eigen/unsupported/Eigen/src/SVD/BDCSVD.h

// 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>
//
// 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 EPSILON 0.0000000000000001

#define ALGOSWAP 16

namespace Eigen {
/** \ingroup SVD_Module
 *
 *
 * \class BDCSVD
 *
 * \brief class Bidiagonal Divide and Conquer SVD
 *
 * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
 * We plan to have a very similar interface to JacobiSVD on this class.
 * It should be used to speed up the calcul of SVD for big matrices. 
 */
template<typename _MatrixType> 
class BDCSVD : public SVDBase<_MatrixType>
{
  typedef SVDBase<_MatrixType> Base;
    
public:
  using Base::rows;
  using Base::cols;
  
  typedef _MatrixType MatrixType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
  typedef typename MatrixType::Index Index;
  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 Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, 
		 MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
  MatrixUType;
  typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, 
		 MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
  MatrixVType;
  typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
  typedef typename internal::plain_row_type<MatrixType>::type RowType;
  typedef typename internal::plain_col_type<MatrixType>::type ColType;
  typedef Matrix<Scalar, Dynamic, Dynamic> MatrixX;
  typedef Matrix<RealScalar, Dynamic, Dynamic> MatrixXr;
  typedef Matrix<RealScalar, Dynamic, 1> VectorType;
  typedef Array<RealScalar, Dynamic, 1> ArrayXr;

  /** \brief Default Constructor.
   *
   * The default constructor is useful in cases in which the user intends to
   * perform decompositions via BDCSVD::compute(const MatrixType&).
   */
  BDCSVD()
    : SVDBase<_MatrixType>::SVDBase(), 
      algoswap(ALGOSWAP), 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)
    : SVDBase<_MatrixType>::SVDBase(), 
      algoswap(ALGOSWAP), 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)
    : SVDBase<_MatrixType>::SVDBase(), 
      algoswap(ALGOSWAP), 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.
   */
  SVDBase<MatrixType>& 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).
   */
  SVDBase<MatrixType>& 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");
    algoswap = s;
  }


  /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
   *
   * \param b the right - hand - side of the equation to solve.
   *
   * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
   *
   * \note SVD solving is implicitly least - squares. Thus, this method serves both purposes of exact solving and least - squares solving.
   * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
   */
  template<typename Rhs>
  inline const internal::solve_retval<BDCSVD, Rhs>
  solve(const MatrixBase<Rhs>& b) const
  {
    eigen_assert(this->m_isInitialized && "BDCSVD is not initialized.");
    eigen_assert(SVDBase<_MatrixType>::computeU() && SVDBase<_MatrixType>::computeV() && 
		 "BDCSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
    return internal::solve_retval<BDCSVD, Rhs>(*this, b.derived());
  }

 
  const MatrixUType& matrixU() const
  {
    eigen_assert(this->m_isInitialized && "SVD is not initialized.");
    if (isTranspose){
      eigen_assert(this->computeV() && "This SVD decomposition didn't compute U. Did you ask for it?");
      return this->m_matrixV;
    }
    else 
    {
      eigen_assert(this->computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
      return this->m_matrixU;
    }
     
  }


  const MatrixVType& matrixV() const
  {
    eigen_assert(this->m_isInitialized && "SVD is not initialized.");
    if (isTranspose){
      eigen_assert(this->computeU() && "This SVD decomposition didn't compute V. Did you ask for it?");
      return this->m_matrixU;
    }
    else
    {
      eigen_assert(this->computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
      return this->m_matrixV;
    }
  }
 
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 ArrayXr& col0, const ArrayXr& diag, VectorType& singVals,
                       ArrayXr& shifts, ArrayXr& mus);
  void perturbCol0(const ArrayXr& col0, const ArrayXr& diag, const VectorType& singVals,
                   const ArrayXr& shifts, const ArrayXr& mus, ArrayXr& zhat);
  void computeSingVecs(const ArrayXr& zhat, const ArrayXr& diag, const VectorType& singVals,
                       const ArrayXr& shifts, const ArrayXr& 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);
  void copyUV(const typename internal::UpperBidiagonalization<MatrixX>::HouseholderUSequenceType& householderU, 
              const typename internal::UpperBidiagonalization<MatrixX>::HouseholderVSequenceType& householderV);

protected:
  MatrixXr m_naiveU, m_naiveV;
  MatrixXr m_computed;
  Index nRec;
  int algoswap;
  bool isTranspose, compU, compV;

public:  
  int m_numIters;
}; //end class BDCSVD


// Methode to allocate ans initialize matrix and attributs
template<typename MatrixType>
void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
{
  isTranspose = (cols > rows);
  if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;
  m_computed = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize );
  if (isTranspose){
    compU = this->computeU();
    compV = this->computeV();    
  } 
  else
  {
    compV = this->computeU();
    compU = this->computeV();   
  }
  if (compU) m_naiveU = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize + 1 );
  else m_naiveU = MatrixXr::Zero(2, this->m_diagSize + 1 );
  
  if (compV) m_naiveV = MatrixXr::Zero(this->m_diagSize, this->m_diagSize);
  

  //should be changed for a cleaner implementation
  if (isTranspose){
    bool aux;
    if (this->computeU()||this->computeV()){
      aux = this->m_computeFullU;
      this->m_computeFullU = this->m_computeFullV;
      this->m_computeFullV = aux;
      aux = this->m_computeThinU;
      this->m_computeThinU = this->m_computeThinV;
      this->m_computeThinV = aux;
    } 
  }
}// end allocate

// Methode which compute the BDCSVD for the int
template<>
SVDBase<Matrix<int, Dynamic, Dynamic> >&
BDCSVD<Matrix<int, Dynamic, Dynamic> >::compute(const MatrixType& matrix, unsigned int computationOptions) {
  allocate(matrix.rows(), matrix.cols(), computationOptions);
  this->m_nonzeroSingularValues = 0;
  m_computed = Matrix<int, Dynamic, Dynamic>::Zero(rows(), cols());
  for (int i=0; i<this->m_diagSize; i++)   {
    this->m_singularValues.coeffRef(i) = 0;
  }
  if (this->m_computeFullU) this->m_matrixU = Matrix<int, Dynamic, Dynamic>::Zero(rows(), rows());
  if (this->m_computeFullV) this->m_matrixV = Matrix<int, Dynamic, Dynamic>::Zero(cols(), cols()); 
  this->m_isInitialized = true;
  return *this;
}


// Methode which compute the BDCSVD
template<typename MatrixType>
SVDBase<MatrixType>&
BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions) 
{
  allocate(matrix.rows(), matrix.cols(), computationOptions);
  using std::abs;

  //**** step 1 Bidiagonalization  isTranspose = (matrix.cols()>matrix.rows()) ;
  MatrixType copy;
  if (isTranspose) copy = matrix.adjoint();
  else copy = matrix;
  
  internal::UpperBidiagonalization<MatrixX> bid(copy);

  //**** step 2 Divide
  m_computed.topRows(this->m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
  m_computed.template bottomRows<1>().setZero();
  divide(0, this->m_diagSize - 1, 0, 0, 0);

  //**** step 3 copy
  for (int i=0; i<this->m_diagSize; i++)   {
    RealScalar a = abs(m_computed.coeff(i, i));
    this->m_singularValues.coeffRef(i) = a;
    if (a == 0){
      this->m_nonzeroSingularValues = i;
      this->m_singularValues.tail(this->m_diagSize - i - 1).setZero();
      break;
    }
    else  if (i == this->m_diagSize - 1)
    {
      this->m_nonzeroSingularValues = i + 1;
      break;
    }
  }
  copyUV(bid.householderU(), bid.householderV());
  this->m_isInitialized = true;
  return *this;
}// end compute


template<typename MatrixType>
void BDCSVD<MatrixType>::copyUV(const typename internal::UpperBidiagonalization<MatrixX>::HouseholderUSequenceType& householderU, 
                                const typename internal::UpperBidiagonalization<MatrixX>::HouseholderVSequenceType& householderV)
{
  // Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
  if (this->computeU()){
    Index Ucols = this->m_computeThinU ? this->m_nonzeroSingularValues : householderU.cols();
    this->m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
    Index blockCols = this->m_computeThinU ? this->m_nonzeroSingularValues : this->m_diagSize;
    this->m_matrixU.block(0, 0, this->m_diagSize, blockCols) = 
        m_naiveV.template cast<Scalar>().block(0, 0, this->m_diagSize, blockCols);
    this->m_matrixU = householderU * this->m_matrixU;
  }
  if (this->computeV()){
    Index Vcols = this->m_computeThinV ? this->m_nonzeroSingularValues : householderV.cols();
    this->m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
    Index blockCols = this->m_computeThinV ? this->m_nonzeroSingularValues : this->m_diagSize;
    this->m_matrixV.block(0, 0, this->m_diagSize, blockCols) = 
        m_naiveU.template cast<Scalar>().block(0, 0, this->m_diagSize, blockCols);
    this->m_matrixV = householderV * this->m_matrixV;
  }
}

// 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 nbRows = nbCols + 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;
  MatrixXr l, f;
  // We use the other algorithm which is more efficient for small 
  // matrices.
  if (n < algoswap){
    JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), 
			  ComputeFullU | (ComputeFullV * compV)) ;
    if (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 (compV) m_naiveV.block(firstRowW, firstColW, n, n).real() << b.matrixV();
    m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
    for (int i=0; i<n; i++)
    {
      m_computed(firstCol + shift + i, firstCol + shift +i) = b.singularValues().coeffRef(i);
    }
    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 (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 (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 (compV) m_naiveV(firstRowW+k, firstColW) = 1;
  if (r0 == 0)
  {
    c0 = 1;
    s0 = 0;
  }
  else
  {
    c0 = alphaK * lambda / r0;
    s0 = betaK * phi / r0;
  }
  if (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();
  }
  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();


  // Second part: try to deflate singular values in combined matrix
  deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);

  // Third part: compute SVD of combined matrix
  MatrixXr UofSVD, VofSVD;
  VectorType singVals;
  computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD);
  if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1) *= UofSVD;
  else m_naiveU.block(0, firstCol, 2, n + 1) *= UofSVD;
  if (compV) m_naiveV.block(firstRowW, firstColW, n, n) *= VofSVD;
  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 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
template <typename MatrixType>
void BDCSVD<MatrixType>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V)
{
  // TODO Get rid of these copies (?)
  ArrayXr col0 = m_computed.block(firstCol, firstCol, n, 1);
  ArrayXr diag = m_computed.block(firstCol, firstCol, n, n).diagonal();
  diag(0) = 0;

  // compute singular values and vectors (in decreasing order)
  singVals.resize(n);
  U.resize(n+1, n+1);
  if (compV) V.resize(n, n);

  if (col0.hasNaN() || diag.hasNaN()) return;

  ArrayXr shifts(n), mus(n), zhat(n);
  computeSingVals(col0, diag, singVals, shifts, mus);
  perturbCol0(col0, diag, singVals, shifts, mus, zhat);
  computeSingVecs(zhat, diag, singVals, shifts, mus, U, V);

  // Reverse order so that singular values in increased order
  singVals.reverseInPlace();
  U.leftCols(n) = U.leftCols(n).rowwise().reverse().eval();
  if (compV) V = V.rowwise().reverse().eval();
}

template <typename MatrixType>
void BDCSVD<MatrixType>::computeSingVals(const ArrayXr& col0, const ArrayXr& diag, 
                                         VectorType& singVals, ArrayXr& shifts, ArrayXr& mus)
{
  using std::abs;
  using std::swap;

  Index n = col0.size();
  for (Index k = 0; k < n; ++k) {
    if (col0(k) == 0) {
      // entry is deflated, so singular value is on diagonal
      singVals(k) = diag(k);
      mus(k) = 0;
      shifts(k) = diag(k);
      continue;
    } 

    // otherwise, use secular equation to find singular value
    RealScalar left = diag(k);
    RealScalar right = (k != n-1) ? diag(k+1) : (diag(n-1) + col0.matrix().norm());

    // first decide whether it's closer to the left end or the right end
    RealScalar mid = left + (right-left) / 2;
    RealScalar fMid = 1 + (col0.square() / ((diag + mid) * (diag - mid))).sum();

    RealScalar shift;
    if (k == n-1 || fMid > 0) shift = left;
    else shift = right;
    
    // measure everything relative to shift
    ArrayXr diagShifted = diag - shift;
    
    // initial guess
    RealScalar muPrev, muCur;
    if (shift == left) {
      muPrev = (right - left) * 0.1;
      if (k == n-1) muCur = right - left;
      else muCur = (right - left) * 0.5; 
    } else {
      muPrev = -(right - left) * 0.1;
      muCur = -(right - left) * 0.5;
    }

    RealScalar fPrev = 1 + (col0.square() / ((diagShifted - muPrev) * (diag + shift + muPrev))).sum();
    RealScalar fCur = 1 + (col0.square() / ((diagShifted - muCur) * (diag + shift + muCur))).sum();
    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 = false;
    while (abs(muCur - muPrev) > 8 * NumTraits<RealScalar>::epsilon() * (std::max)(abs(muCur), abs(muPrev)) && fCur != fPrev && !useBisection) {
      ++m_numIters;

      RealScalar a = (fCur - fPrev) / (1/muCur - 1/muPrev);
      RealScalar b = fCur - a / muCur;

      muPrev = muCur;
      fPrev = fCur;
      muCur = -a / b;
      fCur = 1 + (col0.square() / ((diagShifted - muCur) * (diag + shift + muCur))).sum();
      
      if (shift == left && (muCur < 0 || muCur > right - left)) useBisection = true;
      if (shift == right && (muCur < -(right - left) || muCur > 0)) useBisection = true;
    }

    // fall back on bisection method if rational interpolation did not work
    if (useBisection) {
      RealScalar leftShifted, rightShifted;
      if (shift == left) {
        leftShifted = 1e-30;
        if (k == 0) rightShifted = right - left;
        else rightShifted = (right - left) * 0.6; // theoretically we can take 0.5, but let's be safe
      } else {
        leftShifted = -(right - left) * 0.6;
        rightShifted = -1e-30;
      }
      
      RealScalar fLeft = 1 + (col0.square() / ((diagShifted - leftShifted) * (diag + shift + leftShifted))).sum();
      RealScalar fRight = 1 + (col0.square() / ((diagShifted - rightShifted) * (diag + shift + rightShifted))).sum();
      assert(fLeft * fRight < 0);
      
      while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * (std::max)(abs(leftShifted), abs(rightShifted))) {
        RealScalar midShifted = (leftShifted + rightShifted) / 2;
        RealScalar fMid = 1 + (col0.square() / ((diagShifted - midShifted) * (diag + shift + midShifted))).sum();
        if (fLeft * fMid < 0) {
          rightShifted = midShifted;
          fRight = fMid;
        } 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)
    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 ArrayXr& col0, const ArrayXr& diag, const VectorType& singVals,
    const ArrayXr& shifts, const ArrayXr& mus, ArrayXr& zhat)
{
  Index n = col0.size();
  for (Index k = 0; k < n; ++k) {
    if (col0(k) == 0) 
      zhat(k) = 0;
    else {
      // see equation (3.6)
      using std::sqrt;
      RealScalar tmp = 
        sqrt(
             (singVals(n-1) + diag(k)) * (mus(n-1) + (shifts(n-1) - diag(k)))
             * (
                ((singVals.head(k).array() + diag(k)) * (mus.head(k) + (shifts.head(k) - diag(k))))
                / ((diag.head(k).array() + diag(k)) * (diag.head(k).array() - diag(k)))
               ).prod() 
             * (
                ((singVals.segment(k, n-k-1).array() + diag(k)) * (mus.segment(k, n-k-1) + (shifts.segment(k, n-k-1) - diag(k))))
                / ((diag.tail(n-k-1) + diag(k)) * (diag.tail(n-k-1) - diag(k)))
               ).prod()
             );
      if (col0(k) > 0) zhat(k) = tmp;
      else zhat(k) = -tmp;
    }
  }
}

// compute singular vectors
template <typename MatrixType>
void BDCSVD<MatrixType>::computeSingVecs
   (const ArrayXr& zhat, const ArrayXr& diag, const VectorType& singVals,
    const ArrayXr& shifts, const ArrayXr& mus, MatrixXr& U, MatrixXr& V)
{
  Index n = zhat.size();
  for (Index k = 0; k < n; ++k) {
    if (zhat(k) == 0) {
      U.col(k) = VectorType::Unit(n+1, k);
      if (compV) V.col(k) = VectorType::Unit(n, k);
    } else {
      U.col(k).head(n) = zhat / (((diag - shifts(k)) - mus(k)) * (diag + singVals[k]));
      U(n,k) = 0;
      U.col(k).normalize();
    
      if (compV) {
        V.col(k).tail(n-1) = (diag * zhat / (((diag - shifts(k)) - mus(k)) * (diag + singVals[k]))).tail(n-1);
        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;
  RealScalar c = m_computed(firstCol + shift, firstCol + shift);
  RealScalar s = m_computed(i, firstCol + shift);
  RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
  if (r == 0){
    m_computed(i, i)=0;
    return;
  }
  c/=r;
  s/=r;
  m_computed(firstCol + shift, firstCol + shift) = r;  
  m_computed(i, firstCol + shift) = 0;
  m_computed(i, i) = 0;
  if (compU){
    m_naiveU.col(firstCol).segment(firstCol,size) = 
      c * m_naiveU.col(firstCol).segment(firstCol, size) - 
      s * m_naiveU.col(i).segment(firstCol, size) ;

    m_naiveU.col(i).segment(firstCol, size) = 
      (c + s*s/c) * m_naiveU.col(i).segment(firstCol, size) + 
      (s/c) * m_naiveU.col(firstCol).segment(firstCol,size);
  }
}// end deflation 43


// page 13
// i,j >= 1, i != j and |di - dj| < epsilon * norm2(M)
// We apply two rotations to have zj = 0;
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, firstColm + j - 1);
  RealScalar s = m_computed(firstColm, firstColm + i - 1);
  RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
  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 + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
  m_computed(firstColm + j, firstColm) = 0;
  if (compU){
    m_naiveU.col(firstColu + i).segment(firstColu, size) = 
      c * m_naiveU.col(firstColu + i).segment(firstColu, size) - 
      s * m_naiveU.col(firstColu + j).segment(firstColu, size) ;

    m_naiveU.col(firstColu + j).segment(firstColu, size) = 
      (c + s*s/c) *  m_naiveU.col(firstColu + j).segment(firstColu, size) + 
      (s/c) * m_naiveU.col(firstColu + i).segment(firstColu, size);
  } 
  if (compV){
    m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) = 
      c * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) + 
      s * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) ;

    m_naiveV.col(firstColW + j).segment(firstRowW, size - 1)  = 
      (c + s*s/c) * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) - 
      (s/c) * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1);
  }
}// 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){
  //condition 4.1
  using std::sqrt;
  const Index length = lastCol + 1 - firstCol;
  RealScalar norm1 = m_computed.block(firstCol+shift, firstCol+shift, length, 1).squaredNorm();
  RealScalar norm2 = m_computed.block(firstCol+shift, firstCol+shift, length, length).diagonal().squaredNorm();
  RealScalar EPS = 10 * NumTraits<RealScalar>::epsilon() * sqrt(norm1 + norm2);
  if (m_computed(firstCol + shift, firstCol + shift) < EPS){
    m_computed(firstCol + shift, firstCol + shift) = EPS;
  }

  //condition 4.2
  for (Index i=firstCol + shift + 1;i<=lastCol + shift;i++){
    if (std::abs(m_computed(i, firstCol + shift)) < EPS){
      m_computed(i, firstCol + shift) = 0;
    }
  }

  //condition 4.3
  for (Index i=firstCol + shift + 1;i<=lastCol + shift; i++){
    if (m_computed(i, i) < EPS){
      deflation43(firstCol, shift, i, length);
    }
  }

  //condition 4.4
 
  Index i=firstCol + shift + 1, j=firstCol + shift + k + 1;
  //we stock the final place of each line
  Index *permutation = new Index[length];

  for (Index p =1; p < length; p++) {
    if (i> firstCol + shift + k){
      permutation[p] = j;
      j++;
    } else if (j> lastCol + shift) 
    {
      permutation[p] = i;
      i++;
    }
    else 
    {
      if (m_computed(i, i) < m_computed(j, j)){
        permutation[p] = j;
        j++;
      } 
      else
      {
        permutation[p] = i;
        i++;
      }
    }
  }
  //we do the permutation
  RealScalar aux;
  //we stock the current index of each col
  //and the column of each index
  Index *realInd = new Index[length];
  Index *realCol = new Index[length];
  for (int pos = 0; pos< length; pos++){
    realCol[pos] = pos + firstCol + shift;
    realInd[pos] = pos;
  }
  const Index Zero = firstCol + shift;
  VectorType temp;
  for (int i = 1; i < length - 1; i++){
    const Index I = i + Zero;
    const Index realI = realInd[i];
    const Index j  = permutation[length - i] - Zero;
    const Index J = realCol[j];
    
    //diag displace
    aux = m_computed(I, I); 
    m_computed(I, I) = m_computed(J, J);
    m_computed(J, J) = aux;
    
    //firstrow displace
    aux = m_computed(I, Zero); 
    m_computed(I, Zero) = m_computed(J, Zero);
    m_computed(J, Zero) = aux;

    // change columns
    if (compU) {
      temp = m_naiveU.col(I - shift).segment(firstCol, length + 1);
      m_naiveU.col(I - shift).segment(firstCol, length + 1) << 
        m_naiveU.col(J - shift).segment(firstCol, length + 1);
      m_naiveU.col(J - shift).segment(firstCol, length + 1) << temp;
    } 
    else
    {
      temp = m_naiveU.col(I - shift).segment(0, 2);
      m_naiveU.col(I - shift).segment(0, 2) << 
        m_naiveU.col(J - shift).segment(0, 2);
      m_naiveU.col(J - shift).segment(0, 2) << temp;      
    }
    if (compV) {
      const Index CWI = I + firstColW - Zero;
      const Index CWJ = J + firstColW - Zero;
      temp = m_naiveV.col(CWI).segment(firstRowW, length);
      m_naiveV.col(CWI).segment(firstRowW, length) << m_naiveV.col(CWJ).segment(firstRowW, length);
      m_naiveV.col(CWJ).segment(firstRowW, length) << temp;
    }

    //update real pos
    realCol[realI] = J;
    realCol[j] = I;
    realInd[J - Zero] = realI;
    realInd[I - Zero] = j;
  }
  for (Index i = firstCol + shift + 1; i<lastCol + shift;i++){
    if ((m_computed(i + 1, i + 1) - m_computed(i, i)) < EPS){
      deflation44(firstCol , 
		  firstCol + shift, 
		  firstRowW, 
		  firstColW, 
		  i - Zero, 
		  i + 1 - Zero, 
		  length);
    }
  }
  delete [] permutation;
  delete [] realInd;
  delete [] realCol;
}//end deflation


namespace internal{

template<typename _MatrixType, typename Rhs>
struct solve_retval<BDCSVD<_MatrixType>, Rhs>
  : solve_retval_base<BDCSVD<_MatrixType>, Rhs>
{
  typedef BDCSVD<_MatrixType> BDCSVDType;
  EIGEN_MAKE_SOLVE_HELPERS(BDCSVDType, Rhs)

  template<typename Dest> void evalTo(Dest& dst) const
  {
    eigen_assert(rhs().rows() == dec().rows());
    // A = U S V^*
    // So A^{ - 1} = V S^{ - 1} U^*    
    Index diagSize = (std::min)(dec().rows(), dec().cols());
    typename BDCSVDType::SingularValuesType invertedSingVals(diagSize);
    Index nonzeroSingVals = dec().nonzeroSingularValues();
    invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
    invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
    
    dst = dec().matrixV().leftCols(diagSize)
      * invertedSingVals.asDiagonal()
      * dec().matrixU().leftCols(diagSize).adjoint()
      * rhs();	
    return;
  }
};

} //end namespace internal

  /** \svd_module
   *
   * \return the singular value decomposition of \c *this computed by 
   *  BDC Algorithm
   *
   * \sa class BDCSVD
   */
/*
template<typename Derived>
BDCSVD<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
{
  return BDCSVD<PlainObject>(*this, computationOptions);
}
*/

} // end namespace Eigen

#endif