HouseholderQR.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2010 Vincent Lejeune
//
// 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_QR_H
#define EIGEN_QR_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
template <typename MatrixType_>
struct traits<HouseholderQR<MatrixType_>> : traits<MatrixType_> {
  using XprKind = MatrixXpr;
  using StorageKind = SolverStorage;
  using StorageIndex = int;
  enum { Flags = 0 };
};
 
}  // end namespace internal

template <typename MatrixType_>
class HouseholderQR : public SolverBase<HouseholderQR<MatrixType_>> {
 public:
  using MatrixType = MatrixType_;
  using Base = SolverBase<HouseholderQR>;
  friend class SolverBase<HouseholderQR>;
 
  EIGEN_GENERIC_PUBLIC_INTERFACE(HouseholderQR)
  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };
  using MatrixQType =
      Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags & RowMajorBit) ? RowMajor : ColMajor,
             MaxRowsAtCompileTime, MaxRowsAtCompileTime>;
  using HCoeffsType = typename internal::plain_diag_type<MatrixType>::type;
  using RowVectorType = typename internal::plain_row_type<MatrixType>::type;
  using HouseholderSequenceType =
      HouseholderSequence<MatrixType, internal::remove_all_t<typename HCoeffsType::ConjugateReturnType>>;

  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "HouseHolderQR is not initialized.");
    return Success;
  }

  HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}

  HouseholderQR(Index rows, Index cols)
      : m_qr(rows, cols), m_hCoeffs((std::min)(rows, cols)), m_temp(cols), m_isInitialized(false) {}

  template <typename InputType>
  explicit HouseholderQR(const EigenBase<InputType>& matrix)
      : m_qr(matrix.rows(), matrix.cols()),
        m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_temp(matrix.cols()),
        m_isInitialized(false) {
    compute(matrix.derived());
  }

  template <typename InputType>
  explicit HouseholderQR(EigenBase<InputType>& matrix)
      : m_qr(matrix.derived()),
        m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_temp(matrix.cols()),
        m_isInitialized(false) {
    computeInPlace();
  }
 
#ifdef EIGEN_PARSED_BY_DOXYGEN
  template <typename Rhs>
  inline Solve<HouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

  HouseholderSequenceType householderQ() const {
    eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
    return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
  }

  const MatrixType& matrixQR() const {
    eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
    return m_qr;
  }
 
  template <typename InputType>
  HouseholderQR& compute(const EigenBase<InputType>& matrix) {
    m_qr = matrix.derived();
    computeInPlace();
    return *this;
  }

  typename MatrixType::Scalar determinant() const;

  typename MatrixType::RealScalar absDeterminant() const;

  typename MatrixType::RealScalar logAbsDeterminant() const;

  typename MatrixType::Scalar signDeterminant() const;
 
  inline Index rows() const { return m_qr.rows(); }
  inline Index cols() const { return m_qr.cols(); }

  const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
 
#ifndef EIGEN_PARSED_BY_DOXYGEN
  template <typename RhsType, typename DstType>
  void _solve_impl(const RhsType& rhs, DstType& dst) const;
 
  template <bool Conjugate, typename RhsType, typename DstType>
  void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif
 
 protected:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
 
  void computeInPlace();
 
  MatrixType m_qr;
  HCoeffsType m_hCoeffs;
  RowVectorType m_temp;
  bool m_isInitialized;
};
 
namespace internal {

template <typename HCoeffs, typename Scalar, bool IsComplex>
struct householder_determinant {
  static void run(const HCoeffs& hCoeffs, Scalar& out_det) {
    out_det = Scalar(1);
    Index size = hCoeffs.rows();
    for (Index i = 0; i < size; i++) {
      // For each valid reflection Q_n,
      // det(Q_n) = - conj(h_n) / h_n
      // where h_n is the Householder coefficient.
      if (hCoeffs(i) != Scalar(0)) out_det *= -numext::conj(hCoeffs(i)) / hCoeffs(i);
    }
  }
};

template <typename HCoeffs, typename Scalar>
struct householder_determinant<HCoeffs, Scalar, false> {
  static void run(const HCoeffs& hCoeffs, Scalar& out_det) {
    bool negated = false;
    Index size = hCoeffs.rows();
    for (Index i = 0; i < size; i++) {
      // Each valid reflection negates the determinant.
      if (hCoeffs(i) != Scalar(0)) negated ^= true;
    }
    out_det = negated ? Scalar(-1) : Scalar(1);
  }
};
 
}  // end namespace internal
 
template <typename MatrixType>
typename MatrixType::Scalar HouseholderQR<MatrixType>::determinant() const {
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  Scalar detQ;
  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
  return m_qr.diagonal().prod() * detQ;
}
 
template <typename MatrixType>
typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const {
  using std::abs;
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  return abs(m_qr.diagonal().prod());
}
 
template <typename MatrixType>
typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const {
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  return m_qr.diagonal().cwiseAbs().array().log().sum();
}
 
template <typename MatrixType>
typename MatrixType::Scalar HouseholderQR<MatrixType>::signDeterminant() const {
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  Scalar detQ;
  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
  return detQ * m_qr.diagonal().array().sign().prod();
}
 
namespace internal {

template <typename MatrixQR, typename HCoeffs>
void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0) {
  using Scalar = typename MatrixQR::Scalar;
  using RealScalar = typename MatrixQR::RealScalar;
  Index rows = mat.rows();
  Index cols = mat.cols();
  Index size = (std::min)(rows, cols);
 
  eigen_assert(hCoeffs.size() == size);
 
  using TempType = Matrix<Scalar, MatrixQR::ColsAtCompileTime, 1>;
  TempType tempVector;
  if (tempData == 0) {
    tempVector.resize(cols);
    tempData = tempVector.data();
  }
 
  for (Index k = 0; k < size; ++k) {
    Index remainingRows = rows - k;
    Index remainingCols = cols - k - 1;
 
    RealScalar beta;
    mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
    mat.coeffRef(k, k) = beta;
 
    // apply H to remaining part of mat from the left
    mat.bottomRightCorner(remainingRows, remainingCols)
        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows - 1), hCoeffs.coeffRef(k), tempData + k + 1);
  }
}

template <typename MatrixQR, typename HCoeffs, typename VectorQR>
void householder_qr_inplace_update(MatrixQR& mat, HCoeffs& hCoeffs, const VectorQR& newColumn,
                                   typename MatrixQR::Index k, typename MatrixQR::Scalar* tempData) {
  using Index = typename MatrixQR::Index;
  using RealScalar = typename MatrixQR::RealScalar;
  Index rows = mat.rows();
 
  eigen_assert(k < mat.cols());
  eigen_assert(k < rows);
  eigen_assert(hCoeffs.size() == mat.cols());
  eigen_assert(newColumn.size() == rows);
  eigen_assert(tempData);
 
  // Store new column in mat at column k
  mat.col(k) = newColumn;
  // Apply H = H_1...H_{k-1} on newColumn (skip if k=0)
  for (Index i = 0; i < k; ++i) {
    Index remainingRows = rows - i;
    mat.col(k)
        .tail(remainingRows)
        .applyHouseholderOnTheLeft(mat.col(i).tail(remainingRows - 1), hCoeffs.coeffRef(i), tempData + i + 1);
  }
  // Construct Householder projector in-place in column k
  RealScalar beta;
  mat.col(k).tail(rows - k).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
  mat.coeffRef(k, k) = beta;
}

template <typename MatrixQR, typename HCoeffs, typename MatrixQRScalar = typename MatrixQR::Scalar,
          bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
struct householder_qr_inplace_blocked {
  // This is specialized for LAPACK-supported Scalar types in HouseholderQR_LAPACKE.h
  static void run(MatrixQR& mat, HCoeffs& hCoeffs, Index maxBlockSize = 32, typename MatrixQR::Scalar* tempData = 0) {
    using Scalar = typename MatrixQR::Scalar;
    using BlockType = Block<MatrixQR, Dynamic, Dynamic>;
 
    Index rows = mat.rows();
    Index cols = mat.cols();
    Index size = (std::min)(rows, cols);
 
    using TempType = Matrix<Scalar, Dynamic, 1, ColMajor, MatrixQR::MaxColsAtCompileTime, 1>;
    TempType tempVector;
    if (tempData == 0) {
      tempVector.resize(cols);
      tempData = tempVector.data();
    }
 
    Index blockSize = (std::min)(maxBlockSize, size);
 
    Index k = 0;
    for (k = 0; k < size; k += blockSize) {
      Index bs = (std::min)(size - k, blockSize);  // actual size of the block
      Index tcols = cols - k - bs;                 // trailing columns
      Index brows = rows - k;                      // rows of the block
 
      // partition the matrix:
      //        A00 | A01 | A02
      // mat  = A10 | A11 | A12
      //        A20 | A21 | A22
      // and performs the qr dec of [A11^T A12^T]^T
      // and update [A21^T A22^T]^T using level 3 operations.
      // Finally, the algorithm continues on A22
 
      BlockType A11_21 = mat.block(k, k, brows, bs);
      Block<HCoeffs, Dynamic, 1> hCoeffsSegment = hCoeffs.segment(k, bs);
 
      householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
 
      if (tcols) {
        BlockType A21_22 = mat.block(k, k + bs, brows, tcols);
        apply_block_householder_on_the_left(A21_22, A11_21, hCoeffsSegment, false);  // false == backward
      }
    }
  }
};

template <typename Scalar>
Index householder_qr_panel_width(Index rows, Index cols) {
  const Index size = numext::mini(rows, cols);
  const double m = double(rows), n = double(cols), s = double(size);
  if (m * n * s <= 64.0 * 64.0 * 64.0) return size;
  const double P = m * s - s * s / 2;
  const double Q = m * n * s - (m + n) * s * s / 2 + s * s * s / 3;
  const double target = numext::sqrt(3 * Q / P);
  constexpr Index kPacketSize = packet_traits<Scalar>::size;
  const Index kWidths[] = {8, 16, 24, 32, 48};
  Index width = kPacketSize;
  double distance = NumTraits<double>::highest();
  for (Index w : kWidths) {
    const double d = numext::maxi(double(w) / target, target / double(w));
    if (w % kPacketSize == 0 && d < distance) {
      width = w;
      distance = d;
    }
  }
  return width;
}
 
}  // end namespace internal
 
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename MatrixType_>
template <typename RhsType, typename DstType>
void HouseholderQR<MatrixType_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
  const Index rank = (std::min)(rows(), cols());
 
  typename RhsType::PlainObject c(rhs);
 
  c.applyOnTheLeft(householderQ().setLength(rank).adjoint());
 
  m_qr.topLeftCorner(rank, rank).template triangularView<Upper>().solveInPlace(c.topRows(rank));
 
  dst.topRows(rank) = c.topRows(rank);
  dst.bottomRows(cols() - rank).setZero();
}
 
template <typename MatrixType_>
template <bool Conjugate, typename RhsType, typename DstType>
void HouseholderQR<MatrixType_>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const {
  const Index rank = (std::min)(rows(), cols());
 
  typename RhsType::PlainObject c(rhs);
 
  m_qr.topLeftCorner(rank, rank)
      .template triangularView<Upper>()
      .transpose()
      .template conjugateIf<Conjugate>()
      .solveInPlace(c.topRows(rank));
 
  dst.topRows(rank) = c.topRows(rank);
  dst.bottomRows(rows() - rank).setZero();
 
  dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>());
}
#endif

template <typename MatrixType>
void HouseholderQR<MatrixType>::computeInPlace() {
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = (std::min)(rows, cols);
 
  m_hCoeffs.resize(size);
 
  m_temp.resize(cols);
 
  const Index maxBlockSize = internal::householder_qr_panel_width<Scalar>(rows, cols);
  internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, maxBlockSize, m_temp.data());
 
  m_isInitialized = true;
}

template <typename Derived>
HouseholderQR<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::householderQr() const {
  return HouseholderQR<PlainObject>(eval());
}
 
}  // end namespace Eigen
 
#endif  // EIGEN_QR_H