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 (at) 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);
      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);
    }
    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;
  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 u = d / t;
    rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + numext::abs2(u));
    rot1.s() = rot1.c() * u;
  }
  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 SVDBase<_MatrixType>
{
  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()
      : SVDBase<_MatrixType>::SVDBase()
    {}


    /** \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)
      : SVDBase<_MatrixType>::SVDBase()
    {
      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)
      : SVDBase<_MatrixType>::SVDBase()
    {
      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.
     */
    SVDBase<MatrixType>& compute(const MatrixType& matrix, unsigned int computationOptions);

    /** \brief Method performing the decomposition of given matrix using current options.
     *
     * \param matrix the matrix to decompose
     *
     * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
     */
    SVDBase<MatrixType>& compute(const MatrixType& matrix)
    {
      return compute(matrix, this->m_computationOptions);
    }

    /** \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(this->m_isInitialized && "JacobiSVD is not initialized.");
      eigen_assert(SVDBase<MatrixType>::computeU() && SVDBase<MatrixType>::computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
      return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
    }


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

  protected:
    WorkMatrixType m_workMatrix;

    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)
{
  if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;

  if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
  {
      eigen_assert(!(this->m_computeThinU || this->m_computeThinV) &&
              "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
              "Use the ColPivHouseholderQR preconditioner instead.");
  }

  m_workMatrix.resize(this->m_diagSize, this->m_diagSize);

  if(this->m_cols>this->m_rows) m_qr_precond_morecols.allocate(*this);
  if(this->m_rows>this->m_cols) m_qr_precond_morerows.allocate(*this);
}

template<typename MatrixType, int QRPreconditioner>
SVDBase<MatrixType>&
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,this->m_diagSize,this->m_diagSize);
    if(this->m_computeFullU) this->m_matrixU.setIdentity(this->m_rows,this->m_rows);
    if(this->m_computeThinU) this->m_matrixU.setIdentity(this->m_rows,this->m_diagSize);
    if(this->m_computeFullV) this->m_matrixV.setIdentity(this->m_cols,this->m_cols);
    if(this->m_computeThinV) this->m_matrixV.setIdentity(this->m_cols, this->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 < this->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.
        using std::max;
        RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
                                                                       abs(m_workMatrix.coeff(q,q))));
        if((max)(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(SVDBase<MatrixType>::computeU()) this->m_matrixU.applyOnTheRight(p,q,j_left.transpose());

          m_workMatrix.applyOnTheRight(p,q,j_right);
          if(SVDBase<MatrixType>::computeV()) this->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 < this->m_diagSize; ++i)
  {
    RealScalar a = abs(m_workMatrix.coeff(i,i));
    this->m_singularValues.coeffRef(i) = a;
    if(SVDBase<MatrixType>::computeU() && (a!=RealScalar(0))) this->m_matrixU.col(i) *= this->m_workMatrix.coeff(i,i)/a;
  }

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

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

  this->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^*

    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();
  }
};
} // end namespace internal

/** \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);
}

} // end namespace Eigen

#endif // EIGEN_JACOBISVD_H