text: // 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>
  //
  // This Source Code Form is subject to the terms of the Mozilla
  // Public License v. 2.0. If a copy of the MPL was not distributed
  // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
  
  #ifndef EIGEN_JACOBISVD_H
  #define EIGEN_JACOBISVD_H
  
  namespace Eigen { 
  
  namespace internal {
  // 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 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 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 = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
  > struct qr_preconditioner_impl {};
  
  template<typename MatrixType, int QRPreconditioner, int Case>
  class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
  {
  public:
    typedef typename MatrixType::Index Index;
    void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
    bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
    {
      return false;
    }
  };
  
  /*** preconditioner using FullPivHouseholderQR ***/
  
  template<typename MatrixType>
  class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
  {
  public:
    typedef typename MatrixType::Index Index;
    typedef typename MatrixType::Scalar Scalar;
    enum
    {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
    };
    typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
  
    void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
    {
      if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
      {
        m_qr.~QRType();
        ::new (&m_qr) QRType(svd.rows(), svd.cols());
      }
      if (svd.m_computeFullU) m_workspace.resize(svd.rows());
    }
  
    bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
    {
      if(matrix.rows() > matrix.cols())
      {
        m_qr.compute(matrix);
        svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
        if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
        if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
        return true;
      }
      return false;
    }
  private:
    typedef FullPivHouseholderQR<MatrixType> QRType;
    QRType m_qr;
    WorkspaceType m_workspace;
  };
  
  template<typename MatrixType>
  class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
  {
  public:
    typedef typename MatrixType::Index Index;
    typedef typename MatrixType::Scalar Scalar;
    enum
    {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
      Options = MatrixType::Options
    };
    typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
            TransposeTypeWithSameStorageOrder;
  
    void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
    {
      if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
      {
        m_qr.~QRType();
        ::new (&m_qr) QRType(svd.cols(), svd.rows());
      }
      m_adjoint.resize(svd.cols(), svd.rows());
      if (svd.m_computeFullV) m_workspace.resize(svd.cols());
    }
  
    bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
    {
      if(matrix.cols() > matrix.rows())
      {
        m_adjoint = matrix.adjoint();
        m_qr.compute(m_adjoint);
        svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
        if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
        if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
        return true;
      }
      else return false;
    }
  private:
    typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
    QRType m_qr;
    TransposeTypeWithSameStorageOrder m_adjoint;
    typename internal::plain_row_type<MatrixType>::type m_workspace;
  };
  
  /*** preconditioner using ColPivHouseholderQR ***/
  
  template<typename MatrixType>
  class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
  {
  public:
    typedef typename MatrixType::Index Index;
  
    void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
    {
      if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
      {
        m_qr.~QRType();
        ::new (&m_qr) QRType(svd.rows(), svd.cols());
      }
      if (svd.m_computeFullU) m_workspace.resize(svd.rows());
      else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
    }
  
    bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
    {
      if(matrix.rows() > matrix.cols())
      {
        m_qr.compute(matrix);
        svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
        if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
        else if(svd.m_computeThinU)
        {
          svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
          m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
        }
        if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
        return true;
      }
      return false;
    }
  
  private:
    typedef ColPivHouseholderQR<MatrixType> QRType;
    QRType m_qr;
    typename internal::plain_col_type<MatrixType>::type m_workspace;
  };
  
  template<typename MatrixType>
  class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
  {
  public:
    typedef typename MatrixType::Index Index;
    typedef typename MatrixType::Scalar Scalar;
    enum
    {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
      Options = MatrixType::Options
    };
  
    typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
            TransposeTypeWithSameStorageOrder;
  
    void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
    {
      if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
      {
        m_qr.~QRType();
        ::new (&m_qr) QRType(svd.cols(), svd.rows());
      }
      if (svd.m_computeFullV) m_workspace.resize(svd.cols());
      else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
      m_adjoint.resize(svd.cols(), svd.rows());
    }
  
    bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
    {
      if(matrix.cols() > matrix.rows())
      {
        m_adjoint = matrix.adjoint();
        m_qr.compute(m_adjoint);
  
        svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
        if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
        else if(svd.m_computeThinV)
        {
          svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
          m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
        }
        if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
        return true;
      }
      else return false;
    }
  
  private:
    typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
    QRType m_qr;
    TransposeTypeWithSameStorageOrder m_adjoint;
    typename internal::plain_row_type<MatrixType>::type m_workspace;
  };
  
  /*** preconditioner using HouseholderQR ***/
  
  template<typename MatrixType>
  class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
  {
  public:
    typedef typename MatrixType::Index Index;
  
    void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
    {
      if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
      {
        m_qr.~QRType();
        ::new (&m_qr) QRType(svd.rows(), svd.cols());
      }
      if (svd.m_computeFullU) m_workspace.resize(svd.rows());
      else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
    }
  
    bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
    {
      if(matrix.rows() > matrix.cols())
      {
        m_qr.compute(matrix);
        svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
        if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
        else if(svd.m_computeThinU)
        {
          svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
          m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
        }
        if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
        return true;
      }
      return false;
    }
  private:
    typedef HouseholderQR<MatrixType> QRType;
    QRType m_qr;
    typename internal::plain_col_type<MatrixType>::type m_workspace;
  };
  
  template<typename MatrixType>
  class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
  {
  public:
    typedef typename MatrixType::Index Index;
    typedef typename MatrixType::Scalar Scalar;
    enum
    {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
      Options = MatrixType::Options
    };
  
    typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
            TransposeTypeWithSameStorageOrder;
  
    void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
    {
      if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
      {
        m_qr.~QRType();
        ::new (&m_qr) QRType(svd.cols(), svd.rows());
      }
      if (svd.m_computeFullV) m_workspace.resize(svd.cols());
      else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
      m_adjoint.resize(svd.cols(), svd.rows());
    }
  
    bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
    {
      if(matrix.cols() > matrix.rows())
      {
        m_adjoint = matrix.adjoint();
        m_qr.compute(m_adjoint);
  
        svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
        if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
        else if(svd.m_computeThinV)
        {
          svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
          m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
        }
        if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
        return true;
      }
      else return false;
    }
  
  private:
    typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
    QRType m_qr;
    TransposeTypeWithSameStorageOrder m_adjoint;
    typename internal::plain_row_type<MatrixType>::type m_workspace;
  };
  
  /*** 2x2 SVD implementation
   ***
   *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
   ***/
  
  template<typename MatrixType, int QRPreconditioner>
  struct 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 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)
    {
      using std::sqrt;
      Scalar z;
      JacobiRotation<Scalar> rot;
      RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
      if(n==0)
      {
        z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
        work_matrix.row(p) *= z;
        if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
        if(work_matrix.coeff(q,q)!=Scalar(0))
          z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
        else
          z = Scalar(0);
        work_matrix.row(q) *= z;
        if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
      }
      else
      {
        rot.c() = 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 = 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 = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
          work_matrix.row(q) *= z;
          if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
        }
      }
    }
  };
  
  template<typename MatrixType, typename RealScalar, typename Index>
  void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
                           JacobiRotation<RealScalar> *j_left,
                           JacobiRotation<RealScalar> *j_right)
  {
    using std::sqrt;
    using std::abs;
    Matrix<RealScalar,2,2> m;
    m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
         numext::real(matrix.coeff(q,p)), numext::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() = RealScalar(0);
      rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
    }
    else
    {
      RealScalar t2d2 = numext::hypot(t,d);
      rot1.c() = abs(t)/t2d2;
      rot1.s() = d/t2d2;
      if(t<RealScalar(0))
        rot1.s() = -rot1.s();
    }
    m.applyOnTheLeft(0,1,rot1);
    j_right->makeJacobi(m,0,1);
    *j_left  = rot1 * j_right->transpose();
  }
  
  } // end namespace internal
  
  /** \ingroup SVD_Module
    *
    *
    * \class JacobiSVD
    *
    * \brief Two-sided Jacobi SVD decomposition of a rectangular 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 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, 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),
          m_isAllocated(false),
          m_usePrescribedThreshold(false),
          m_computationOptions(0),
          m_rows(-1), m_cols(-1), m_diagSize(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 JacobiSVD()
        */
      JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
        : m_isInitialized(false),
          m_isAllocated(false),
          m_usePrescribedThreshold(false),
          m_computationOptions(0),
          m_rows(-1), m_cols(-1)
      {
        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)
        : m_isInitialized(false),
          m_isAllocated(false),
          m_usePrescribedThreshold(false),
          m_computationOptions(0),
          m_rows(-1), m_cols(-1)
      {
        compute(matrix, computationOptions);
      }
  
      /** \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.
       */
      JacobiSVD& 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).
       */
      JacobiSVD& compute(const MatrixType& matrix)
      {
        return compute(matrix, m_computationOptions);
      }
  
      /** \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
      {
        eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
        eigen_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
      {
        eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
        eigen_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.  Singular values are always sorted in decreasing order.
       */
      const SingularValuesType& singularValues() const
      {
        eigen_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 internal::solve_retval<JacobiSVD, Rhs>
      solve(const MatrixBase<Rhs>& b) const
      {
        eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
        eigen_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
        return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
      }
  
      /** \returns the number of singular values that are not exactly 0 */
      Index nonzeroSingularValues() const
      {
        eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
        return m_nonzeroSingularValues;
      }
      
      /** \returns the rank of the matrix of which \c *this is the SVD.
        *
        * \note This method has to determine which singular values should be considered nonzero.
        *       For that, it uses the threshold value that you can control by calling
        *       setThreshold(const RealScalar&).
        */
      inline Index rank() const
      {
        using std::abs;
        eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
        if(m_singularValues.size()==0) return 0;
        RealScalar premultiplied_threshold = m_singularValues.coeff(0) * threshold();
        Index i = m_nonzeroSingularValues-1;
        while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i;
        return i+1;
      }
      
      /** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(),
        * which need to determine when singular values are to be considered nonzero.
        * This is not used for the SVD decomposition itself.
        *
        * When it needs to get the threshold value, Eigen calls threshold().
        * The default is \c NumTraits<Scalar>::epsilon()
        *
        * \param threshold The new value to use as the threshold.
        *
        * A singular value will be considered nonzero if its value is strictly greater than
        *  \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$.
        *
        * If you want to come back to the default behavior, call setThreshold(Default_t)
        */
      JacobiSVD& setThreshold(const RealScalar& threshold)
      {
        m_usePrescribedThreshold = true;
        m_prescribedThreshold = threshold;
        return *this;
      }
  
      /** Allows to come back to the default behavior, letting Eigen use its default formula for
        * determining the threshold.
        *
        * You should pass the special object Eigen::Default as parameter here.
        * \code svd.setThreshold(Eigen::Default); \endcode
        *
        * See the documentation of setThreshold(const RealScalar&).
        */
      JacobiSVD& setThreshold(Default_t)
      {
        m_usePrescribedThreshold = false;
        return *this;
      }
  
      /** Returns the threshold that will be used by certain methods such as rank().
        *
        * See the documentation of setThreshold(const RealScalar&).
        */
      RealScalar threshold() const
      {
        eigen_assert(m_isInitialized || m_usePrescribedThreshold);
        return m_usePrescribedThreshold ? m_prescribedThreshold
                                        : (std::max<Index>)(1,m_diagSize)*NumTraits<Scalar>::epsilon();
      }
  
      inline Index rows() const { return m_rows; }
      inline Index cols() const { return m_cols; }
  
    private:
      void allocate(Index rows, Index cols, unsigned int computationOptions);
  
    protected:
      MatrixUType m_matrixU;
      MatrixVType m_matrixV;
      SingularValuesType m_singularValues;
      WorkMatrixType m_workMatrix;
      bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold;
      bool m_computeFullU, m_computeThinU;
      bool m_computeFullV, m_computeThinV;
      unsigned int m_computationOptions;
      Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
      RealScalar m_prescribedThreshold;
  
      template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
      friend struct internal::svd_precondition_2x2_block_to_be_real;
      template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
      friend struct internal::qr_preconditioner_impl;
  
      internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
      internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
  };
  
  template<typename MatrixType, int QRPreconditioner>
  void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
  {
    eigen_assert(rows >= 0 && cols >= 0);
  
    if (m_isAllocated &&
        rows == m_rows &&
        cols == m_cols &&
        computationOptions == m_computationOptions)
    {
      return;
    }
  
    m_rows = rows;
    m_cols = cols;
    m_isInitialized = false;
    m_isAllocated = true;
    m_computationOptions = computationOptions;
    m_computeFullU = (computationOptions & ComputeFullU) != 0;
    m_computeThinU = (computationOptions & ComputeThinU) != 0;
    m_computeFullV = (computationOptions & ComputeFullV) != 0;
    m_computeThinV = (computationOptions & ComputeThinV) != 0;
    eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
    eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
    eigen_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)
    {
        eigen_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);
    if(RowsAtCompileTime==Dynamic)
      m_matrixU.resize(m_rows, m_computeFullU ? m_rows
                              : m_computeThinU ? m_diagSize
                              : 0);
    if(ColsAtCompileTime==Dynamic)
      m_matrixV.resize(m_cols, m_computeFullV ? m_cols
                              : m_computeThinV ? m_diagSize
                              : 0);
    m_workMatrix.resize(m_diagSize, m_diagSize);
    
    if(m_cols>m_rows) m_qr_precond_morecols.allocate(*this);
    if(m_rows>m_cols) m_qr_precond_morerows.allocate(*this);
  }
  
  template<typename MatrixType, int QRPreconditioner>
  JacobiSVD<MatrixType, QRPreconditioner>&
  JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
  {
    using std::abs;
    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 = RealScalar(2) * NumTraits<Scalar>::epsilon();
  
    // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
    const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
  
    /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
  
    if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.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);
    }
    
    // Scaling factor to reducover/under-flows
    RealScalar scale = m_workMatrix.cwiseAbs().maxCoeff();
    if(scale==RealScalar(0)) scale = RealScalar(1);
    m_workMatrix /= scale;
  
    /*** 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. Similarly, small denormal numbers are considered zero.
          RealScalar threshold = numext::maxi(considerAsZero, precision * numext::maxi(abs(m_workMatrix.coeff(p,p)),
                                                                                       abs(m_workMatrix.coeff(q,q))));
          if(numext::maxi(abs(m_workMatrix.coeff(p,q)),abs(m_workMatrix.coeff(q,p))) > threshold)
          {
            finished = false;
  
            // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
            internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
            JacobiRotation<RealScalar> j_left, j_right;
            internal::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 = 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;
    }
    
    m_singularValues *= scale;
  
    /*** 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;
  }
  
  namespace internal {
  template<typename _MatrixType, int QRPreconditioner, typename Rhs>
  struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
    : 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
    {
      eigen_assert(rhs().rows() == dec().rows());
  
      // A = U S V^*
      // So A^{-1} = V S^{-1} U^*
  
      Matrix<Scalar, Dynamic, Rhs::ColsAtCompileTime, 0, _MatrixType::MaxRowsAtCompileTime, Rhs::MaxColsAtCompileTime> tmp;
      Index rank = dec().rank();
      
      tmp.noalias() = dec().matrixU().leftCols(rank).adjoint() * rhs();
      tmp = dec().singularValues().head(rank).asDiagonal().inverse() * tmp;
      dst = dec().matrixV().leftCols(rank) * tmp;
    }
  };
  } // end namespace internal
  
  #ifndef __CUDACC__
  /** \svd_module
    *
    * \return the singular value decomposition of \c *this computed by two-sided
    * Jacobi transformations.
    *
    * \sa class JacobiSVD
    */
  template<typename Derived>
  JacobiSVD<typename MatrixBase<Derived>::PlainObject>
  MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
  {
    return JacobiSVD<PlainObject>(*this, computationOptions);
  }
  #endif // __CUDACC__
  
  } // end namespace Eigen
  
  #endif // EIGEN_JACOBISVD_H
  