Eigen/src/Householder/HouseholderSequence.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// SPDX-License-Identifier: MPL-2.0

#ifndef EIGEN_HOUSEHOLDER_SEQUENCE_H
#define EIGEN_HOUSEHOLDER_SEQUENCE_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

/** \ingroup Householder_Module
 * \householder_module
 * \class HouseholderSequence
 * \brief Sequence of Householder reflections acting on subspaces with decreasing size
 * \tparam VectorsType type of matrix containing the Householder vectors
 * \tparam CoeffsType  type of vector containing the Householder coefficients
 * \tparam Side        either OnTheLeft (the default) or OnTheRight
 *
 * This class represents a product sequence of Householder reflections where the first Householder reflection
 * acts on the whole space, the second Householder reflection leaves the one-dimensional subspace spanned by
 * the first unit vector invariant, the third Householder reflection leaves the two-dimensional subspace
 * spanned by the first two unit vectors invariant, and so on up to the last reflection which leaves all but
 * one dimensions invariant and acts only on the last dimension. Such sequences of Householder reflections
 * are used in several algorithms to zero out certain parts of a matrix. Indeed, the methods
 * HessenbergDecomposition::matrixQ(), Tridiagonalization::matrixQ(), HouseholderQR::householderQ(),
 * and ColPivHouseholderQR::householderQ() all return a %HouseholderSequence.
 *
 * More precisely, the class %HouseholderSequence represents an \f$ n \times n \f$ matrix \f$ H \f$ of the
 * form \f$ H = \prod_{i=0}^{n-1} H_i \f$ where the i-th Householder reflection is \f$ H_i = I - h_i v_i
 * v_i^* \f$. The i-th Householder coefficient \f$ h_i \f$ is a scalar and the i-th Householder vector \f$
 * v_i \f$ is a vector of the form
 * \f[
 * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
 * \f]
 * The last \f$ n-i \f$ entries of \f$ v_i \f$ are called the essential part of the Householder vector.
 *
 * Typical usages are listed below, where H is a HouseholderSequence:
 * \code
 * A.applyOnTheRight(H);             // A = A * H
 * A.applyOnTheLeft(H);              // A = H * A
 * A.applyOnTheRight(H.adjoint());   // A = A * H^*
 * A.applyOnTheLeft(H.adjoint());    // A = H^* * A
 * MatrixXd Q = H;                   // conversion to a dense matrix
 * \endcode
 * In addition to the adjoint, you can also apply the inverse (=adjoint), the transpose, and the conjugate operators.
 *
 * See the documentation for HouseholderSequence(const VectorsType&, const CoeffsType&) for an example.
 *
 * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
 */

namespace internal {

template <typename VectorsType, typename CoeffsType, int Side>
struct traits<HouseholderSequence<VectorsType, CoeffsType, Side> > {
  using Scalar = typename VectorsType::Scalar;
  using StorageIndex = typename VectorsType::StorageIndex;
  using StorageKind = typename VectorsType::StorageKind;
  enum {
    RowsAtCompileTime =
        Side == OnTheLeft ? traits<VectorsType>::RowsAtCompileTime : traits<VectorsType>::ColsAtCompileTime,
    ColsAtCompileTime = RowsAtCompileTime,
    MaxRowsAtCompileTime =
        Side == OnTheLeft ? traits<VectorsType>::MaxRowsAtCompileTime : traits<VectorsType>::MaxColsAtCompileTime,
    MaxColsAtCompileTime = MaxRowsAtCompileTime,
    Flags = 0
  };
};

struct HouseholderSequenceShape {};

template <typename VectorsType, typename CoeffsType, int Side>
struct evaluator_traits<HouseholderSequence<VectorsType, CoeffsType, Side> >
    : public evaluator_traits_base<HouseholderSequence<VectorsType, CoeffsType, Side> > {
  using Shape = HouseholderSequenceShape;
};

template <typename VectorsType, typename CoeffsType, int Side>
struct hseq_side_dependent_impl {
  using EssentialVectorType = Block<const VectorsType, Dynamic, 1>;
  using HouseholderSequenceType = HouseholderSequence<VectorsType, CoeffsType, OnTheLeft>;
  static EIGEN_DEVICE_FUNC inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k) {
    Index start = k + 1 + h.m_shift;
    return Block<const VectorsType, Dynamic, 1>(h.m_vectors, start, k, h.rows() - start, 1);
  }
};

template <typename VectorsType, typename CoeffsType>
struct hseq_side_dependent_impl<VectorsType, CoeffsType, OnTheRight> {
  using EssentialVectorType = Transpose<Block<const VectorsType, 1, Dynamic>>;
  using HouseholderSequenceType = HouseholderSequence<VectorsType, CoeffsType, OnTheRight>;
  static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k) {
    Index start = k + 1 + h.m_shift;
    return Block<const VectorsType, 1, Dynamic>(h.m_vectors, k, start, 1, h.rows() - start).transpose();
  }
};

template <typename OtherScalarType, typename MatrixType>
struct matrix_type_times_scalar_type {
  using ResultScalar = typename ScalarBinaryOpTraits<OtherScalarType, typename MatrixType::Scalar>::ReturnType;
  using Type = Matrix<ResultScalar, MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime, 0,
                      MatrixType::MaxRowsAtCompileTime, MatrixType::MaxColsAtCompileTime>;
};

}  // end namespace internal

template <typename VectorsType, typename CoeffsType, int Side>
class HouseholderSequence : public EigenBase<HouseholderSequence<VectorsType, CoeffsType, Side> > {
  using EssentialVectorType =
      typename internal::hseq_side_dependent_impl<VectorsType, CoeffsType, Side>::EssentialVectorType;

 public:
  enum {
    RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime,
    ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime,
    MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime
  };
  using Scalar = typename internal::traits<HouseholderSequence>::Scalar;

  using ConjugateReturnType = HouseholderSequence<
      std::conditional_t<NumTraits<Scalar>::IsComplex,
                         internal::remove_all_t<typename VectorsType::ConjugateReturnType>, VectorsType>,
      std::conditional_t<NumTraits<Scalar>::IsComplex, internal::remove_all_t<typename CoeffsType::ConjugateReturnType>,
                         CoeffsType>,
      Side>;

  using AdjointReturnType = HouseholderSequence<
      VectorsType,
      std::conditional_t<NumTraits<Scalar>::IsComplex, internal::remove_all_t<typename CoeffsType::ConjugateReturnType>,
                         CoeffsType>,
      Side>;

  using TransposeReturnType = HouseholderSequence<
      std::conditional_t<NumTraits<Scalar>::IsComplex,
                         internal::remove_all_t<typename VectorsType::ConjugateReturnType>, VectorsType>,
      CoeffsType, Side>;

  using ConstHouseholderSequence =
      HouseholderSequence<std::add_const_t<VectorsType>, std::add_const_t<CoeffsType>, Side>;

  /** \brief Constructor.
   * \param[in]  v      %Matrix containing the essential parts of the Householder vectors
   * \param[in]  h      Vector containing the Householder coefficients
   *
   * Constructs the Householder sequence with coefficients given by \p h and vectors given by \p v. The
   * i-th Householder coefficient \f$ h_i \f$ is given by \p h(i) and the essential part of the i-th
   * Householder vector \f$ v_i \f$ is given by \p v(k,i) with \p k > \p i (the subdiagonal part of the
   * i-th column). If \p v has fewer columns than rows, then the Householder sequence contains as many
   * Householder reflections as there are columns.
   *
   * \note The %HouseholderSequence object stores \p v and \p h by reference.
   *
   * Example: \include HouseholderSequence_HouseholderSequence.cpp
   * Output: \verbinclude HouseholderSequence_HouseholderSequence.out
   *
   * \sa setLength(), setShift()
   */
  EIGEN_DEVICE_FUNC HouseholderSequence(const VectorsType& v, const CoeffsType& h)
      : m_vectors(v), m_coeffs(h), m_reverse(false), m_length(v.diagonalSize()), m_shift(0) {}

  /** \brief Copy constructor. */
  EIGEN_DEVICE_FUNC HouseholderSequence(const HouseholderSequence& other)
      : m_vectors(other.m_vectors),
        m_coeffs(other.m_coeffs),
        m_reverse(other.m_reverse),
        m_length(other.m_length),
        m_shift(other.m_shift) {}

  /** \brief Number of rows of transformation viewed as a matrix.
   * \returns Number of rows
   * \details This equals the dimension of the space that the transformation acts on.
   */
  EIGEN_DEVICE_FUNC constexpr Index rows() const noexcept {
    return Side == OnTheLeft ? m_vectors.rows() : m_vectors.cols();
  }

  /** \brief Number of columns of transformation viewed as a matrix.
   * \returns Number of columns
   * \details This equals the dimension of the space that the transformation acts on.
   */
  EIGEN_DEVICE_FUNC constexpr Index cols() const noexcept { return rows(); }

  /** \brief Essential part of a Householder vector.
   * \param[in]  k  Index of Householder reflection
   * \returns    Vector containing non-trivial entries of k-th Householder vector
   *
   * This function returns the essential part of the Householder vector \f$ v_i \f$. This is a vector of
   * length \f$ n-i \f$ containing the last \f$ n-i \f$ entries of the vector
   * \f[
   * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
   * \f]
   * The index \f$ i \f$ equals \p k + shift(), corresponding to the k-th column of the matrix \p v
   * passed to the constructor.
   *
   * \sa setShift(), shift()
   */
  EIGEN_DEVICE_FUNC const EssentialVectorType essentialVector(Index k) const {
    eigen_assert(k >= 0 && k < m_length);
    return internal::hseq_side_dependent_impl<VectorsType, CoeffsType, Side>::essentialVector(*this, k);
  }

  /** \brief %Transpose of the Householder sequence. */
  TransposeReturnType transpose() const {
    return TransposeReturnType(m_vectors.conjugate(), m_coeffs)
        .setReverseFlag(!m_reverse)
        .setLength(m_length)
        .setShift(m_shift);
  }

  /** \brief Complex conjugate of the Householder sequence. */
  ConjugateReturnType conjugate() const {
    return ConjugateReturnType(m_vectors.conjugate(), m_coeffs.conjugate())
        .setReverseFlag(m_reverse)
        .setLength(m_length)
        .setShift(m_shift);
  }

  /** \returns a %HouseholderSequence over the vectors and coefficients of \c *this, complex-conjugated if
   *           Cond==true.
   *
   * \warning Unlike conjugate(), this function does not propagate the state of \c *this: the returned sequence uses
   *          the defaults of the two-argument constructor, that is, not reversed, full length and zero shift.
   */
  template <bool Cond>
  EIGEN_DEVICE_FUNC inline std::conditional_t<Cond, ConjugateReturnType, ConstHouseholderSequence> conjugateIf() const {
    using ReturnType = std::conditional_t<Cond, ConjugateReturnType, ConstHouseholderSequence>;
    return ReturnType(m_vectors.template conjugateIf<Cond>(), m_coeffs.template conjugateIf<Cond>());
  }

  /** \brief Adjoint (conjugate transpose) of the Householder sequence. */
  AdjointReturnType adjoint() const {
    return AdjointReturnType(m_vectors, m_coeffs.conjugate())
        .setReverseFlag(!m_reverse)
        .setLength(m_length)
        .setShift(m_shift);
  }

  /** \brief Inverse of the Householder sequence (equals the adjoint). */
  AdjointReturnType inverse() const { return adjoint(); }

  /** \internal */
  template <typename DestType>
  inline EIGEN_DEVICE_FUNC void evalTo(DestType& dst) const {
    Matrix<Scalar, DestType::RowsAtCompileTime, 1, AutoAlign | ColMajor, DestType::MaxRowsAtCompileTime, 1> workspace(
        rows());
    evalTo(dst, workspace);
  }

  /** \internal */
  template <typename Dest, typename Workspace>
  EIGEN_DEVICE_FUNC void evalTo(Dest& dst, Workspace& workspace) const {
    workspace.resize(rows());
    Index vecs = m_length;
    if (internal::is_same_dense(dst, m_vectors)) {
      // in-place
      dst.diagonal().setOnes();
      dst.template triangularView<StrictlyUpper>().setZero();
      for (Index k = vecs - 1; k >= 0; --k) {
        Index cornerSize = rows() - k - m_shift;
        if (m_reverse)
          dst.bottomRightCorner(cornerSize, cornerSize)
              .applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data());
        else
          dst.bottomRightCorner(cornerSize, cornerSize)
              .applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data());

        // clear the off diagonal vector
        dst.col(k).tail(rows() - k - 1).setZero();
      }
      // clear the remaining columns if needed
      for (Index k = 0; k < cols() - vecs; ++k) dst.col(k).tail(rows() - k - 1).setZero();
    } else if (m_length > BlockSize) {
      dst.setIdentity(rows(), rows());
      applyThisOnTheLeft(dst, workspace, true);
    } else {
      dst.setIdentity(rows(), rows());
      for (Index k = vecs - 1; k >= 0; --k) {
        Index cornerSize = rows() - k - m_shift;
        if (m_reverse)
          dst.bottomRightCorner(cornerSize, cornerSize)
              .applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data());
        else
          dst.bottomRightCorner(cornerSize, cornerSize)
              .applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data());
      }
    }
  }

  /** \internal */
  template <typename Dest>
  inline void applyThisOnTheRight(Dest& dst) const {
    Matrix<Scalar, 1, Dest::RowsAtCompileTime, RowMajor, 1, Dest::MaxRowsAtCompileTime> workspace(dst.rows());
    applyThisOnTheRight(dst, workspace);
  }

  /** \internal */
  template <typename Dest, typename Workspace>
  inline void applyThisOnTheRight(Dest& dst, Workspace& workspace) const {
    // Use the block path when the reflectors are long enough for GEMM to outperform GEMV.
    // The threshold on rows() (the reflector length) is higher than for the left-side path because
    // the right-side block application has more overhead from the tmp = mat * V product layout.
    if (m_length >= BlockSize && rows() - m_shift >= 4 * BlockSize) {
      applyBlockOnTheRight(dst);
    } else {
      workspace.resize(dst.rows());
      for (Index k = 0; k < m_length; ++k) {
        Index actual_k = m_reverse ? m_length - k - 1 : k;
        dst.rightCols(rows() - m_shift - actual_k)
            .applyHouseholderOnTheRight(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
      }
    }
  }

 private:
  /** \internal Block Householder application on the right, kept out-of-line
   * to avoid template bloat pessimizing the scalar path above. */
  template <typename Dest>
  EIGEN_DONT_INLINE void applyBlockOnTheRight(Dest& dst) const {
    // Make sure we have at least 2 useful blocks, otherwise it is point-less:
    Index blockSize = m_length < Index(2 * BlockSize) ? (m_length + 1) / 2 : Index(BlockSize);
    for (Index i = 0; i < m_length; i += blockSize) {
      // Right-side application processes blocks in opposite order to left-side:
      // forward (m_reverse=false): first block first; reversed: last block first.
      Index end = m_reverse ? m_length - i : (std::min)(m_length, i + blockSize);
      Index k = m_reverse ? (std::max)(Index(0), end - blockSize) : i;
      Index bs = end - k;
      Index start = k + m_shift;

      using SubVectorsType = Block<internal::remove_all_t<VectorsType>, Dynamic, Dynamic>;
      SubVectorsType sub_vecs1(m_vectors.const_cast_derived(), Side == OnTheRight ? k : start,
                               Side == OnTheRight ? start : k, Side == OnTheRight ? bs : m_vectors.rows() - start,
                               Side == OnTheRight ? m_vectors.cols() - start : bs);
      std::conditional_t<Side == OnTheRight, Transpose<SubVectorsType>, SubVectorsType&> sub_vecs(sub_vecs1);

      Index dstCols = rows() - m_shift - k;
      auto sub_dst = dst.rightCols(dstCols);
      internal::apply_block_householder_on_the_right(sub_dst, sub_vecs, m_coeffs.segment(k, bs), !m_reverse);
    }
  }

 public:
  /** \internal */
  template <typename Dest>
  inline void applyThisOnTheLeft(Dest& dst, bool inputIsIdentity = false) const {
    Matrix<Scalar, 1, Dest::ColsAtCompileTime, RowMajor, 1, Dest::MaxColsAtCompileTime> workspace;
    applyThisOnTheLeft(dst, workspace, inputIsIdentity);
  }

  /** \internal */
  template <typename Dest, typename Workspace>
  inline void applyThisOnTheLeft(Dest& dst, Workspace& workspace, bool inputIsIdentity = false) const {
    if (inputIsIdentity && m_reverse) inputIsIdentity = false;
    // if the entries are large enough, then apply the reflectors by block
    if (m_length >= BlockSize && dst.cols() > 1) {
      // Make sure we have at least 2 useful blocks, otherwise it is point-less:
      Index blockSize = m_length < Index(2 * BlockSize) ? (m_length + 1) / 2 : Index(BlockSize);
      for (Index i = 0; i < m_length; i += blockSize) {
        Index end = m_reverse ? (std::min)(m_length, i + blockSize) : m_length - i;
        Index k = m_reverse ? i : (std::max)(Index(0), end - blockSize);
        Index bs = end - k;
        Index start = k + m_shift;

        using SubVectorsType = Block<internal::remove_all_t<VectorsType>, Dynamic, Dynamic>;
        SubVectorsType sub_vecs1(m_vectors.const_cast_derived(), Side == OnTheRight ? k : start,
                                 Side == OnTheRight ? start : k, Side == OnTheRight ? bs : m_vectors.rows() - start,
                                 Side == OnTheRight ? m_vectors.cols() - start : bs);
        std::conditional_t<Side == OnTheRight, Transpose<SubVectorsType>, SubVectorsType&> sub_vecs(sub_vecs1);

        Index dstRows = rows() - m_shift - k;

        if (inputIsIdentity) {
          Block<Dest, Dynamic, Dynamic> sub_dst = dst.bottomRightCorner(dstRows, dstRows);
          apply_block_householder_on_the_left(sub_dst, sub_vecs, m_coeffs.segment(k, bs), !m_reverse);
        } else {
          auto sub_dst = dst.bottomRows(dstRows);
          apply_block_householder_on_the_left(sub_dst, sub_vecs, m_coeffs.segment(k, bs), !m_reverse);
        }
      }
    } else {
      workspace.resize(dst.cols());
      for (Index k = 0; k < m_length; ++k) {
        Index actual_k = m_reverse ? k : m_length - k - 1;
        Index dstRows = rows() - m_shift - actual_k;

        if (inputIsIdentity) {
          Block<Dest, Dynamic, Dynamic> sub_dst = dst.bottomRightCorner(dstRows, dstRows);
          sub_dst.applyHouseholderOnTheLeft(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
        } else {
          auto sub_dst = dst.bottomRows(dstRows);
          sub_dst.applyHouseholderOnTheLeft(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
        }
      }
    }
  }

  /** \brief Computes the product of a Householder sequence with a matrix.
   * \param[in]  other  %Matrix being multiplied.
   * \returns    Expression object representing the product.
   *
   * This function computes \f$ HM \f$ where \f$ H \f$ is the Householder sequence represented by \p *this
   * and \f$ M \f$ is the matrix \p other.
   */
  template <typename OtherDerived>
  typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator*(
      const MatrixBase<OtherDerived>& other) const {
    typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type res(
        other.template cast<typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::ResultScalar>());
    applyThisOnTheLeft(res, internal::is_identity<OtherDerived>::value && res.rows() == res.cols());
    return res;
  }

  template <typename VectorsType_, typename CoeffsType_, int Side_>
  friend struct internal::hseq_side_dependent_impl;

  /** \brief Sets the length of the Householder sequence.
   * \param [in]  length  New value for the length.
   *
   * By default, the length \f$ n \f$ of the Householder sequence \f$ H = H_0 H_1 \ldots H_{n-1} \f$ is set
   * to the number of columns of the matrix \p v passed to the constructor, or the number of rows if that
   * is smaller. After this function is called, the length equals \p length.
   *
   * \sa length()
   */
  EIGEN_DEVICE_FUNC HouseholderSequence& setLength(Index length) {
    m_length = length;
    return *this;
  }

  /** \brief Sets the shift of the Householder sequence.
   * \param [in]  shift  New value for the shift.
   *
   * By default, a %HouseholderSequence object represents \f$ H = H_0 H_1 \ldots H_{n-1} \f$ and the i-th
   * column of the matrix \p v passed to the constructor corresponds to the i-th Householder
   * reflection. After this function is called, the object represents \f$ H = H_{\mathrm{shift}}
   * H_{\mathrm{shift}+1} \ldots H_{n-1} \f$ and the i-th column of \p v corresponds to the (shift+i)-th
   * Householder reflection.
   *
   * \sa shift()
   */
  EIGEN_DEVICE_FUNC HouseholderSequence& setShift(Index shift) {
    m_shift = shift;
    return *this;
  }

  EIGEN_DEVICE_FUNC Index length() const {
    return m_length;
  } /**< \brief Returns the length of the Householder sequence. */

  EIGEN_DEVICE_FUNC Index shift() const {
    return m_shift;
  } /**< \brief Returns the shift of the Householder sequence. */

  /* Necessary for .adjoint() and .conjugate() */
  template <typename VectorsType2, typename CoeffsType2, int Side2>
  friend class HouseholderSequence;

 protected:
  /** \internal
   * \brief Sets the reverse flag.
   * \param [in]  reverse  New value of the reverse flag.
   *
   * By default, the reverse flag is not set. If the reverse flag is set, then this object represents
   * \f$ H^r = H_{n-1} \ldots H_1 H_0 \f$ instead of \f$ H = H_0 H_1 \ldots H_{n-1} \f$.
   * \note For real valued HouseholderSequence this is equivalent to transposing \f$ H \f$.
   *
   * \sa reverseFlag(), transpose(), adjoint()
   */
  HouseholderSequence& setReverseFlag(bool reverse) {
    m_reverse = reverse;
    return *this;
  }

  bool reverseFlag() const { return m_reverse; } /**< \internal \brief Returns the reverse flag. */

  typename VectorsType::Nested m_vectors;
  typename CoeffsType::Nested m_coeffs;
  bool m_reverse;
  Index m_length;
  Index m_shift;
  enum { BlockSize = 48 };
};

/** \brief Computes the product of a matrix with a Householder sequence.
 * \param[in]  other  %Matrix being multiplied.
 * \param[in]  h      %HouseholderSequence being multiplied.
 * \returns    Expression object representing the product.
 *
 * This function computes \f$ MH \f$ where \f$ M \f$ is the matrix \p other and \f$ H \f$ is the
 * Householder sequence represented by \p h.
 */
template <typename OtherDerived, typename VectorsType, typename CoeffsType, int Side>
typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar, OtherDerived>::Type operator*(
    const MatrixBase<OtherDerived>& other, const HouseholderSequence<VectorsType, CoeffsType, Side>& h) {
  typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar, OtherDerived>::Type res(
      other.template cast<typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,
                                                                           OtherDerived>::ResultScalar>());
  h.applyThisOnTheRight(res);
  return res;
}

/** \ingroup Householder_Module
 * \householder_module
 * \brief Convenience function for constructing a Householder sequence.
 * \returns A HouseholderSequence constructed from the specified arguments.
 */
template <typename VectorsType, typename CoeffsType>
HouseholderSequence<VectorsType, CoeffsType> householderSequence(const VectorsType& v, const CoeffsType& h) {
  return HouseholderSequence<VectorsType, CoeffsType, OnTheLeft>(v, h);
}

/** \ingroup Householder_Module
 * \householder_module
 * \brief Convenience function for constructing a Householder sequence.
 * \returns A HouseholderSequence constructed from the specified arguments.
 * \details This function differs from householderSequence() in that the template argument \p OnTheSide of
 * the constructed HouseholderSequence is set to OnTheRight, instead of the default OnTheLeft.
 */
template <typename VectorsType, typename CoeffsType>
HouseholderSequence<VectorsType, CoeffsType, OnTheRight> rightHouseholderSequence(const VectorsType& v,
                                                                                  const CoeffsType& h) {
  return HouseholderSequence<VectorsType, CoeffsType, OnTheRight>(v, h);
}

}  // end namespace Eigen

#endif  // EIGEN_HOUSEHOLDER_SEQUENCE_H