text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  // 
  // We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD"
  // research report written by Ming Gu and Stanley C.Eisenstat
  // The code variable names correspond to the names they used in their 
  // report
  //
  // Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
  // Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
  // Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
  // Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
  // Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
  // Copyright (C) 2014 Gael Guennebaud <gael.guennebaud@inria.fr>
  //
  // 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_BDCSVD_H
  #define EIGEN_BDCSVD_H
  // #define EIGEN_BDCSVD_DEBUG_VERBOSE
  // #define EIGEN_BDCSVD_SANITY_CHECKS
  namespace Eigen {
  
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
  IOFormat bdcsvdfmt(8, 0, ", ", "\n", "  [", "]");
  #endif
    
  template<typename _MatrixType> class BDCSVD;
  
  namespace internal {
  
  template<typename _MatrixType> 
  struct traits<BDCSVD<_MatrixType> >
  {
    typedef _MatrixType MatrixType;
  };  
  
  } // end namespace internal
    
    
  /** \ingroup SVD_Module
   *
   *
   * \class BDCSVD
   *
   * \brief class Bidiagonal Divide and Conquer SVD
   *
   * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition
   *
   */
  template<typename _MatrixType> 
  class BDCSVD : public SVDBase<BDCSVD<_MatrixType> >
  {
    typedef SVDBase<BDCSVD> Base;
      
  public:
    using Base::rows;
    using Base::cols;
    using Base::computeU;
    using Base::computeV;
    
    typedef _MatrixType MatrixType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
    enum {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime, 
      ColsAtCompileTime = MatrixType::ColsAtCompileTime, 
      DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime), 
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, 
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, 
      MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime), 
      MatrixOptions = MatrixType::Options
    };
  
    typedef typename Base::MatrixUType MatrixUType;
    typedef typename Base::MatrixVType MatrixVType;
    typedef typename Base::SingularValuesType SingularValuesType;
    
    typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> MatrixX;
    typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr;
    typedef Matrix<RealScalar, Dynamic, 1> VectorType;
    typedef Array<RealScalar, Dynamic, 1> ArrayXr;
    typedef Array<Index,1,Dynamic> ArrayXi;
    typedef Ref<ArrayXr> ArrayRef;
    typedef Ref<ArrayXi> IndicesRef;
  
    /** \brief Default Constructor.
     *
     * The default constructor is useful in cases in which the user intends to
     * perform decompositions via BDCSVD::compute(const MatrixType&).
     */
    BDCSVD() : m_algoswap(16), m_numIters(0)
    {}
  
  
    /** \brief Default Constructor with memory preallocation
     *
     * Like the default constructor but with preallocation of the internal data
     * according to the specified problem size.
     * \sa BDCSVD()
     */
    BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
      : m_algoswap(16), m_numIters(0)
    {
      allocate(rows, cols, computationOptions);
    }
  
    /** \brief Constructor performing the decomposition of given matrix.
     *
     * \param matrix the matrix to decompose
     * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
     *                           By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU, 
     *                           #ComputeFullV, #ComputeThinV.
     *
     * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
     * available with the (non - default) FullPivHouseholderQR preconditioner.
     */
    BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
      : m_algoswap(16), m_numIters(0)
    {
      compute(matrix, computationOptions);
    }
  
    ~BDCSVD() 
    {
    }
    
    /** \brief Method performing the decomposition of given matrix using custom options.
     *
     * \param matrix the matrix to decompose
     * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
     *                           By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU, 
     *                           #ComputeFullV, #ComputeThinV.
     *
     * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
     * available with the (non - default) FullPivHouseholderQR preconditioner.
     */
    BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
  
    /** \brief Method performing the decomposition of given matrix using current options.
     *
     * \param matrix the matrix to decompose
     *
     * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
     */
    BDCSVD& compute(const MatrixType& matrix)
    {
      return compute(matrix, this->m_computationOptions);
    }
  
    void setSwitchSize(int s) 
    {
      eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
      m_algoswap = s;
    }
   
  private:
    void allocate(Index rows, Index cols, unsigned int computationOptions);
    void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift);
    void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V);
    void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals, ArrayRef shifts, ArrayRef mus);
    void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat);
    void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V);
    void deflation43(Index firstCol, Index shift, Index i, Index size);
    void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
    void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
    template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
    void copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naivev);
    void structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1);
    static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift);
  
  protected:
    MatrixXr m_naiveU, m_naiveV;
    MatrixXr m_computed;
    Index m_nRec;
    ArrayXr m_workspace;
    ArrayXi m_workspaceI;
    int m_algoswap;
    bool m_isTranspose, m_compU, m_compV;
    
    using Base::m_singularValues;
    using Base::m_diagSize;
    using Base::m_computeFullU;
    using Base::m_computeFullV;
    using Base::m_computeThinU;
    using Base::m_computeThinV;
    using Base::m_matrixU;
    using Base::m_matrixV;
    using Base::m_isInitialized;
    using Base::m_nonzeroSingularValues;
  
  public:  
    int m_numIters;
  }; //end class BDCSVD
  
  
  // Method to allocate and initialize matrix and attributes
  template<typename MatrixType>
  void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
  {
    m_isTranspose = (cols > rows);
  
    if (Base::allocate(rows, cols, computationOptions))
      return;
    
    m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize );
    m_compU = computeV();
    m_compV = computeU();
    if (m_isTranspose)
      std::swap(m_compU, m_compV);
    
    if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 );
    else         m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 );
    
    if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize);
    
    m_workspace.resize((m_diagSize+1)*(m_diagSize+1)*3);
    m_workspaceI.resize(3*m_diagSize);
  }// end allocate
  
  template<typename MatrixType>
  BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions) 
  {
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "\n\n\n======================================================================================================================\n\n\n";
  #endif
    allocate(matrix.rows(), matrix.cols(), computationOptions);
    using std::abs;
    
    //**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return
    if(matrix.cols() < m_algoswap)
    {
      // FIXME this line involves temporaries
      JacobiSVD<MatrixType> jsvd(matrix,computationOptions);
      if(computeU()) m_matrixU = jsvd.matrixU();
      if(computeV()) m_matrixV = jsvd.matrixV();
      m_singularValues = jsvd.singularValues();
      m_nonzeroSingularValues = jsvd.nonzeroSingularValues();
      m_isInitialized = true;
      return *this;
    }
    
    //**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
    RealScalar scale = matrix.cwiseAbs().maxCoeff();
    if(scale==RealScalar(0)) scale = RealScalar(1);
    MatrixX copy;
    if (m_isTranspose) copy = matrix.adjoint()/scale;
    else               copy = matrix/scale;
    
    //**** step 1 - Bidiagonalization
    // FIXME this line involves temporaries
    internal::UpperBidiagonalization<MatrixX> bid(copy);
  
    //**** step 2 - Divide & Conquer
    m_naiveU.setZero();
    m_naiveV.setZero();
    // FIXME this line involves a temporary matrix
    m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
    m_computed.template bottomRows<1>().setZero();
    divide(0, m_diagSize - 1, 0, 0, 0);
  
    //**** step 3 - Copy singular values and vectors
    for (int i=0; i<m_diagSize; i++)
    {
      RealScalar a = abs(m_computed.coeff(i, i));
      m_singularValues.coeffRef(i) = a * scale;
      if (a == 0)
      {
        m_nonzeroSingularValues = i;
        m_singularValues.tail(m_diagSize - i - 1).setZero();
        break;
      }
      else if (i == m_diagSize - 1)
      {
        m_nonzeroSingularValues = i + 1;
        break;
      }
    }
  
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
  //   std::cout << "m_naiveU\n" << m_naiveU << "\n\n";
  //   std::cout << "m_naiveV\n" << m_naiveV << "\n\n";
  #endif
    if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
    else              copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
  
    m_isInitialized = true;
    return *this;
  }// end compute
  
  
  template<typename MatrixType>
  template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
  void BDCSVD<MatrixType>::copyUV(const HouseholderU &householderU, const HouseholderV &householderV, const NaiveU &naiveU, const NaiveV &naiveV)
  {
    // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
    if (computeU())
    {
      Index Ucols = m_computeThinU ? m_diagSize : householderU.cols();
      m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
      m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
      householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer
    }
    if (computeV())
    {
      Index Vcols = m_computeThinV ? m_diagSize : householderV.cols();
      m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
      m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
      householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer
    }
  }
  
  /** \internal
    * Performs A = A * B exploiting the special structure of the matrix A. Splitting A as:
    *  A = [A1]
    *      [A2]
    * such that A1.rows()==n1, then we assume that at least half of the columns of A1 and A2 are zeros.
    * We can thus pack them prior to the the matrix product. However, this is only worth the effort if the matrix is large
    * enough.
    */
  template<typename MatrixType>
  void BDCSVD<MatrixType>::structured_update(Block<MatrixXr,Dynamic,Dynamic> A, const MatrixXr &B, Index n1)
  {
    Index n = A.rows();
    if(n>100)
    {
      // If the matrices are large enough, let's exploit the sparse structure of A by
      // splitting it in half (wrt n1), and packing the non-zero columns.
      Index n2 = n - n1;
      Map<MatrixXr> A1(m_workspace.data()      , n1, n);
      Map<MatrixXr> A2(m_workspace.data()+ n1*n, n2, n);
      Map<MatrixXr> B1(m_workspace.data()+  n*n, n,  n);
      Map<MatrixXr> B2(m_workspace.data()+2*n*n, n,  n);
      Index k1=0, k2=0;
      for(Index j=0; j<n; ++j)
      {
        if( (A.col(j).head(n1).array()!=0).any() )
        {
          A1.col(k1) = A.col(j).head(n1);
          B1.row(k1) = B.row(j);
          ++k1;
        }
        if( (A.col(j).tail(n2).array()!=0).any() )
        {
          A2.col(k2) = A.col(j).tail(n2);
          B2.row(k2) = B.row(j);
          ++k2;
        }
      }
    
      A.topRows(n1).noalias()    = A1.leftCols(k1) * B1.topRows(k1);
      A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2);
    }
    else
    {
      Map<MatrixXr,Aligned> tmp(m_workspace.data(),n,n);
      tmp.noalias() = A*B;
      A = tmp;
    }
  }
  
  // The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the 
  // place of the submatrix we are currently working on.
  
  //@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
  //@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU; 
  // lastCol + 1 - firstCol is the size of the submatrix.
  //@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section 1 for more information on W)
  //@param firstRowW : Same as firstRowW with the column.
  //@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix 
  // to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
  template<typename MatrixType>
  void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift)
  {
    // requires rows = cols + 1;
    using std::pow;
    using std::sqrt;
    using std::abs;
    const Index n = lastCol - firstCol + 1;
    const Index k = n/2;
    RealScalar alphaK;
    RealScalar betaK; 
    RealScalar r0; 
    RealScalar lambda, phi, c0, s0;
    VectorType l, f;
    // We use the other algorithm which is more efficient for small 
    // matrices.
    if (n < m_algoswap)
    {
      // FIXME this line involves temporaries
      JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0));
      if (m_compU)
        m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU();
      else 
      {
        m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0);
        m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n);
      }
      if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV();
      m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
      m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n);
      return;
    }
    // We use the divide and conquer algorithm
    alphaK =  m_computed(firstCol + k, firstCol + k);
    betaK = m_computed(firstCol + k + 1, firstCol + k);
    // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
    // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the 
    // right submatrix before the left one. 
    divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
    divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
  
    if (m_compU)
    {
      lambda = m_naiveU(firstCol + k, firstCol + k);
      phi = m_naiveU(firstCol + k + 1, lastCol + 1);
    } 
    else 
    {
      lambda = m_naiveU(1, firstCol + k);
      phi = m_naiveU(0, lastCol + 1);
    }
    r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi));
    if (m_compU)
    {
      l = m_naiveU.row(firstCol + k).segment(firstCol, k);
      f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
    } 
    else 
    {
      l = m_naiveU.row(1).segment(firstCol, k);
      f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
    }
    if (m_compV) m_naiveV(firstRowW+k, firstColW) = 1;
    if (r0 == 0)
    {
      c0 = 1;
      s0 = 0;
    }
    else
    {
      c0 = alphaK * lambda / r0;
      s0 = betaK * phi / r0;
    }
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
    
    if (m_compU)
    {
      MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));     
      // we shiftW Q1 to the right
      for (Index i = firstCol + k - 1; i >= firstCol; i--) 
        m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1);
      // we shift q1 at the left with a factor c0
      m_naiveU.col(firstCol).segment( firstCol, k + 1) = (q1 * c0);
      // last column = q1 * - s0
      m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * ( - s0));
      // first column = q2 * s0
      m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) = m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0; 
      // q2 *= c0
      m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
    } 
    else 
    {
      RealScalar q1 = m_naiveU(0, firstCol + k);
      // we shift Q1 to the right
      for (Index i = firstCol + k - 1; i >= firstCol; i--) 
        m_naiveU(0, i + 1) = m_naiveU(0, i);
      // we shift q1 at the left with a factor c0
      m_naiveU(0, firstCol) = (q1 * c0);
      // last column = q1 * - s0
      m_naiveU(0, lastCol + 1) = (q1 * ( - s0));
      // first column = q2 * s0
      m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0; 
      // q2 *= c0
      m_naiveU(1, lastCol + 1) *= c0;
      m_naiveU.row(1).segment(firstCol + 1, k).setZero();
      m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
    }
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
    
    m_computed(firstCol + shift, firstCol + shift) = r0;
    m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real();
    m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real();
  
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    ArrayXr tmp1 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
  #endif
    // Second part: try to deflate singular values in combined matrix
    deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    ArrayXr tmp2 = (m_computed.block(firstCol+shift, firstCol+shift, n, n)).jacobiSvd().singularValues();
    std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n";
    std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n";
    std::cout << "err:      " << ((tmp1-tmp2).abs()>1e-12*tmp2.abs()).transpose() << "\n";
    static int count = 0;
    std::cout << "# " << ++count << "\n\n";
    assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm());
  //   assert(count<681);
  //   assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all());
  #endif
    
    // Third part: compute SVD of combined matrix
    MatrixXr UofSVD, VofSVD;
    VectorType singVals;
    computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD);
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(UofSVD.allFinite());
    assert(VofSVD.allFinite());
  #endif
    
    if (m_compU)
      structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n+2)/2);
    else
    {
      Map<Matrix<RealScalar,2,Dynamic>,Aligned> tmp(m_workspace.data(),2,n+1);
      tmp.noalias() = m_naiveU.middleCols(firstCol, n+1) * UofSVD;
      m_naiveU.middleCols(firstCol, n + 1) = tmp;
    }
    
    if (m_compV)  structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n+1)/2);
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
    
    m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero();
    m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals;
  }// end divide
  
  // Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in
  // the first column and on the diagonal and has undergone deflation, so diagonal is in increasing
  // order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except
  // that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order.
  //
  // TODO Opportunities for optimization: better root finding algo, better stopping criterion, better
  // handling of round-off errors, be consistent in ordering
  // For instance, to solve the secular equation using FMM, see http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf
  template <typename MatrixType>
  void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V)
  {
    ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n);
    m_workspace.head(n) =  m_computed.block(firstCol, firstCol, n, n).diagonal();
    ArrayRef diag = m_workspace.head(n);
    diag(0) = 0;
  
    // Allocate space for singular values and vectors
    singVals.resize(n);
    U.resize(n+1, n+1);
    if (m_compV) V.resize(n, n);
  
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    if (col0.hasNaN() || diag.hasNaN())
      std::cout << "\n\nHAS NAN\n\n";
  #endif
    
    // Many singular values might have been deflated, the zero ones have been moved to the end,
    // but others are interleaved and we must ignore them at this stage.
    // To this end, let's compute a permutation skipping them:
    Index actual_n = n;
    while(actual_n>1 && diag(actual_n-1)==0) --actual_n;
    Index m = 0; // size of the deflated problem
    for(Index k=0;k<actual_n;++k)
      if(col0(k)!=0)
        m_workspaceI(m++) = k;
    Map<ArrayXi> perm(m_workspaceI.data(),m);
    
    Map<ArrayXr> shifts(m_workspace.data()+1*n, n);
    Map<ArrayXr> mus(m_workspace.data()+2*n, n);
    Map<ArrayXr> zhat(m_workspace.data()+3*n, n);
  
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "computeSVDofM using:\n";
    std::cout << "  z: " << col0.transpose() << "\n";
    std::cout << "  d: " << diag.transpose() << "\n";
  #endif
    
    // Compute singVals, shifts, and mus
    computeSingVals(col0, diag, perm, singVals, shifts, mus);
    
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "  j:        " << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() << "\n\n";
    std::cout << "  sing-val: " << singVals.transpose() << "\n";
    std::cout << "  mu:       " << mus.transpose() << "\n";
    std::cout << "  shift:    " << shifts.transpose() << "\n";
    
    {
      Index actual_n = n;
      while(actual_n>1 && col0(actual_n-1)==0) --actual_n;
      std::cout << "\n\n    mus:    " << mus.head(actual_n).transpose() << "\n\n";
      std::cout << "    check1 (expect0) : " << ((singVals.array()-(shifts+mus)) / singVals.array()).head(actual_n).transpose() << "\n\n";
      std::cout << "    check2 (>0)      : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n";
      std::cout << "    check3 (>0)      : " << ((diag.segment(1,actual_n-1)-singVals.head(actual_n-1).array()) / singVals.head(actual_n-1).array()).transpose() << "\n\n\n";
      std::cout << "    check4 (>0)      : " << ((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).transpose() << "\n\n\n";
    }
  #endif
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(singVals.allFinite());
    assert(mus.allFinite());
    assert(shifts.allFinite());
  #endif
    
    // Compute zhat
    perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat);
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "  zhat: " << zhat.transpose() << "\n";
  #endif
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(zhat.allFinite());
  #endif
    
    computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V);
    
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() << "\n";
    std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() << "\n";
  #endif
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(U.allFinite());
    assert(V.allFinite());
    assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 1e-14 * n);
    assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 1e-14 * n);
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
    
    // Because of deflation, the singular values might not be completely sorted.
    // Fortunately, reordering them is a O(n) problem
    for(Index i=0; i<actual_n-1; ++i)
    {
      if(singVals(i)>singVals(i+1))
      {
        using std::swap;
        swap(singVals(i),singVals(i+1));
        U.col(i).swap(U.col(i+1));
        if(m_compV) V.col(i).swap(V.col(i+1));
      }
    }
    
    // Reverse order so that singular values in increased order
    // Because of deflation, the zeros singular-values are already at the end
    singVals.head(actual_n).reverseInPlace();
    U.leftCols(actual_n).rowwise().reverseInPlace();
    if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace();
    
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    JacobiSVD<MatrixXr> jsvd(m_computed.block(firstCol, firstCol, n, n) );
    std::cout << "  * j:        " << jsvd.singularValues().transpose() << "\n\n";
    std::cout << "  * sing-val: " << singVals.transpose() << "\n";
  //   std::cout << "  * err:      " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n";
  #endif
  }
  
  template <typename MatrixType>
  typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::secularEq(RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const ArrayRef& diagShifted, RealScalar shift)
  {
    Index m = perm.size();
    RealScalar res = 1;
    for(Index i=0; i<m; ++i)
    {
      Index j = perm(i);
      res += numext::abs2(col0(j)) / ((diagShifted(j) - mu) * (diag(j) + shift + mu));
    }
    return res;
  }
  
  template <typename MatrixType>
  void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm,
                                           VectorType& singVals, ArrayRef shifts, ArrayRef mus)
  {
    using std::abs;
    using std::swap;
  
    Index n = col0.size();
    Index actual_n = n;
    while(actual_n>1 && col0(actual_n-1)==0) --actual_n;
  
    for (Index k = 0; k < n; ++k)
    {
      if (col0(k) == 0 || actual_n==1)
      {
        // if col0(k) == 0, then entry is deflated, so singular value is on diagonal
        // if actual_n==1, then the deflated problem is already diagonalized
        singVals(k) = k==0 ? col0(0) : diag(k);
        mus(k) = 0;
        shifts(k) = k==0 ? col0(0) : diag(k);
        continue;
      } 
  
      // otherwise, use secular equation to find singular value
      RealScalar left = diag(k);
      RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm());
      if(k==actual_n-1)
        right = (diag(actual_n-1) + col0.matrix().norm());
      else
      {
        // Skip deflated singular values
        Index l = k+1;
        while(col0(l)==0) { ++l; eigen_internal_assert(l<actual_n); }
        right = diag(l);
      }
  
      // first decide whether it's closer to the left end or the right end
      RealScalar mid = left + (right-left) / 2;
      RealScalar fMid = secularEq(mid, col0, diag, perm, diag, 0);
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
      std::cout << right-left << "\n";
      std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, diag-left, left) << " " << secularEq(mid-right, col0, diag, perm, diag-right, right)   << "\n";
      std::cout << "     = " << secularEq(0.1*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.2*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.3*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.4*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.49*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.5*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.51*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.6*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.7*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.8*(left+right), col0, diag, perm, diag, 0)
                << " "       << secularEq(0.9*(left+right), col0, diag, perm, diag, 0) << "\n";
  #endif
      RealScalar shift = (k == actual_n-1 || fMid > 0) ? left : right;
      
      // measure everything relative to shift
      Map<ArrayXr> diagShifted(m_workspace.data()+4*n, n);
      diagShifted = diag - shift;
      
      // initial guess
      RealScalar muPrev, muCur;
      if (shift == left)
      {
        muPrev = (right - left) * 0.1;
        if (k == actual_n-1) muCur = right - left;
        else                 muCur = (right - left) * 0.5; 
      }
      else
      {
        muPrev = -(right - left) * 0.1;
        muCur = -(right - left) * 0.5;
      }
  
      RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift);
      RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift);
      if (abs(fPrev) < abs(fCur))
      {
        swap(fPrev, fCur);
        swap(muPrev, muCur);
      }
  
      // rational interpolation: fit a function of the form a / mu + b through the two previous
      // iterates and use its zero to compute the next iterate
      bool useBisection = fPrev*fCur>0;
      while (fCur!=0 && abs(muCur - muPrev) > 8 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) && abs(fCur - fPrev)>NumTraits<RealScalar>::epsilon() && !useBisection)
      {
        ++m_numIters;
  
        // Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples.
        RealScalar a = (fCur - fPrev) / (1/muCur - 1/muPrev);
        RealScalar b = fCur - a / muCur;
        // And find mu such that f(mu)==0:
        RealScalar muZero = -a/b;
        RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift);
        
        muPrev = muCur;
        fPrev = fCur;
        muCur = muZero;
        fCur = fZero;
        
        
        if (shift == left  && (muCur < 0 || muCur > right - left)) useBisection = true;
        if (shift == right && (muCur < -(right - left) || muCur > 0)) useBisection = true;
        if (abs(fCur)>abs(fPrev)) useBisection = true;
      }
  
      // fall back on bisection method if rational interpolation did not work
      if (useBisection)
      {
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
        std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n";
  #endif
        RealScalar leftShifted, rightShifted;
        if (shift == left)
        {
          leftShifted = RealScalar(1)/NumTraits<RealScalar>::highest();
          // I don't understand why the case k==0 would be special there:
          // if (k == 0) rightShifted = right - left; else 
          rightShifted = (k==actual_n-1) ? right : ((right - left) * 0.6); // theoretically we can take 0.5, but let's be safe
        }
        else
        {
          leftShifted = -(right - left) * 0.6;
          rightShifted = -RealScalar(1)/NumTraits<RealScalar>::highest();
        }
        
        RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift);
  
  #if defined EIGEN_INTERNAL_DEBUGGING || defined EIGEN_BDCSVD_DEBUG_VERBOSE
        RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift);
  #endif
  
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
        if(!(fLeft * fRight<0))
          std::cout << k << " : " <<  fLeft << " * " << fRight << " == " << fLeft * fRight << "  ;  " << left << " - " << right << " -> " <<  leftShifted << " " << rightShifted << "   shift=" << shift << "\n";
  #endif
        eigen_internal_assert(fLeft * fRight < 0);
        
        while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted)))
        {
          RealScalar midShifted = (leftShifted + rightShifted) / 2;
          fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
          if (fLeft * fMid < 0)
          {
            rightShifted = midShifted;
          }
          else
          {
            leftShifted = midShifted;
            fLeft = fMid;
          }
        }
  
        muCur = (leftShifted + rightShifted) / 2;
      }
        
      singVals[k] = shift + muCur;
      shifts[k] = shift;
      mus[k] = muCur;
  
      // perturb singular value slightly if it equals diagonal entry to avoid division by zero later
      // (deflation is supposed to avoid this from happening)
      // - this does no seem to be necessary anymore -
  //     if (singVals[k] == left) singVals[k] *= 1 + NumTraits<RealScalar>::epsilon();
  //     if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon();
    }
  }
  
  
  // zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1)
  template <typename MatrixType>
  void BDCSVD<MatrixType>::perturbCol0
     (const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
      const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat)
  {
    using std::sqrt;
    Index n = col0.size();
    Index m = perm.size();
    if(m==0)
    {
      zhat.setZero();
      return;
    }
    Index last = perm(m-1);
    // The offset permits to skip deflated entries while computing zhat
    for (Index k = 0; k < n; ++k)
    {
      if (col0(k) == 0) // deflated
        zhat(k) = 0;
      else
      {
        // see equation (3.6)
        RealScalar dk = diag(k);
        RealScalar prod = (singVals(last) + dk) * (mus(last) + (shifts(last) - dk));
  
        for(Index l = 0; l<m; ++l)
        {
          Index i = perm(l);
          if(i!=k)
          {
            Index j = i<k ? i : perm(l-1);
            prod *= ((singVals(j)+dk) / ((diag(i)+dk))) * ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk)));
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
            if(i!=k && std::abs(((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) - 1) > 0.9 )
              std::cout << "     " << ((singVals(j)+dk)*(mus(j)+(shifts(j)-dk)))/((diag(i)+dk)*(diag(i)-dk)) << " == (" << (singVals(j)+dk) << " * " << (mus(j)+(shifts(j)-dk))
                         << ") / (" << (diag(i)+dk) << " * " << (diag(i)-dk) << ")\n";
  #endif
          }
        }
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
        std::cout << "zhat(" << k << ") =  sqrt( " << prod << ")  ;  " << (singVals(last) + dk) << " * " << mus(last) + shifts(last) << " - " << dk << "\n";
  #endif
        RealScalar tmp = sqrt(prod);
        zhat(k) = col0(k) > 0 ? tmp : -tmp;
      }
    }
  }
  
  // compute singular vectors
  template <typename MatrixType>
  void BDCSVD<MatrixType>::computeSingVecs
     (const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef &perm, const VectorType& singVals,
      const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V)
  {
    Index n = zhat.size();
    Index m = perm.size();
    
    for (Index k = 0; k < n; ++k)
    {
      if (zhat(k) == 0)
      {
        U.col(k) = VectorType::Unit(n+1, k);
        if (m_compV) V.col(k) = VectorType::Unit(n, k);
      }
      else
      {
        U.col(k).setZero();
        for(Index l=0;l<m;++l)
        {
          Index i = perm(l);
          U(i,k) = zhat(i)/(((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
        }
        U(n,k) = 0;      
        U.col(k).normalize();
      
        if (m_compV)
        {
          V.col(k).setZero();
          for(Index l=1;l<m;++l)
          {
            Index i = perm(l);
            V(i,k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k)) )/( (diag(i) + singVals[k]));
          }
          V(0,k) = -1;
          V.col(k).normalize();
        }
      }
    }
    U.col(n) = VectorType::Unit(n+1, n);
  }
  
  
  // page 12_13
  // i >= 1, di almost null and zi non null.
  // We use a rotation to zero out zi applied to the left of M
  template <typename MatrixType>
  void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size)
  {
    using std::abs;
    using std::sqrt;
    using std::pow;
    Index start = firstCol + shift;
    RealScalar c = m_computed(start, start);
    RealScalar s = m_computed(start+i, start);
    RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
    if (r == 0)
    {
      m_computed(start+i, start+i) = 0;
      return;
    }
    m_computed(start,start) = r;  
    m_computed(start+i, start) = 0;
    m_computed(start+i, start+i) = 0;
    
    JacobiRotation<RealScalar> J(c/r,-s/r);
    if (m_compU)  m_naiveU.middleRows(firstCol, size+1).applyOnTheRight(firstCol, firstCol+i, J);
    else          m_naiveU.applyOnTheRight(firstCol, firstCol+i, J);
  }// end deflation 43
  
  
  // page 13
  // i,j >= 1, i!=j and |di - dj| < epsilon * norm2(M)
  // We apply two rotations to have zj = 0;
  // TODO deflation44 is still broken and not properly tested
  template <typename MatrixType>
  void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size)
  {
    using std::abs;
    using std::sqrt;
    using std::conj;
    using std::pow;
    RealScalar c = m_computed(firstColm+i, firstColm);
    RealScalar s = m_computed(firstColm+j, firstColm);
    RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; "
      << m_computed(firstColm + i-1, firstColm)  << " "
      << m_computed(firstColm + i, firstColm)  << " "
      << m_computed(firstColm + i+1, firstColm) << " "
      << m_computed(firstColm + i+2, firstColm) << "\n";
    std::cout << m_computed(firstColm + i-1, firstColm + i-1)  << " "
      << m_computed(firstColm + i, firstColm+i)  << " "
      << m_computed(firstColm + i+1, firstColm+i+1) << " "
      << m_computed(firstColm + i+2, firstColm+i+2) << "\n";
  #endif
    if (r==0)
    {
      m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
      return;
    }
    c/=r;
    s/=r;
    m_computed(firstColm + i, firstColm) = r;  
    m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
    m_computed(firstColm + j, firstColm) = 0;
  
    JacobiRotation<RealScalar> J(c,-s);
    if (m_compU)  m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J);
    else          m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J);
    if (m_compV)  m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J);
  }// end deflation 44
  
  
  // acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
  template <typename MatrixType>
  void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift)
  {
    using std::sqrt;
    using std::abs;
    const Index length = lastCol + 1 - firstCol;
    
    Block<MatrixXr,Dynamic,1> col0(m_computed, firstCol+shift, firstCol+shift, length, 1);
    Diagonal<MatrixXr> fulldiag(m_computed);
    VectorBlock<Diagonal<MatrixXr>,Dynamic> diag(fulldiag, firstCol+shift, length);
    
    RealScalar maxDiag = diag.tail((std::max)(Index(1),length-1)).cwiseAbs().maxCoeff();
    RealScalar epsilon_strict = NumTraits<RealScalar>::epsilon() * maxDiag;
    RealScalar epsilon_coarse = 8 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag);
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
  
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE  
    std::cout << "\ndeflate:" << diag.head(k+1).transpose() << "  |  " << diag.segment(k+1,length-k-1).transpose() << "\n";
  #endif
    
    //condition 4.1
    if (diag(0) < epsilon_coarse)
    { 
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
      std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n";
  #endif
      diag(0) = epsilon_coarse;
    }
  
    //condition 4.2
    for (Index i=1;i<length;++i)
      if (abs(col0(i)) < epsilon_strict)
      {
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
        std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict << "  (diag(" << i << ")=" << diag(i) << ")\n";
  #endif
        col0(i) = 0;
      }
  
    //condition 4.3
    for (Index i=1;i<length; i++)
      if (diag(i) < epsilon_coarse)
      {
  #ifdef  EIGEN_BDCSVD_DEBUG_VERBOSE
        std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) << " < " << epsilon_coarse << "\n";
  #endif
        deflation43(firstCol, shift, i, length);
      }
  
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "to be sorted: " << diag.transpose() << "\n\n";
  #endif
    {
      // Check for total deflation
      // If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting
      bool total_deflation = (col0.tail(length-1).array()==RealScalar(0)).all();
      
      // Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge.
      // First, compute the respective permutation.
      Index *permutation = m_workspaceI.data();
      {
        permutation[0] = 0;
        Index p = 1;
        
        // Move deflated diagonal entries at the end.
        for(Index i=1; i<length; ++i)
          if(diag(i)==0)
            permutation[p++] = i;
          
        Index i=1, j=k+1;
        for( ; p < length; ++p)
        {
               if (i > k)             permutation[p] = j++;
          else if (j >= length)       permutation[p] = i++;
          else if (diag(i) < diag(j)) permutation[p] = j++;
          else                        permutation[p] = i++;
        }
      }
      
      // If we have a total deflation, then we have to insert diag(0) at the right place
      if(total_deflation)
      {
        for(Index i=1; i<length; ++i)
        {
          Index pi = permutation[i];
          if(diag(pi)==0 || diag(0)<diag(pi))
            permutation[i-1] = permutation[i];
          else
          {
            permutation[i-1] = 0;
            break;
          }
        }
      }
      
      // Current index of each col, and current column of each index
      Index *realInd = m_workspaceI.data()+length;
      Index *realCol = m_workspaceI.data()+2*length;
      
      for(int pos = 0; pos< length; pos++)
      {
        realCol[pos] = pos;
        realInd[pos] = pos;
      }
      
      for(Index i = total_deflation?0:1; i < length; i++)
      {
        const Index pi = permutation[length - (total_deflation ? i+1 : i)];
        const Index J = realCol[pi];
        
        using std::swap;
        // swap diagonal and first column entries:
        swap(diag(i), diag(J));
        if(i!=0 && J!=0) swap(col0(i), col0(J));
  
        // change columns
        if (m_compU) m_naiveU.col(firstCol+i).segment(firstCol, length + 1).swap(m_naiveU.col(firstCol+J).segment(firstCol, length + 1));
        else         m_naiveU.col(firstCol+i).segment(0, 2)                .swap(m_naiveU.col(firstCol+J).segment(0, 2));
        if (m_compV) m_naiveV.col(firstColW + i).segment(firstRowW, length).swap(m_naiveV.col(firstColW + J).segment(firstRowW, length));
  
        //update real pos
        const Index realI = realInd[i];
        realCol[realI] = J;
        realCol[pi] = i;
        realInd[J] = realI;
        realInd[i] = pi;
      }
    }
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n";
    std::cout << "      : " << col0.transpose() << "\n\n";
  #endif
      
    //condition 4.4
    {
      Index i = length-1;
      while(i>0 && (diag(i)==0 || col0(i)==0)) --i;
      for(; i>1;--i)
         if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag )
        {
  #ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
          std::cout << "deflation 4.4 with i = " << i << " because " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()*diag(i) << "\n";
  #endif
          eigen_internal_assert(abs(diag(i) - diag(i-1))<epsilon_coarse && " diagonal entries are not properly sorted");
          deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i-1, i, length);
        }
    }
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    for(Index j=2;j<length;++j)
      assert(diag(j-1)<=diag(j) || diag(j)==0);
  #endif
    
  #ifdef EIGEN_BDCSVD_SANITY_CHECKS
    assert(m_naiveU.allFinite());
    assert(m_naiveV.allFinite());
    assert(m_computed.allFinite());
  #endif
  }//end deflation
  
  #ifndef __CUDACC__
  /** \svd_module
    *
    * \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm
    *
    * \sa class BDCSVD
    */
  template<typename Derived>
  BDCSVD<typename MatrixBase<Derived>::PlainObject>
  MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
  {
    return BDCSVD<PlainObject>(*this, computationOptions);
  }
  #endif
  
  } // end namespace Eigen
  
  #endif
  