eigen/Eigen/src/SVD/JacobiSVD.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.

#ifndef EIGEN_JACOBISVD_H
#define EIGEN_JACOBISVD_H

// forward declaration (needed by ICC)
// the empty body is required by MSVC
template<typename MatrixType, int QRPreconditioner,
         bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
struct ei_svd_precondition_2x2_block_to_be_real {};


/*** QR preconditioners (R-SVD)
 ***
 *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
 *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
 *** JacobiSVD which by itself is only able to work on square matrices.
 ***/

enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };

template<typename MatrixType, int QRPreconditioner, int Case>
struct ei_qr_preconditioner_should_do_anything
{
  enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
             MatrixType::ColsAtCompileTime != Dynamic &&
             MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
         b = MatrixType::RowsAtCompileTime != Dynamic &&
             MatrixType::ColsAtCompileTime != Dynamic &&
             MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
         ret = !( (QRPreconditioner == NoQRPreconditioner) ||
                  (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
                  (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
  };
};

template<typename MatrixType, int QRPreconditioner, int Case,
         bool DoAnything = ei_qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
> struct ei_qr_preconditioner_impl {};

template<typename MatrixType, int QRPreconditioner, int Case>
struct ei_qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
{
  static bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
  {
    return false;
  }
};

/*** preconditioner using FullPivHouseholderQR ***/

template<typename MatrixType>
struct ei_qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
{
  static bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
  {
    if(matrix.rows() > matrix.cols())
    {
      FullPivHouseholderQR<MatrixType> qr(matrix);
      svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
      if(svd.m_computeFullU) svd.m_matrixU = qr.matrixQ();
      if(svd.computeV()) svd.m_matrixV = qr.colsPermutation();
      return true;
    }
    return false;
  }
};

template<typename MatrixType>
struct ei_qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
{
  static bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
  {
    if(matrix.cols() > matrix.rows())
    {
      typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::RowsAtCompileTime,
                     MatrixType::Options, MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime>
              TransposeTypeWithSameStorageOrder;
      FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> qr(matrix.adjoint());
      svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
      if(svd.m_computeFullV) svd.m_matrixV = qr.matrixQ();
      if(svd.computeU()) svd.m_matrixU = qr.colsPermutation();
      return true;
    }
    else return false;
  }
};

/*** preconditioner using ColPivHouseholderQR ***/

template<typename MatrixType>
struct ei_qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
{
  static bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
  {
    if(matrix.rows() > matrix.cols())
    {
      ColPivHouseholderQR<MatrixType> qr(matrix);
      svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
      if(svd.m_computeFullU) svd.m_matrixU = qr.householderQ();
      else if(svd.m_computeThinU) {
        svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
        qr.householderQ().applyThisOnTheLeft(svd.m_matrixU);
      }
      if(svd.computeV()) svd.m_matrixV = qr.colsPermutation();
      return true;
    }
    return false;
  }
};

template<typename MatrixType>
struct ei_qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
{
  static bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
  {
    if(matrix.cols() > matrix.rows())
    {
      typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::RowsAtCompileTime,
                     MatrixType::Options, MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime>
              TransposeTypeWithSameStorageOrder;
      ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> qr(matrix.adjoint());
      svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
      if(svd.m_computeFullV) svd.m_matrixV = qr.householderQ();
      else if(svd.m_computeThinV) {
        svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
        qr.householderQ().applyThisOnTheLeft(svd.m_matrixV);
      }
      if(svd.computeU()) svd.m_matrixU = qr.colsPermutation();
      return true;
    }
    else return false;
  }
};

/*** preconditioner using HouseholderQR ***/

template<typename MatrixType>
struct ei_qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
{
  static bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
  {
    if(matrix.rows() > matrix.cols())
    {
      HouseholderQR<MatrixType> qr(matrix);
      svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
      if(svd.m_computeFullU) svd.m_matrixU = qr.householderQ();
      else if(svd.m_computeThinU) {
        svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
        qr.householderQ().applyThisOnTheLeft(svd.m_matrixU);
      }
      if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
      return true;
    }
    return false;
  }
};

template<typename MatrixType>
struct ei_qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
{
  static bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
  {
    if(matrix.cols() > matrix.rows())
    {
      typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::RowsAtCompileTime,
                     MatrixType::Options, MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime>
              TransposeTypeWithSameStorageOrder;
      HouseholderQR<TransposeTypeWithSameStorageOrder> qr(matrix.adjoint());
      svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
      if(svd.m_computeFullV) svd.m_matrixV = qr.householderQ();
      else if(svd.m_computeThinV) {
        svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
        qr.householderQ().applyThisOnTheLeft(svd.m_matrixV);
      }
      if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
      return true;
    }
    else return false;
  }
};



/** \ingroup SVD_Module
  *
  *
  * \class JacobiSVD
  *
  * \brief Two-sided Jacobi SVD decomposition of a square matrix
  *
  * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
  * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
  *                        for the R-SVD step for non-square matrices. See discussion of possible values below.
  *
  * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
  *   \f[ A = U S V^* \f]
  * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
  * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
  * and right \em singular \em vectors of \a A respectively.
  *
  * Singular values are always sorted in decreasing order.
  *
  * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly.
  *
  * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
  * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
  * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
  * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
  *
  * Here's an example demonstrating basic usage:
  * \include JacobiSVD_basic.cpp
  * Output: \verbinclude JacobiSVD_basic.out
  * 
  * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
  * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and
  * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
  * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
  *
  * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
  * terminate in finite (and reasonable) time.
  * 
  * The possible values for QRPreconditioner are:
  * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
  * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
  *     Contrary to other QRs, it doesn't allow computing thin unitaries.
  * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
  *     This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
  *     is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
  *     process is more reliable than the optimized bidiagonal SVD iterations.
  * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
  *     JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
  *     faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
  *     if QR preconditioning is needed before applying it anyway.
  *
  * \sa MatrixBase::jacobiSvd()
  */
template<typename _MatrixType, int QRPreconditioner> class JacobiSVD
{
  public:

    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 ei_plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
    typedef typename ei_plain_row_type<MatrixType>::type RowType;
    typedef typename ei_plain_col_type<MatrixType>::type ColType;
    typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
                   MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
            WorkMatrixType;

    /** \brief Default Constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via JacobiSVD::compute(const MatrixType&).
      */
    JacobiSVD() : m_isInitialized(false) {}


    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem size.
      * \sa JacobiSVD()
      */
    JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 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.
     */
    JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
    {
      compute(matrix, computationOptions);
    }

    /** \brief Method 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.
     */
    JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions = 0);

    /** \returns the \a U matrix.
     *
     * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
     * the U matrix is n-by-n if you asked for ComputeFullU, and is n-by-m if you asked for ComputeThinU.
     *
     * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed.
     *
     * This method asserts that you asked for \a U to be computed.
     */
    const MatrixUType& matrixU() const
    {
      ei_assert(m_isInitialized && "JacobiSVD is not initialized.");
      ei_assert(computeU() && "This JacobiSVD decomposition didn't compute U. Did you ask for it?");
      return m_matrixU;
    }

    /** \returns the \a V matrix.
     *
     * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
     * the V matrix is p-by-p if you asked for ComputeFullV, and is p-by-m if you asked for ComputeThinV.
     *
     * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed.
     *
     * This method asserts that you asked for \a V to be computed.
     */
    const MatrixVType& matrixV() const
    {
      ei_assert(m_isInitialized && "JacobiSVD is not initialized.");
      ei_assert(computeV() && "This JacobiSVD decomposition didn't compute V. Did you ask for it?");
      return m_matrixV;
    }

    /** \returns the vector of singular values.
     *
     * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the
     * returned vector has size \a m.
     */
    const SingularValuesType& singularValues() const
    {
      ei_assert(m_isInitialized && "JacobiSVD is not initialized.");
      return m_singularValues;
    }

    /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
    inline bool computeU() const { return m_computeFullU || m_computeThinU; }
    /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
    inline bool computeV() const { return m_computeFullV || m_computeThinV; }

    /** \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 ei_solve_retval<JacobiSVD, Rhs>
    solve(const MatrixBase<Rhs>& b) const
    {
      ei_assert(m_isInitialized && "JacobiSVD is not initialized.");
      ei_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
      return ei_solve_retval<JacobiSVD, Rhs>(*this, b.derived());
    }

    /** \returns the number of singular values that are not exactly 0 */
    Index nonzeroSingularValues() const
    {
      ei_assert(m_isInitialized && "JacobiSVD is not initialized.");
      return m_nonzeroSingularValues;
    }

    inline Index rows() const { return m_rows; }
    inline Index cols() const { return m_cols; }

  private:
    void allocate(Index rows, Index cols, unsigned int computationOptions = 0);

  protected:
    MatrixUType m_matrixU;
    MatrixVType m_matrixV;
    SingularValuesType m_singularValues;
    WorkMatrixType m_workMatrix;
    bool m_isInitialized;
    bool m_computeFullU, m_computeThinU;
    bool m_computeFullV, m_computeThinV;
    Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;

    template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
    friend struct ei_svd_precondition_2x2_block_to_be_real;
    template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
    friend struct ei_qr_preconditioner_impl;
};

template<typename MatrixType, int QRPreconditioner>
void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
{
  m_rows = rows;
  m_cols = cols;
  m_isInitialized = false;
  m_computeFullU = (computationOptions & ComputeFullU) != 0;
  m_computeThinU = (computationOptions & ComputeThinU) != 0;
  m_computeFullV = (computationOptions & ComputeFullV) != 0;
  m_computeThinV = (computationOptions & ComputeThinV) != 0;
  ei_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
  ei_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
  ei_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
              "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
  if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
  {
      ei_assert(!(m_computeThinU || m_computeThinV) &&
              "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
              "Use the ColPivHouseholderQR preconditioner instead.");
  }
  m_diagSize = std::min(m_rows, m_cols);
  m_singularValues.resize(m_diagSize);
  m_matrixU.resize(m_rows, m_computeFullU ? m_rows
                          : m_computeThinU ? m_diagSize
                          : 0);
  m_matrixV.resize(m_cols, m_computeFullV ? m_cols
                          : m_computeThinV ? m_diagSize
                          : 0);
  m_workMatrix.resize(m_diagSize, m_diagSize);
}


template<typename MatrixType, int QRPreconditioner>
struct ei_svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
{
  typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
  typedef typename SVD::Index Index;
  static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
};

template<typename MatrixType, int QRPreconditioner>
struct ei_svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
{
  typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef typename SVD::Index Index;
  static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
  {
    Scalar z;
    JacobiRotation<Scalar> rot;
    RealScalar n = ei_sqrt(ei_abs2(work_matrix.coeff(p,p)) + ei_abs2(work_matrix.coeff(q,p)));
    if(n==0)
    {
      z = ei_abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
      work_matrix.row(p) *= z;
      if(svd.computeU()) svd.m_matrixU.col(p) *= ei_conj(z);
      z = ei_abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
      work_matrix.row(q) *= z;
      if(svd.computeU()) svd.m_matrixU.col(q) *= ei_conj(z);
    }
    else
    {
      rot.c() = ei_conj(work_matrix.coeff(p,p)) / n;
      rot.s() = work_matrix.coeff(q,p) / n;
      work_matrix.applyOnTheLeft(p,q,rot);
      if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
      if(work_matrix.coeff(p,q) != Scalar(0))
      {
        Scalar z = ei_abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
        work_matrix.col(q) *= z;
        if(svd.computeV()) svd.m_matrixV.col(q) *= z;
      }
      if(work_matrix.coeff(q,q) != Scalar(0))
      {
        z = ei_abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
        work_matrix.row(q) *= z;
        if(svd.computeU()) svd.m_matrixU.col(q) *= ei_conj(z);
      }
    }
  }
};

template<typename MatrixType, typename RealScalar, typename Index>
void ei_real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
                            JacobiRotation<RealScalar> *j_left,
                            JacobiRotation<RealScalar> *j_right)
{
  Matrix<RealScalar,2,2> m;
  m << ei_real(matrix.coeff(p,p)), ei_real(matrix.coeff(p,q)),
       ei_real(matrix.coeff(q,p)), ei_real(matrix.coeff(q,q));
  JacobiRotation<RealScalar> rot1;
  RealScalar t = m.coeff(0,0) + m.coeff(1,1);
  RealScalar d = m.coeff(1,0) - m.coeff(0,1);
  if(t == RealScalar(0))
  {
    rot1.c() = 0;
    rot1.s() = d > 0 ? 1 : -1;
  }
  else
  {
    RealScalar u = d / t;
    rot1.c() = RealScalar(1) / ei_sqrt(1 + ei_abs2(u));
    rot1.s() = rot1.c() * u;
  }
  m.applyOnTheLeft(0,1,rot1);
  j_right->makeJacobi(m,0,1);
  *j_left  = rot1 * j_right->transpose();
}

template<typename MatrixType, int QRPreconditioner>
JacobiSVD<MatrixType, QRPreconditioner>&
JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
{
  allocate(matrix.rows(), matrix.cols(), computationOptions);

  // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
  // only worsening the precision of U and V as we accumulate more rotations
  const RealScalar precision = 2 * NumTraits<Scalar>::epsilon();

  /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */

  if(!ei_qr_preconditioner_impl<MatrixType, QRPreconditioner, PreconditionIfMoreColsThanRows>::run(*this, matrix)
  && !ei_qr_preconditioner_impl<MatrixType, QRPreconditioner, PreconditionIfMoreRowsThanCols>::run(*this, matrix))
  {
    m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize);
    if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
    if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
    if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
    if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
  }

  /*** step 2. The main Jacobi SVD iteration. ***/

  bool finished = false;
  while(!finished)
  {
    finished = true;

    // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix

    for(Index p = 1; p < m_diagSize; ++p)
    {
      for(Index q = 0; q < p; ++q)
      {
        // if this 2x2 sub-matrix is not diagonal already...
        // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
        // keep us iterating forever.
        if(std::max(ei_abs(m_workMatrix.coeff(p,q)),ei_abs(m_workMatrix.coeff(q,p)))
            > std::max(ei_abs(m_workMatrix.coeff(p,p)),ei_abs(m_workMatrix.coeff(q,q)))*precision)
        {
          finished = false;

          // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
          ei_svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
          JacobiRotation<RealScalar> j_left, j_right;
          ei_real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);

          // accumulate resulting Jacobi rotations
          m_workMatrix.applyOnTheLeft(p,q,j_left);
          if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());

          m_workMatrix.applyOnTheRight(p,q,j_right);
          if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
        }
      }
    }
  }

  /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/

  for(Index i = 0; i < m_diagSize; ++i)
  {
    RealScalar a = ei_abs(m_workMatrix.coeff(i,i));
    m_singularValues.coeffRef(i) = a;
    if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
  }

  /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/

  m_nonzeroSingularValues = m_diagSize;
  for(Index i = 0; i < m_diagSize; i++)
  {
    Index pos;
    RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
    if(maxRemainingSingularValue == RealScalar(0))
    {
      m_nonzeroSingularValues = i;
      break;
    }
    if(pos)
    {
      pos += i;
      std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
      if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
      if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
    }
  }

  m_isInitialized = true;
  return *this;
}

template<typename _MatrixType, int QRPreconditioner, typename Rhs>
struct ei_solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
  : ei_solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
{
  typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
  EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)

  template<typename Dest> void evalTo(Dest& dst) const
  {
    ei_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 JacobiSVDType::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();
  }
};

template<typename Derived>
JacobiSVD<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
{
  return JacobiSVD<PlainObject>(*this, computationOptions);
}



#endif // EIGEN_JACOBISVD_H