SparseQR.h

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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012-2013 Desire Nuentsa <desire.nuentsa_wakam@inria.fr>
// Copyright (C) 2012-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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_SPARSE_QR_H
#define EIGEN_SPARSE_QR_H

namespace Eigen {

template<typename MatrixType, typename OrderingType> class SparseQR;
template<typename SparseQRType> struct SparseQRMatrixQReturnType;
template<typename SparseQRType> struct SparseQRMatrixQTransposeReturnType;
template<typename SparseQRType, typename Derived> struct SparseQR_QProduct;
namespace internal {
  template <typename SparseQRType> struct traits<SparseQRMatrixQReturnType<SparseQRType> >
  {
    typedef typename SparseQRType::MatrixType ReturnType;
    typedef typename ReturnType::StorageIndex StorageIndex;
    typedef typename ReturnType::StorageKind StorageKind;
    enum {
      RowsAtCompileTime = Dynamic,
      ColsAtCompileTime = Dynamic
    };
  };
  template <typename SparseQRType> struct traits<SparseQRMatrixQTransposeReturnType<SparseQRType> >
  {
    typedef typename SparseQRType::MatrixType ReturnType;
  };
  template <typename SparseQRType, typename Derived> struct traits<SparseQR_QProduct<SparseQRType, Derived> >
  {
    typedef typename Derived::PlainObject ReturnType;
  };
} // End namespace internal

template<typename _MatrixType, typename _OrderingType>
class SparseQR : public SparseSolverBase<SparseQR<_MatrixType,_OrderingType> >
{
  protected:
    typedef SparseSolverBase<SparseQR<_MatrixType,_OrderingType> > Base;
    using Base::m_isInitialized;
  public:
    using Base::_solve_impl;
    typedef _MatrixType MatrixType;
    typedef _OrderingType OrderingType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef typename MatrixType::StorageIndex StorageIndex;
    typedef SparseMatrix<Scalar,ColMajor,StorageIndex> QRMatrixType;
    typedef Matrix<StorageIndex, Dynamic, 1> IndexVector;
    typedef Matrix<Scalar, Dynamic, 1> ScalarVector;
    typedef PermutationMatrix<Dynamic, Dynamic, StorageIndex> PermutationType;

    enum {
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    
  public:
    SparseQR () :  m_analysisIsok(false), m_lastError(""), m_useDefaultThreshold(true),m_isQSorted(false),m_isEtreeOk(false)
    { }
    
    explicit SparseQR(const MatrixType& mat) : m_analysisIsok(false), m_lastError(""), m_useDefaultThreshold(true),m_isQSorted(false),m_isEtreeOk(false)
    {
      compute(mat);
    }
    
    void compute(const MatrixType& mat)
    {
      analyzePattern(mat);
      factorize(mat);
    }
    void analyzePattern(const MatrixType& mat);
    void factorize(const MatrixType& mat);
    
    inline Index rows() const { return m_pmat.rows(); }
    
    inline Index cols() const { return m_pmat.cols();}
    
    const QRMatrixType& matrixR() const { return m_R; }
    
    Index rank() const
    {
      eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
      return m_nonzeropivots; 
    }
    
    SparseQRMatrixQReturnType<SparseQR> matrixQ() const 
    { return SparseQRMatrixQReturnType<SparseQR>(*this); }
    
    const PermutationType& colsPermutation() const
    { 
      eigen_assert(m_isInitialized && "Decomposition is not initialized.");
      return m_outputPerm_c;
    }
    
    std::string lastErrorMessage() const { return m_lastError; }
    
    template<typename Rhs, typename Dest>
    bool _solve_impl(const MatrixBase<Rhs> &B, MatrixBase<Dest> &dest) const
    {
      eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
      eigen_assert(this->rows() == B.rows() && "SparseQR::solve() : invalid number of rows in the right hand side matrix");

      Index rank = this->rank();
      
      // Compute Q^T * b;
      typename Dest::PlainObject y, b;
      y = this->matrixQ().transpose() * B; 
      b = y;
      
      // Solve with the triangular matrix R
      y.resize((std::max<Index>)(cols(),y.rows()),y.cols());
      y.topRows(rank) = this->matrixR().topLeftCorner(rank, rank).template triangularView<Upper>().solve(b.topRows(rank));
      y.bottomRows(y.rows()-rank).setZero();
      
      // Apply the column permutation
      if (m_perm_c.size())  dest = colsPermutation() * y.topRows(cols());
      else                  dest = y.topRows(cols());
      
      m_info = Success;
      return true;
    }

    void setPivotThreshold(const RealScalar& threshold)
    {
      m_useDefaultThreshold = false;
      m_threshold = threshold;
    }
    
    template<typename Rhs>
    inline const Solve<SparseQR, Rhs> solve(const MatrixBase<Rhs>& B) const 
    {
      eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
      eigen_assert(this->rows() == B.rows() && "SparseQR::solve() : invalid number of rows in the right hand side matrix");
      return Solve<SparseQR, Rhs>(*this, B.derived());
    }
    template<typename Rhs>
    inline const Solve<SparseQR, Rhs> solve(const SparseMatrixBase<Rhs>& B) const
    {
          eigen_assert(m_isInitialized && "The factorization should be called first, use compute()");
          eigen_assert(this->rows() == B.rows() && "SparseQR::solve() : invalid number of rows in the right hand side matrix");
          return Solve<SparseQR, Rhs>(*this, B.derived());
    }
    
    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "Decomposition is not initialized.");
      return m_info;
    }


    inline void _sort_matrix_Q()
    {
      if(this->m_isQSorted) return;
      // The matrix Q is sorted during the transposition
      SparseMatrix<Scalar, RowMajor, Index> mQrm(this->m_Q);
      this->m_Q = mQrm;
      this->m_isQSorted = true;
    }

    
  protected:
    bool m_analysisIsok;
    bool m_factorizationIsok;
    mutable ComputationInfo m_info;
    std::string m_lastError;
    QRMatrixType m_pmat;            // Temporary matrix
    QRMatrixType m_R;               // The triangular factor matrix
    QRMatrixType m_Q;               // The orthogonal reflectors
    ScalarVector m_hcoeffs;         // The Householder coefficients
    PermutationType m_perm_c;       // Fill-reducing  Column  permutation
    PermutationType m_pivotperm;    // The permutation for rank revealing
    PermutationType m_outputPerm_c; // The final column permutation
    RealScalar m_threshold;         // Threshold to determine null Householder reflections
    bool m_useDefaultThreshold;     // Use default threshold
    Index m_nonzeropivots;          // Number of non zero pivots found
    IndexVector m_etree;            // Column elimination tree
    IndexVector m_firstRowElt;      // First element in each row
    bool m_isQSorted;               // whether Q is sorted or not
    bool m_isEtreeOk;               // whether the elimination tree match the initial input matrix
    
    template <typename, typename > friend struct SparseQR_QProduct;
    
};

template <typename MatrixType, typename OrderingType>
void SparseQR<MatrixType,OrderingType>::analyzePattern(const MatrixType& mat)
{
  eigen_assert(mat.isCompressed() && "SparseQR requires a sparse matrix in compressed mode. Call .makeCompressed() before passing it to SparseQR");
  // Copy to a column major matrix if the input is rowmajor
  typename internal::conditional<MatrixType::IsRowMajor,QRMatrixType,const MatrixType&>::type matCpy(mat);
  // Compute the column fill reducing ordering
  OrderingType ord; 
  ord(matCpy, m_perm_c); 
  Index n = mat.cols();
  Index m = mat.rows();
  Index diagSize = (std::min)(m,n);
  
  if (!m_perm_c.size())
  {
    m_perm_c.resize(n);
    m_perm_c.indices().setLinSpaced(n, 0,StorageIndex(n-1));
  }
  
  // Compute the column elimination tree of the permuted matrix
  m_outputPerm_c = m_perm_c.inverse();
  internal::coletree(matCpy, m_etree, m_firstRowElt, m_outputPerm_c.indices().data());
  m_isEtreeOk = true;
  
  m_R.resize(m, n);
  m_Q.resize(m, diagSize);
  
  // Allocate space for nonzero elements : rough estimation
  m_R.reserve(2*mat.nonZeros()); //FIXME Get a more accurate estimation through symbolic factorization with the etree
  m_Q.reserve(2*mat.nonZeros());
  m_hcoeffs.resize(diagSize);
  m_analysisIsok = true;
}

template <typename MatrixType, typename OrderingType>
void SparseQR<MatrixType,OrderingType>::factorize(const MatrixType& mat)
{
  using std::abs;
  
  eigen_assert(m_analysisIsok && "analyzePattern() should be called before this step");
  StorageIndex m = StorageIndex(mat.rows());
  StorageIndex n = StorageIndex(mat.cols());
  StorageIndex diagSize = (std::min)(m,n);
  IndexVector mark((std::max)(m,n)); mark.setConstant(-1);  // Record the visited nodes
  IndexVector Ridx(n), Qidx(m);                             // Store temporarily the row indexes for the current column of R and Q
  Index nzcolR, nzcolQ;                                     // Number of nonzero for the current column of R and Q
  ScalarVector tval(m);                                     // The dense vector used to compute the current column
  RealScalar pivotThreshold = m_threshold;
  
  m_R.setZero();
  m_Q.setZero();
  m_pmat = mat;
  if(!m_isEtreeOk)
  {
    m_outputPerm_c = m_perm_c.inverse();
    internal::coletree(m_pmat, m_etree, m_firstRowElt, m_outputPerm_c.indices().data());
    m_isEtreeOk = true;
  }

  m_pmat.uncompress(); // To have the innerNonZeroPtr allocated
  
  // Apply the fill-in reducing permutation lazily:
  {
    // If the input is row major, copy the original column indices,
    // otherwise directly use the input matrix
    // 
    IndexVector originalOuterIndicesCpy;
    const StorageIndex *originalOuterIndices = mat.outerIndexPtr();
    if(MatrixType::IsRowMajor)
    {
      originalOuterIndicesCpy = IndexVector::Map(m_pmat.outerIndexPtr(),n+1);
      originalOuterIndices = originalOuterIndicesCpy.data();
    }
    
    for (int i = 0; i < n; i++)
    {
      Index p = m_perm_c.size() ? m_perm_c.indices()(i) : i;
      m_pmat.outerIndexPtr()[p] = originalOuterIndices[i]; 
      m_pmat.innerNonZeroPtr()[p] = originalOuterIndices[i+1] - originalOuterIndices[i]; 
    }
  }
  
  /* Compute the default threshold as in MatLab, see:
   * Tim Davis, "Algorithm 915, SuiteSparseQR: Multifrontal Multithreaded Rank-Revealing
   * Sparse QR Factorization, ACM Trans. on Math. Soft. 38(1), 2011, Page 8:3 
   */
  if(m_useDefaultThreshold) 
  {
    RealScalar max2Norm = 0.0;
    for (int j = 0; j < n; j++) max2Norm = numext::maxi(max2Norm, m_pmat.col(j).norm());
    if(max2Norm==RealScalar(0))
      max2Norm = RealScalar(1);
    pivotThreshold = 20 * (m + n) * max2Norm * NumTraits<RealScalar>::epsilon();
  }
  
  // Initialize the numerical permutation
  m_pivotperm.setIdentity(n);
  
  StorageIndex nonzeroCol = 0; // Record the number of valid pivots
  m_Q.startVec(0);

  // Left looking rank-revealing QR factorization: compute a column of R and Q at a time
  for (StorageIndex col = 0; col < n; ++col)
  {
    mark.setConstant(-1);
    m_R.startVec(col);
    mark(nonzeroCol) = col;
    Qidx(0) = nonzeroCol;
    nzcolR = 0; nzcolQ = 1;
    bool found_diag = nonzeroCol>=m;
    tval.setZero(); 
    
    // Symbolic factorization: find the nonzero locations of the column k of the factors R and Q, i.e.,
    // all the nodes (with indexes lower than rank) reachable through the column elimination tree (etree) rooted at node k.
    // Note: if the diagonal entry does not exist, then its contribution must be explicitly added,
    // thus the trick with found_diag that permits to do one more iteration on the diagonal element if this one has not been found.
    for (typename QRMatrixType::InnerIterator itp(m_pmat, col); itp || !found_diag; ++itp)
    {
      StorageIndex curIdx = nonzeroCol;
      if(itp) curIdx = StorageIndex(itp.row());
      if(curIdx == nonzeroCol) found_diag = true;
      
      // Get the nonzeros indexes of the current column of R
      StorageIndex st = m_firstRowElt(curIdx); // The traversal of the etree starts here
      if (st < 0 )
      {
        m_lastError = "Empty row found during numerical factorization";
        m_info = InvalidInput;
        return;
      }

      // Traverse the etree 
      Index bi = nzcolR;
      for (; mark(st) != col; st = m_etree(st))
      {
        Ridx(nzcolR) = st;  // Add this row to the list,
        mark(st) = col;     // and mark this row as visited
        nzcolR++;
      }

      // Reverse the list to get the topological ordering
      Index nt = nzcolR-bi;
      for(Index i = 0; i < nt/2; i++) std::swap(Ridx(bi+i), Ridx(nzcolR-i-1));
       
      // Copy the current (curIdx,pcol) value of the input matrix
      if(itp) tval(curIdx) = itp.value();
      else    tval(curIdx) = Scalar(0);
      
      // Compute the pattern of Q(:,k)
      if(curIdx > nonzeroCol && mark(curIdx) != col ) 
      {
        Qidx(nzcolQ) = curIdx;  // Add this row to the pattern of Q,
        mark(curIdx) = col;     // and mark it as visited
        nzcolQ++;
      }
    }

    // Browse all the indexes of R(:,col) in reverse order
    for (Index i = nzcolR-1; i >= 0; i--)
    {
      Index curIdx = Ridx(i);
      
      // Apply the curIdx-th householder vector to the current column (temporarily stored into tval)
      Scalar tdot(0);
      
      // First compute q' * tval
      tdot = m_Q.col(curIdx).dot(tval);

      tdot *= m_hcoeffs(curIdx);
      
      // Then update tval = tval - q * tau
      // FIXME: tval -= tdot * m_Q.col(curIdx) should amount to the same (need to check/add support for efficient "dense ?= sparse")
      for (typename QRMatrixType::InnerIterator itq(m_Q, curIdx); itq; ++itq)
        tval(itq.row()) -= itq.value() * tdot;

      // Detect fill-in for the current column of Q
      if(m_etree(Ridx(i)) == nonzeroCol)
      {
        for (typename QRMatrixType::InnerIterator itq(m_Q, curIdx); itq; ++itq)
        {
          StorageIndex iQ = StorageIndex(itq.row());
          if (mark(iQ) != col)
          {
            Qidx(nzcolQ++) = iQ;  // Add this row to the pattern of Q,
            mark(iQ) = col;       // and mark it as visited
          }
        }
      }
    } // End update current column
    
    Scalar tau = RealScalar(0);
    RealScalar beta = 0;
    
    if(nonzeroCol < diagSize)
    {
      // Compute the Householder reflection that eliminate the current column
      // FIXME this step should call the Householder module.
      Scalar c0 = nzcolQ ? tval(Qidx(0)) : Scalar(0);
      
      // First, the squared norm of Q((col+1):m, col)
      RealScalar sqrNorm = 0.;
      for (Index itq = 1; itq < nzcolQ; ++itq) sqrNorm += numext::abs2(tval(Qidx(itq)));
      if(sqrNorm == RealScalar(0) && numext::imag(c0) == RealScalar(0))
      {
        beta = numext::real(c0);
        tval(Qidx(0)) = 1;
      }
      else
      {
        using std::sqrt;
        beta = sqrt(numext::abs2(c0) + sqrNorm);
        if(numext::real(c0) >= RealScalar(0))
          beta = -beta;
        tval(Qidx(0)) = 1;
        for (Index itq = 1; itq < nzcolQ; ++itq)
          tval(Qidx(itq)) /= (c0 - beta);
        tau = numext::conj((beta-c0) / beta);
          
      }
    }

    // Insert values in R
    for (Index  i = nzcolR-1; i >= 0; i--)
    {
      Index curIdx = Ridx(i);
      if(curIdx < nonzeroCol) 
      {
        m_R.insertBackByOuterInnerUnordered(col, curIdx) = tval(curIdx);
        tval(curIdx) = Scalar(0.);
      }
    }

    if(nonzeroCol < diagSize && abs(beta) >= pivotThreshold)
    {
      m_R.insertBackByOuterInner(col, nonzeroCol) = beta;
      // The householder coefficient
      m_hcoeffs(nonzeroCol) = tau;
      // Record the householder reflections
      for (Index itq = 0; itq < nzcolQ; ++itq)
      {
        Index iQ = Qidx(itq);
        m_Q.insertBackByOuterInnerUnordered(nonzeroCol,iQ) = tval(iQ);
        tval(iQ) = Scalar(0.);
      }
      nonzeroCol++;
      if(nonzeroCol<diagSize)
        m_Q.startVec(nonzeroCol);
    }
    else
    {
      // Zero pivot found: move implicitly this column to the end
      for (Index j = nonzeroCol; j < n-1; j++) 
        std::swap(m_pivotperm.indices()(j), m_pivotperm.indices()[j+1]);
      
      // Recompute the column elimination tree
      internal::coletree(m_pmat, m_etree, m_firstRowElt, m_pivotperm.indices().data());
      m_isEtreeOk = false;
    }
  }
  
  m_hcoeffs.tail(diagSize-nonzeroCol).setZero();
  
  // Finalize the column pointers of the sparse matrices R and Q
  m_Q.finalize();
  m_Q.makeCompressed();
  m_R.finalize();
  m_R.makeCompressed();
  m_isQSorted = false;

  m_nonzeropivots = nonzeroCol;
  
  if(nonzeroCol<n)
  {
    // Permute the triangular factor to put the 'dead' columns to the end
    QRMatrixType tempR(m_R);
    m_R = tempR * m_pivotperm;
    
    // Update the column permutation
    m_outputPerm_c = m_outputPerm_c * m_pivotperm;
  }
  
  m_isInitialized = true; 
  m_factorizationIsok = true;
  m_info = Success;
}

template <typename SparseQRType, typename Derived>
struct SparseQR_QProduct : ReturnByValue<SparseQR_QProduct<SparseQRType, Derived> >
{
  typedef typename SparseQRType::QRMatrixType MatrixType;
  typedef typename SparseQRType::Scalar Scalar;
  // Get the references 
  SparseQR_QProduct(const SparseQRType& qr, const Derived& other, bool transpose) : 
  m_qr(qr),m_other(other),m_transpose(transpose) {}
  inline Index rows() const { return m_transpose ? m_qr.rows() : m_qr.cols(); }
  inline Index cols() const { return m_other.cols(); }
  
  // Assign to a vector
  template<typename DesType>
  void evalTo(DesType& res) const
  {
    Index m = m_qr.rows();
    Index n = m_qr.cols();
    Index diagSize = (std::min)(m,n);
    res = m_other;
    if (m_transpose)
    {
      eigen_assert(m_qr.m_Q.rows() == m_other.rows() && "Non conforming object sizes");
      //Compute res = Q' * other column by column
      for(Index j = 0; j < res.cols(); j++){
        for (Index k = 0; k < diagSize; k++)
        {
          Scalar tau = Scalar(0);
          tau = m_qr.m_Q.col(k).dot(res.col(j));
          if(tau==Scalar(0)) continue;
          tau = tau * m_qr.m_hcoeffs(k);
          res.col(j) -= tau * m_qr.m_Q.col(k);
        }
      }
    }
    else
    {
      eigen_assert(m_qr.m_Q.rows() == m_other.rows() && "Non conforming object sizes");
      // Compute res = Q * other column by column
      for(Index j = 0; j < res.cols(); j++)
      {
        for (Index k = diagSize-1; k >=0; k--)
        {
          Scalar tau = Scalar(0);
          tau = m_qr.m_Q.col(k).dot(res.col(j));
          if(tau==Scalar(0)) continue;
          tau = tau * m_qr.m_hcoeffs(k);
          res.col(j) -= tau * m_qr.m_Q.col(k);
        }
      }
    }
  }
  
  const SparseQRType& m_qr;
  const Derived& m_other;
  bool m_transpose;
};

template<typename SparseQRType>
struct SparseQRMatrixQReturnType : public EigenBase<SparseQRMatrixQReturnType<SparseQRType> >
{  
  typedef typename SparseQRType::Scalar Scalar;
  typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
  enum {
    RowsAtCompileTime = Dynamic,
    ColsAtCompileTime = Dynamic
  };
  explicit SparseQRMatrixQReturnType(const SparseQRType& qr) : m_qr(qr) {}
  template<typename Derived>
  SparseQR_QProduct<SparseQRType, Derived> operator*(const MatrixBase<Derived>& other)
  {
    return SparseQR_QProduct<SparseQRType,Derived>(m_qr,other.derived(),false);
  }
  SparseQRMatrixQTransposeReturnType<SparseQRType> adjoint() const
  {
    return SparseQRMatrixQTransposeReturnType<SparseQRType>(m_qr);
  }
  inline Index rows() const { return m_qr.rows(); }
  inline Index cols() const { return (std::min)(m_qr.rows(),m_qr.cols()); }
  // To use for operations with the transpose of Q
  SparseQRMatrixQTransposeReturnType<SparseQRType> transpose() const
  {
    return SparseQRMatrixQTransposeReturnType<SparseQRType>(m_qr);
  }
  const SparseQRType& m_qr;
};

template<typename SparseQRType>
struct SparseQRMatrixQTransposeReturnType
{
  explicit SparseQRMatrixQTransposeReturnType(const SparseQRType& qr) : m_qr(qr) {}
  template<typename Derived>
  SparseQR_QProduct<SparseQRType,Derived> operator*(const MatrixBase<Derived>& other)
  {
    return SparseQR_QProduct<SparseQRType,Derived>(m_qr,other.derived(), true);
  }
  const SparseQRType& m_qr;
};

namespace internal {
  
template<typename SparseQRType>
struct evaluator_traits<SparseQRMatrixQReturnType<SparseQRType> >
{
  typedef typename SparseQRType::MatrixType MatrixType;
  typedef typename storage_kind_to_evaluator_kind<typename MatrixType::StorageKind>::Kind Kind;
  typedef SparseShape Shape;
};

template< typename DstXprType, typename SparseQRType>
struct Assignment<DstXprType, SparseQRMatrixQReturnType<SparseQRType>, internal::assign_op<typename DstXprType::Scalar,typename DstXprType::Scalar>, Sparse2Sparse>
{
  typedef SparseQRMatrixQReturnType<SparseQRType> SrcXprType;
  typedef typename DstXprType::Scalar Scalar;
  typedef typename DstXprType::StorageIndex StorageIndex;
  static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar,Scalar> &/*func*/)
  {
    typename DstXprType::PlainObject idMat(src.m_qr.rows(), src.m_qr.rows());
    idMat.setIdentity();
    // Sort the sparse householder reflectors if needed
    const_cast<SparseQRType *>(&src.m_qr)->_sort_matrix_Q();
    dst = SparseQR_QProduct<SparseQRType, DstXprType>(src.m_qr, idMat, false);
  }
};

template< typename DstXprType, typename SparseQRType>
struct Assignment<DstXprType, SparseQRMatrixQReturnType<SparseQRType>, internal::assign_op<typename DstXprType::Scalar,typename DstXprType::Scalar>, Sparse2Dense>
{
  typedef SparseQRMatrixQReturnType<SparseQRType> SrcXprType;
  typedef typename DstXprType::Scalar Scalar;
  typedef typename DstXprType::StorageIndex StorageIndex;
  static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<Scalar,Scalar> &/*func*/)
  {
    dst = src.m_qr.matrixQ() * DstXprType::Identity(src.m_qr.rows(), src.m_qr.rows());
  }
};

} // end namespace internal

} // end namespace Eigen

#endif