R2, D>,
        {
            assert_eq!(
                self.qr.nrows(),
                b.nrows(),
                "QR solve matrix dimension mismatch."
            );
            assert!(
                self.qr.is_square(),
                "QR solve: unable to solve a non-square system."
            );

            self.q_tr_mul(b);
            self.solve_upper_triangular_mut(b)
        }

        // FIXME: duplicate code from the `solve` module.
        fn solve_upper_triangular_mut<R2: Dim, C2: Dim, S2>(
            &self,
            b: &mut Matrix<N, R2, C2, S2>,
        ) -> bool
        where
            S2: StorageMut<N, R2, C2>,
            ShapeConstraint: SameNumberOfRows<R2, D>,
        {
            let dim = self.qr.nrows();

            for k in 0..b.ncols() {
                let mut b = b.column_mut(k);
                for i in (0..dim).rev() {
                    let coeff;

                    unsafe {
                        let diag = self.diag.vget_unchecked(i).modulus();

                        if diag.is_zero() {
                            return false;
                        }

                        coeff = b.vget_unchecked(i).unscale(diag);
                        *b.vget_unchecked_mut(i) = coeff;
                    }

                    b.rows_range_mut(..i)
                        .axpy(-coeff, &self.qr.slice_range(..i, i), N::one());
                }
            }

            true
        }

        /// Computes the inverse of the decomposed matrix.
        ///
        /// Returns `None` if the decomposed matrix is not invertible.
        pub fn try_inverse(&self) -> Option<MatrixN<N, D>> {
            assert!(
                self.qr.is_square(),
                "QR inverse: unable to compute the inverse of a non-square matrix."
            );

            // FIXME: is there a less naive method ?
            let (nrows, ncols) = self.qr.data.shape();
            let mut res = MatrixN::identity_generic(nrows, ncols);

            if self.solve_mut(&mut res) {
                Some(res)
            } else {
                None
            }
        }

        /// Indicates if the decomposed matrix is invertible.
        pub fn is_invertible(&self) -> bool {
            assert!(
                self.qr.is_square(),
                "QR: unable to test the invertibility of a non-square matrix."
            );

            for i in 0..self.diag.len() {
                if self.diag[i].is_zero() {
                    return false;
                }
            }

            true
        }

        // /// Computes the determinant of the decomposed matrix.
        // pub fn determinant(&self) -> N {
        //     let dim = self.qr.nrows();
        //     assert!(self.qr.is_square(), "QR determinant: unable to compute the determinant of a non-square matrix.");

        //     let mut res = N::one();
        //     for i in 0 .. dim {
        //         res *= unsafe { *self.diag.vget_unchecked(i) };
        //     }

        //     res self.q_determinant()
        // }
    }

    impl<N: ComplexField, R: DimMin<C>, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S>
    where DefaultAllocator: Allocator<N, R, C> + Allocator<N, R> + Allocator<N, DimMinimum<R, C>>
    {
        /// Computes the QR decomposition of this matrix.
        pub fn qr(self) -> QR<N, R, C> {
            QR::new(self.into_owned())
        }
    }

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