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

    use alga::general::ComplexField;
    use crate::allocator::Allocator;
    use crate::base::{DefaultAllocator, MatrixMN, MatrixN, SquareMatrix, VectorN};
    use crate::dimension::{DimDiff, DimSub, U1};
    use crate::storage::Storage;

    use crate::linalg::householder;

    /// Hessenberg decomposition of a general matrix.
    #[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
    #[cfg_attr(
        feature = "serde-serialize",
        serde(bound(
            serialize = "DefaultAllocator: Allocator<N, D, D> +
                               Allocator<N, DimDiff<D, U1>>,
             MatrixN<N, D>: Serialize,
             VectorN<N, DimDiff<D, U1>>: Serialize"
        ))
    )]
    #[cfg_attr(
        feature = "serde-serialize",
        serde(bound(
            deserialize = "DefaultAllocator: Allocator<N, D, D> +
                               Allocator<N, DimDiff<D, U1>>,
             MatrixN<N, D>: Deserialize<'de>,
             VectorN<N, DimDiff<D, U1>>: Deserialize<'de>"
        ))
    )]
    #[derive(Clone, Debug)]
    pub struct Hessenberg<N: ComplexField, D: DimSub<U1>>
    where DefaultAllocator: Allocator<N, D, D> + Allocator<N, DimDiff<D, U1>>
    {
        hess: MatrixN<N, D>,
        subdiag: VectorN<N, DimDiff<D, U1>>,
    }

    impl<N: ComplexField, D: DimSub<U1>> Copy for Hessenberg<N, D>
    where
        DefaultAllocator: Allocator<N, D, D> + Allocator<N, DimDiff<D, U1>>,
        MatrixN<N, D>: Copy,
        VectorN<N, DimDiff<D, U1>>: Copy,
    {}

    impl<N: ComplexField, D: DimSub<U1>> Hessenberg<N, D>
    where DefaultAllocator: Allocator<N, D, D> + Allocator<N, D> + Allocator<N, DimDiff<D, U1>>
    {
        /// Computes the Hessenberg decomposition using householder reflections.
        pub fn new(hess: MatrixN<N, D>) -> Self {
            let mut work = unsafe { MatrixMN::new_uninitialized_generic(hess.data.shape().0, U1) };
            Self::new_with_workspace(hess, &mut work)
        }

        /// Computes the Hessenberg decomposition using householder reflections.
        ///
        /// The workspace containing `D` elements must be provided but its content does not have to be
        /// initialized.
        pub fn new_with_workspace(mut hess: MatrixN<N, D>, work: &mut VectorN<N, D>) -> Self {
            assert!(
                hess.is_square(),
                "Cannot compute the hessenberg decomposition of a non-square matrix."
            );

            let dim = hess.data.shape().0;

            assert!(
                dim.value() != 0,
                "Cannot compute the hessenberg decomposition of an empty matrix."
            );
            assert_eq!(
                dim.value(),
                work.len(),
                "Hessenberg: invalid workspace size."
            );

            let mut subdiag = unsafe { MatrixMN::new_uninitialized_generic(dim.sub(U1), U1) };

            if dim.value() == 0 {
                return Hessenberg { hess, subdiag };
            }

            for ite in 0..dim.value() - 1 {
                householder::clear_column_unchecked(&mut hess, &mut subdiag[ite], ite, 1, Some(work));
            }

            Hessenberg { hess, subdiag }
        }

        /// Retrieves `(q, h)` with `q` the orthogonal matrix of this decomposition and `h` the
        /// hessenberg matrix.
        #[inline]
        pub fn unpack(self) -> (MatrixN<N, D>, MatrixN<N, D>) {
            let q = self.q();

            (q, self.unpack_h())
        }

        /// Retrieves the upper trapezoidal submatrix `H` of this decomposition.
        #[inline]
        pub fn unpack_h(mut self) -> MatrixN<N, D> {
            let dim = self.hess.nrows();
            self.hess.fill_lower_triangle(N::zero(), 2);
            self.hess
                .slice_mut((1, 0), (dim - 1, dim - 1))
                .set_partial_diagonal(self.subdiag.iter().map(|e| N::from_real(e.modulus())));
            self.hess
        }

        // FIXME: add a h that moves out of self.
        /// Retrieves the upper trapezoidal submatrix `H` of this decomposition.
        ///
        /// This is less efficient than `.unpack_h()` as it allocates a new matrix.
        #[inline]
        pub fn h(&self) -> MatrixN<N, D> {
            let dim = self.hess.nrows();
            let mut res = self.hess.clone();
            res.fill_lower_triangle(N::zero(), 2);
            res.slice_mut((1, 0), (dim - 1, dim - 1))
                .set_partial_diagonal(self.subdiag.iter().map(|e| N::from_real(e.modulus())));
            res
        }

        /// Computes the orthogonal matrix `Q` of this decomposition.
        pub fn q(&self) -> MatrixN<N, D> {
            householder::assemble_q(&self.hess, self.subdiag.as_slice())
        }

        #[doc(hidden)]
        pub fn hess_internal(&self) -> &MatrixN<N, D> {
            &self.hess
        }
    }

    impl<N: ComplexField, D: DimSub<U1>, S: Storage<N, D, D>> SquareMatrix<N, D, S>
    where DefaultAllocator: Allocator<N, D, D> + Allocator<N, D> + Allocator<N, DimDiff<D, U1>>
    {
        /// Computes the Hessenberg decomposition of this matrix using householder reflections.
        pub fn hessenberg(self) -> Hessenberg<N, D> {
            Hessenberg::new(self.into_owned())
        }
    }

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