HouseholderQR.h

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// 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/.
 
#ifndef EIGEN_QR_H
#define EIGEN_QR_H
 
#include "./InternalHeaderCheck.h"
 
namespace Eigen { 
 
namespace internal {
template<typename MatrixType_> struct traits<HouseholderQR<MatrixType_> >
 : traits<MatrixType_>
{
  typedef MatrixXpr XprKind;
  typedef SolverStorage StorageKind;
  typedef int StorageIndex;
  enum { Flags = 0 };
};
 
} // end namespace internal
 
template<typename MatrixType_> class HouseholderQR
        : public SolverBase<HouseholderQR<MatrixType_> >
{
  public:
 
    typedef MatrixType_ MatrixType;
    typedef SolverBase<HouseholderQR> Base;
    friend class SolverBase<HouseholderQR>;
 
    EIGEN_GENERIC_PUBLIC_INTERFACE(HouseholderQR)
    enum {
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
    typedef HouseholderSequence<MatrixType,internal::remove_all_t<typename HCoeffsType::ConjugateReturnType>> HouseholderSequenceType;
 
    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 const 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;
 
    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();
}
 
namespace internal {
 
template<typename MatrixQR, typename HCoeffs>
void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
{
  typedef typename MatrixQR::Scalar Scalar;
  typedef typename MatrixQR::RealScalar RealScalar;
  Index rows = mat.rows();
  Index cols = mat.cols();
  Index size = (std::min)(rows,cols);
 
  eigen_assert(hCoeffs.size() == size);
 
  typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
  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 m_qr from the left
    mat.bottomRightCorner(remainingRows, remainingCols)
        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
  }
}
 
// TODO: add a corresponding public API for updating a QR factorization
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) {
  typedef typename MatrixQR::Index Index;
  typedef typename MatrixQR::RealScalar 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)
  {
    typedef typename MatrixQR::Scalar Scalar;
    typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
 
    Index rows = mat.rows();
    Index cols = mat.cols();
    Index size = (std::min)(rows, cols);
 
    typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
    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 continue 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
      }
    }
  }
};
 
} // 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);
 
  internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());
 
  m_isInitialized = true;
}
 
template<typename Derived>
const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::householderQr() const
{
  return HouseholderQR<PlainObject>(eval());
}
 
} // end namespace Eigen
 
#endif // EIGEN_QR_H