Reafctoring in D&C SVD unsupported module: clean and merge the SVDBase class to Eigen/SVD, rm copy/pasted JacobiSVD.h file

unsupported/Eigen/src/BDCSVD/BDCSVD.h

@@ -0,0 +1,949 @@
+// 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 {
+
+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
+ *
+ * \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<BDCSVD<_MatrixType> >
+{
+  typedef SVDBase<BDCSVD> 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() : 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)
+    : 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)
+    : 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.
+   */
+  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");
+    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(computeU() && 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;
+    }
+  }
+  
+  using Base::computeU;
+  using Base::computeV;
+ 
+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 (Base::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<>
+BDCSVD<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>
+BDCSVD<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

unsupported/Eigen/src/BDCSVD/CMakeLists.txt

@@ -0,0 +1,6 @@
+FILE(GLOB Eigen_BDCSVD_SRCS "*.h")
+
+INSTALL(FILES
+  ${Eigen_BDCSVD_SRCS}
+  DESTINATION ${INCLUDE_INSTALL_DIR}unsupported/Eigen/src/BDCSVD COMPONENT Devel
+  )

unsupported/Eigen/src/BDCSVD/TODOBdcsvd.txt

@@ -0,0 +1,29 @@
+TO DO LIST
+
+
+
+(optional optimization) - do all the allocations in the allocate part 
+                        - support static matrices
+                        - return a error at compilation time when using integer matrices (int, long, std::complex<int>, ...)
+
+to finish the algorithm :
+			-implement the last part of the algorithm as described on the reference paper. 
+			    You may find more information on that part on this paper
+
+			-to replace the call to JacobiSVD at the end of the divide algorithm, just after the call to 
+			    deflation.
+
+(suggested step by step resolution)
+                       0) comment the call to Jacobi in the last part of the divide method and everything right after
+                               until the end of the method. What is commented can be a guideline to steps 3) 4) and 6)
+                       1) solve the secular equation (Characteristic equation) on the values that are not null (zi!=0 and di!=0), after the deflation
+                               wich should be uncommented in the divide method
+                       2) remember the values of the singular values that are already computed (zi=0)
+                       3) assign the singular values found in m_computed at the right places (with the ones found in step 2) )
+                               in decreasing order
+                       4) set the firstcol to zero (except the first element) in m_computed
+                       5) compute all the singular vectors when CompV is set to true and only the left vectors when
+                               CompV is set to false
+                       6) multiply naiveU and naiveV to the right by the matrices found, only naiveU when CompV is set to
+                               false, /!\ if CompU is false NaiveU has only 2 rows
+                       7) delete everything commented in step 0)

unsupported/Eigen/src/BDCSVD/doneInBDCSVD.txt

@@ -0,0 +1,21 @@
+This unsupported package is about a divide and conquer algorithm to compute SVD.
+
+The implementation follows as closely as possible the following reference paper : 
+http://www.cs.yale.edu/publications/techreports/tr933.pdf
+
+The code documentation uses the same names for variables as the reference paper. The code, deflation included, is
+working  but there are a few things that could be optimised as explained in the TODOBdsvd. 
+
+In the code comments were put at the line where would be the third step of the algorithm so one could simply add the call 
+of a function doing the last part of the algorithm and that would not require any knowledge of the part we implemented.
+
+In the TODOBdcsvd we explain what is the main difficulty of the last part and suggest a reference paper to help solve it.
+
+The implemented has trouble with fixed size matrices. 
+
+In the actual implementation, it returns matrices of zero when ask to do a svd on an int matrix. 
+
+
+Paper for the third part:
+http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf
+