ColPivHouseholderQR.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 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/.
 
#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
 
namespace Eigen { 
 
namespace internal {
template<typename _MatrixType> struct traits<ColPivHouseholderQR<_MatrixType> >
 : traits<_MatrixType>
{
  enum { Flags = 0 };
};
 
} // end namespace internal
 
template<typename _MatrixType> class ColPivHouseholderQR
{
  public:
 
    typedef _MatrixType MatrixType;
    enum {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      Options = MatrixType::Options,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    // FIXME should be int
    typedef typename MatrixType::StorageIndex StorageIndex;
    typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
    typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
    typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
    typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
    typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
    typedef typename MatrixType::PlainObject PlainObject;
    
  private:
    
    typedef typename PermutationType::StorageIndex PermIndexType;
    
  public:
 
    ColPivHouseholderQR()
      : m_qr(),
        m_hCoeffs(),
        m_colsPermutation(),
        m_colsTranspositions(),
        m_temp(),
        m_colSqNorms(),
        m_isInitialized(false),
        m_usePrescribedThreshold(false) {}
 
    ColPivHouseholderQR(Index rows, Index cols)
      : m_qr(rows, cols),
        m_hCoeffs((std::min)(rows,cols)),
        m_colsPermutation(PermIndexType(cols)),
        m_colsTranspositions(cols),
        m_temp(cols),
        m_colSqNorms(cols),
        m_isInitialized(false),
        m_usePrescribedThreshold(false) {}
 
    template<typename InputType>
    explicit ColPivHouseholderQR(const EigenBase<InputType>& matrix)
      : m_qr(matrix.rows(), matrix.cols()),
        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
        m_colsPermutation(PermIndexType(matrix.cols())),
        m_colsTranspositions(matrix.cols()),
        m_temp(matrix.cols()),
        m_colSqNorms(matrix.cols()),
        m_isInitialized(false),
        m_usePrescribedThreshold(false)
    {
      compute(matrix.derived());
    }
 
    template<typename Rhs>
    inline const Solve<ColPivHouseholderQR, Rhs>
    solve(const MatrixBase<Rhs>& b) const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return Solve<ColPivHouseholderQR, Rhs>(*this, b.derived());
    }
 
    HouseholderSequenceType householderQ() const;
    HouseholderSequenceType matrixQ() const
    {
      return householderQ(); 
    }
 
    const MatrixType& matrixQR() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_qr;
    }
    
    const MatrixType& matrixR() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_qr;
    }
    
    template<typename InputType>
    ColPivHouseholderQR& compute(const EigenBase<InputType>& matrix);
 
    const PermutationType& colsPermutation() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_colsPermutation;
    }
 
    typename MatrixType::RealScalar absDeterminant() const;
 
    typename MatrixType::RealScalar logAbsDeterminant() const;
 
    inline Index rank() const
    {
      using std::abs;
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
      Index result = 0;
      for(Index i = 0; i < m_nonzero_pivots; ++i)
        result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
      return result;
    }
 
    inline Index dimensionOfKernel() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return cols() - rank();
    }
 
    inline bool isInjective() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return rank() == cols();
    }
 
    inline bool isSurjective() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return rank() == rows();
    }
 
    inline bool isInvertible() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return isInjective() && isSurjective();
    }
 
    inline const Inverse<ColPivHouseholderQR> inverse() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return Inverse<ColPivHouseholderQR>(*this);
    }
 
    inline Index rows() const { return m_qr.rows(); }
    inline Index cols() const { return m_qr.cols(); }
    
    const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
 
    ColPivHouseholderQR& setThreshold(const RealScalar& threshold)
    {
      m_usePrescribedThreshold = true;
      m_prescribedThreshold = threshold;
      return *this;
    }
 
    ColPivHouseholderQR& setThreshold(Default_t)
    {
      m_usePrescribedThreshold = false;
      return *this;
    }
 
    RealScalar threshold() const
    {
      eigen_assert(m_isInitialized || m_usePrescribedThreshold);
      return m_usePrescribedThreshold ? m_prescribedThreshold
      // this formula comes from experimenting (see "LU precision tuning" thread on the list)
      // and turns out to be identical to Higham's formula used already in LDLt.
                                      : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
    }
 
    inline Index nonzeroPivots() const
    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_nonzero_pivots;
    }
 
    RealScalar maxPivot() const { return m_maxpivot; }
    
    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "Decomposition is not initialized.");
      return Success;
    }
    
    #ifndef EIGEN_PARSED_BY_DOXYGEN
    template<typename RhsType, typename DstType>
    EIGEN_DEVICE_FUNC
    void _solve_impl(const RhsType &rhs, DstType &dst) const;
    #endif
 
  protected:
    
    static void check_template_parameters()
    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }
    
    void computeInPlace();
    
    MatrixType m_qr;
    HCoeffsType m_hCoeffs;
    PermutationType m_colsPermutation;
    IntRowVectorType m_colsTranspositions;
    RowVectorType m_temp;
    RealRowVectorType m_colSqNorms;
    bool m_isInitialized, m_usePrescribedThreshold;
    RealScalar m_prescribedThreshold, m_maxpivot;
    Index m_nonzero_pivots;
    Index m_det_pq;
};
 
template<typename MatrixType>
typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
{
  using std::abs;
  eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
{
  eigen_assert(m_isInitialized && "ColPivHouseholderQR 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>
template<typename InputType>
ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const EigenBase<InputType>& matrix)
{
  check_template_parameters();
  
  // the column permutation is stored as int indices, so just to be sure:
  eigen_assert(matrix.cols()<=NumTraits<int>::highest());
 
  m_qr = matrix;
  
  computeInPlace();
  
  return *this;
}
 
template<typename MatrixType>
void ColPivHouseholderQR<MatrixType>::computeInPlace()
{
  using std::abs;
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = m_qr.diagonalSize();
  
  m_hCoeffs.resize(size);
 
  m_temp.resize(cols);
 
  m_colsTranspositions.resize(m_qr.cols());
  Index number_of_transpositions = 0;
 
  m_colSqNorms.resize(cols);
  for(Index k = 0; k < cols; ++k)
    m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
 
  RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);
 
  m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
  m_maxpivot = RealScalar(0);
 
  for(Index k = 0; k < size; ++k)
  {
    // first, we look up in our table m_colSqNorms which column has the biggest squared norm
    Index biggest_col_index;
    RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
    biggest_col_index += k;
 
    // since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
    // the actual squared norm of the selected column.
    // Note that not doing so does result in solve() sometimes returning inf/nan values
    // when running the unit test with 1000 repetitions.
    biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();
 
    // we store that back into our table: it can't hurt to correct our table.
    m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;
 
    // Track the number of meaningful pivots but do not stop the decomposition to make
    // sure that the initial matrix is properly reproduced. See bug 941.
    if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
      m_nonzero_pivots = k;
 
    // apply the transposition to the columns
    m_colsTranspositions.coeffRef(k) = biggest_col_index;
    if(k != biggest_col_index) {
      m_qr.col(k).swap(m_qr.col(biggest_col_index));
      std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.coeffRef(biggest_col_index));
      ++number_of_transpositions;
    }
 
    // generate the householder vector, store it below the diagonal
    RealScalar beta;
    m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
 
    // apply the householder transformation to the diagonal coefficient
    m_qr.coeffRef(k,k) = beta;
 
    // remember the maximum absolute value of diagonal coefficients
    if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
 
    // apply the householder transformation
    m_qr.bottomRightCorner(rows-k, cols-k-1)
        .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
 
    // update our table of squared norms of the columns
    m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
  }
 
  m_colsPermutation.setIdentity(PermIndexType(cols));
  for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
    m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
 
  m_det_pq = (number_of_transpositions%2) ? -1 : 1;
  m_isInitialized = true;
}
 
#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename _MatrixType>
template<typename RhsType, typename DstType>
void ColPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
{
  eigen_assert(rhs.rows() == rows());
 
  const Index nonzero_pivots = nonzeroPivots();
 
  if(nonzero_pivots == 0)
  {
    dst.setZero();
    return;
  }
 
  typename RhsType::PlainObject c(rhs);
 
  // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
  c.applyOnTheLeft(householderSequence(m_qr, m_hCoeffs)
                    .setLength(nonzero_pivots)
                    .transpose()
    );
 
  m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
      .template triangularView<Upper>()
      .solveInPlace(c.topRows(nonzero_pivots));
 
  for(Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
  for(Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
}
#endif
 
namespace internal {
 
template<typename DstXprType, typename MatrixType, typename Scalar>
struct Assignment<DstXprType, Inverse<ColPivHouseholderQR<MatrixType> >, internal::assign_op<Scalar>, Dense2Dense, Scalar>
{
  typedef ColPivHouseholderQR<MatrixType> QrType;
  typedef Inverse<QrType> SrcXprType;
  static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar> &)
  {    
    dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
  }
};
 
} // end namespace internal
 
template<typename MatrixType>
typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
  ::householderQ() const
{
  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
  return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
}
 
#ifndef __CUDACC__
template<typename Derived>
const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::colPivHouseholderQr() const
{
  return ColPivHouseholderQR<PlainObject>(eval());
}
#endif // __CUDACC__
 
} // end namespace Eigen
 
#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H