Two-sided Jacobi SVD decomposition of a rectangular matrix. More...

#include <src/eigen/Eigen/src/SVD/JacobiSVD.h>

Inheritance diagram for Eigen::JacobiSVD< _MatrixType, QRPreconditioner >:

Collaboration diagram for Eigen::JacobiSVD< _MatrixType, QRPreconditioner >:

Public Types
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 _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef NumTraits< typenameMatrixType::Scalar >::Real	RealScalar

typedef Base::MatrixUType	MatrixUType

typedef Base::MatrixVType	MatrixVType

typedef Base::SingularValuesType	SingularValuesType

typedef internal::plain_row_type< MatrixType >::type	RowType

typedef internal::plain_col_type< MatrixType >::type	ColType

typedef Matrix< Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime >	WorkMatrixType

enum

typedef MatrixType::StorageIndex	StorageIndex

typedef Eigen::Index	Index

Public Member Functions
	JacobiSVD ()
	Default Constructor.

	JacobiSVD (Index rows, Index cols, unsigned int computationOptions=0)
	Default Constructor with memory preallocation.

	JacobiSVD (const MatrixType &matrix, unsigned int computationOptions=0)
	Constructor performing the decomposition of given matrix.

JacobiSVD &	compute (const MatrixType &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix using custom options.

JacobiSVD &	compute (const MatrixType &matrix)
	Method performing the decomposition of given matrix using current options.

bool	computeU () const

bool	computeV () const

Index	rows () const

Index	cols () const

Index	rank () const

JacobiSVD< _MatrixType, QRPreconditioner > &	derived ()

const JacobiSVD< _MatrixType, QRPreconditioner > &	derived () const

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

const SingularValuesType &	singularValues () const

Index	nonzeroSingularValues () const

JacobiSVD< _MatrixType, QRPreconditioner > &	setThreshold (const RealScalar &threshold)

JacobiSVD< _MatrixType, QRPreconditioner > &	setThreshold (Default_t)

RealScalar	threshold () const

const Solve< JacobiSVD< _MatrixType, QRPreconditioner >, Rhs >	solve (const MatrixBase< Rhs > &b) const

EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const

void	_solve_impl (const RhsType &rhs, DstType &dst) const

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
WorkMatrixType	m_workMatrix

internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows >	m_qr_precond_morecols

internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols >	m_qr_precond_morerows

MatrixType	m_scaledMatrix

MatrixUType	m_matrixU

MatrixVType	m_matrixV

SingularValuesType	m_singularValues

bool	m_isInitialized

bool	m_isAllocated

bool	m_usePrescribedThreshold

bool	m_computeFullU

bool	m_computeThinU

bool	m_computeFullV

bool	m_computeThinV

unsigned int	m_computationOptions

Index	m_nonzeroSingularValues

Index	m_rows

Index	m_cols

Index	m_diagSize

RealScalar	m_prescribedThreshold

Private Types
typedef SVDBase< JacobiSVD >	Base

Private Member Functions
void	allocate (Index rows, Index cols, unsigned int computationOptions)

Friends
template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex>
struct	internal::svd_precondition_2x2_block_to_be_real

template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything>
struct	internal::qr_preconditioner_impl

Detailed Description

template<typename _MatrixType, int QRPreconditioner>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >

Two-sided Jacobi SVD decomposition of a rectangular matrix.

Template Parameters

_MatrixType	the type of the matrix of which we are computing the SVD decomposition
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 as a product

$A = U S V^*$

where U is a n-by-n unitary, V is a p-by-p unitary, and 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 singular values of A and the columns of U and V are known as the left and right singular vectors of A respectively.

Singular values are always sorted in decreasing order.

This JacobiSVD decomposition computes only the singular values by default. If you want U or V, you need to ask for them explicitly.

You can ask for only thin U or V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting m be the smaller value among n and p, there are only m singular vectors; the remaining columns of U and V do not correspond to actual singular vectors. Asking for thin U or V means asking for only their m first columns to be formed. So U is then a n-by-m matrix, and V is then a p-by-m matrix. Notice that thin U and V are all you need for (least squares) solving.

Here's an example demonstrating basic usage:

Output:

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 $ O(n^2p) $ where n is the smaller dimension and 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:

ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. Contrary to other QRs, it doesn't allow computing thin unitaries.
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.
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.

See also: MatrixBase::jacobiSvd()

Member Typedef Documentation

◆ Base

template<typename _MatrixType , int QRPreconditioner>

typedef SVDBase<JacobiSVD> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Base

private

◆ ColType

template<typename _MatrixType , int QRPreconditioner>

typedef internal::plain_col_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::ColType

◆ Index

typedef Eigen::Index Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::Index

inherited

◆ MatrixType

template<typename _MatrixType , int QRPreconditioner>

typedef _MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType

◆ MatrixUType

template<typename _MatrixType , int QRPreconditioner>

typedef Base::MatrixUType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixUType

◆ MatrixVType

template<typename _MatrixType , int QRPreconditioner>

typedef Base::MatrixVType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixVType

◆ RealScalar

template<typename _MatrixType , int QRPreconditioner>

typedef NumTraits<typenameMatrixType::Scalar>::Real Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RealScalar

◆ RowType

template<typename _MatrixType , int QRPreconditioner>

typedef internal::plain_row_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RowType

◆ Scalar

template<typename _MatrixType , int QRPreconditioner>

typedef MatrixType::Scalar Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Scalar

◆ SingularValuesType

template<typename _MatrixType , int QRPreconditioner>

typedef Base::SingularValuesType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::SingularValuesType

◆ StorageIndex

typedef MatrixType::StorageIndex Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::StorageIndex

inherited

◆ WorkMatrixType

template<typename _MatrixType , int QRPreconditioner>

typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::WorkMatrixType

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

inherited

       {
    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
  };

◆ anonymous enum

template<typename _MatrixType , int QRPreconditioner>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

         {
      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
    };

Constructor & Destructor Documentation

◆ JacobiSVD() [1/3]

template<typename _MatrixType , int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD ( )

inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via JacobiSVD::compute(const MatrixType&).

522 {}

◆ JacobiSVD() [2/3]

template<typename _MatrixType , int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions = `0`
	)

inline

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: JacobiSVD()

    {
      allocate(rows, cols, computationOptions);
    }

◆ JacobiSVD() [3/3]

template<typename _MatrixType , int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	const MatrixType &	matrix,
		unsigned int	computationOptions = `0`
	)

inlineexplicit

Constructor performing the decomposition of given matrix.

Parameters

matrix	the matrix to decompose
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.

    {
      compute(matrix, computationOptions);
    }

Member Function Documentation

◆ _solve_impl() [1/2]

EIGEN_DEVICE_FUNC void Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

inherited

◆ _solve_impl() [2/2]

void Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

inherited

{
  eigen_assert(rhs.rows() == rows());
 
  // A = U S V^*
  // So A^{-1} = V S^{-1} U^*
 
  Matrix<Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime, RhsType::MaxColsAtCompileTime> tmp;
  Index l_rank = rank();
  tmp.noalias() =  m_matrixU.leftCols(l_rank).adjoint() * rhs;
  tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
  dst = m_matrixV.leftCols(l_rank) * tmp;
}

◆ allocate()

template<typename MatrixType , int QRPreconditioner>

void Eigen::JacobiSVD< MatrixType, QRPreconditioner >::allocate	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions
	)

private

{
  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);
  if(m_rows!=m_cols)  m_scaledMatrix.resize(rows,cols);
}

References Eigen::ComputeFullU, Eigen::ComputeFullV, Eigen::ComputeThinU, Eigen::ComputeThinV, Eigen::Dynamic, eigen_assert, EIGEN_IMPLIES, and Eigen::FullPivHouseholderQRPreconditioner.

◆ check_template_parameters()

static void Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::check_template_parameters ( )

inlinestaticprotectedinherited

  {
    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
  }

◆ cols()

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::cols ( ) const

inline

195{ return m_cols; }

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate().

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◆ compute() [1/2]

template<typename _MatrixType , int QRPreconditioner>

JacobiSVD & Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::compute ( const MatrixType & matrix )

inline

Method performing the decomposition of given matrix using current options.

Parameters

matrix the matrix to decompose

This method uses the current computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).

    {
      return compute(matrix, m_computationOptions);
    }

◆ compute() [2/2]

template<typename MatrixType , int QRPreconditioner>

JacobiSVD< MatrixType, QRPreconditioner > & Eigen::JacobiSVD< MatrixType, QRPreconditioner >::compute	(	const MatrixType &	matrix,
		unsigned int	computationOptions
	)

Method performing the decomposition of given matrix using custom options.

Parameters

matrix	the matrix to decompose
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.

{
  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 denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
  const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
 
  // Scaling factor to reduce over/under-flows
  RealScalar scale = matrix.cwiseAbs().maxCoeff();
  if(scale==RealScalar(0)) scale = RealScalar(1);
  
  /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
 
  if(m_rows!=m_cols)
  {
    m_scaledMatrix = matrix / scale;
    m_qr_precond_morecols.run(*this, m_scaledMatrix);
    m_qr_precond_morerows.run(*this, m_scaledMatrix);
  }
  else
  {
    m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale;
    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. ***/
  RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff();
 
  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<RealScalar>(considerAsZero, precision * maxDiagEntry);
        if(abs(m_workMatrix.coeff(p,q))>threshold || 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
          // the complex to real operation returns true if the updated 2x2 block is not already diagonal
          if(internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q, maxDiagEntry))
          {
            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);
 
            // keep track of the largest diagonal coefficient
            maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p,p)), abs(m_workMatrix.coeff(q,q))));
          }
        }
      }
    }
  }
 
  /*** 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)
  {
    // For a complex matrix, some diagonal coefficients might note have been
    // treated by svd_precondition_2x2_block_to_be_real, and the imaginary part
    // of some diagonal entry might not be null.
    if(NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i,i)))>considerAsZero)
    {
      RealScalar a = abs(m_workMatrix.coeff(i,i));
      m_singularValues.coeffRef(i) = abs(a);
      if(computeU()) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
    }
    else
    {
      // m_workMatrix.coeff(i,i) is already real, no difficulty:
      RealScalar a = numext::real(m_workMatrix.coeff(i,i));
      m_singularValues.coeffRef(i) = abs(a);
      if(computeU() && (a<RealScalar(0))) m_matrixU.col(i) = -m_matrixU.col(i);
    }
  }
  
  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;
}

References Eigen::internal::real_2x2_jacobi_svd(), scale(), and Eigen::JacobiRotation< Scalar >::transpose().

Referenced by igl::min_quad_dense_precompute(), and igl::polar_svd().

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◆ computeU()

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::computeU ( ) const

inline

Returns: true if U (full or thin) is asked for in this SVD decomposition

190{ return m_computeFullU || m_computeThinU; }

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), and Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run().

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◆ computeV()

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::computeV ( ) const

inline

Returns: true if V (full or thin) is asked for in this SVD decomposition

192{ return m_computeFullV || m_computeThinV; }

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), and Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run().

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◆ derived() [1/2]

JacobiSVD< _MatrixType, QRPreconditioner > & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::derived ( )

inlineinherited

71{ return *static_cast<Derived*>(this); }

◆ derived() [2/2]

const JacobiSVD< _MatrixType, QRPreconditioner > & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::derived ( ) const

inlineinherited

72{ return *static_cast<const Derived*>(this); }

◆ matrixU()

const MatrixUType & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::matrixU ( ) const

inlineinherited

Returns: the U matrix.

For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the U matrix is n-by-n if you asked for ComputeFullU , and is n-by-m if you asked for ComputeThinU .

The m first columns of U are the left singular vectors of the matrix being decomposed.

This method asserts that you asked for U to be computed.

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    eigen_assert(computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
    return m_matrixU;
  }

◆ matrixV()

const MatrixVType & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::matrixV ( ) const

inlineinherited

Returns: the V matrix.

For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the V matrix is p-by-p if you asked for ComputeFullV , and is p-by-m if you asked for ComputeThinV .

The m first columns of V are the right singular vectors of the matrix being decomposed.

This method asserts that you asked for V to be computed.

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    eigen_assert(computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
    return m_matrixV;
  }

◆ nonzeroSingularValues()

Index Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::nonzeroSingularValues ( ) const

inlineinherited

Returns: the number of singular values that are not exactly 0

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    return m_nonzeroSingularValues;
  }

◆ rank()

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::rank ( ) const

inline

Returns: the rank of the matrix of which *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&).

  {
    using std::abs;
    eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
    if(m_singularValues.size()==0) return 0;
    RealScalar premultiplied_threshold = numext::maxi<RealScalar>(m_singularValues.coeff(0) * threshold(), (std::numeric_limits<RealScalar>::min)());
    Index i = m_nonzeroSingularValues-1;
    while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i;
    return i+1;
  }

Referenced by igl::null().

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◆ rows()

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::rows ( ) const

inline

194{ return m_rows; }

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate().

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◆ setThreshold() [1/2]

JacobiSVD< _MatrixType, QRPreconditioner > & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::setThreshold ( const RealScalar & threshold )

inlineinherited

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 NumTraits<Scalar>::epsilon()

Parameters

threshold The new value to use as the threshold.

A singular value will be considered nonzero if its value is strictly greater than $\vert singular value \vert \leqslant threshold \times \vert max singular value \vert$ .

If you want to come back to the default behavior, call setThreshold(Default_t)

  {
    m_usePrescribedThreshold = true;
    m_prescribedThreshold = threshold;
    return derived();
  }

◆ setThreshold() [2/2]

JacobiSVD< _MatrixType, QRPreconditioner > & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::setThreshold ( Default_t )

inlineinherited

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.

svd.setThreshold(Eigen::Default);

Eigen::Default

@ Default

Definition Constants.h:352

See the documentation of setThreshold(const RealScalar&).

  {
    m_usePrescribedThreshold = false;
    return derived();
  }

◆ singularValues()

const SingularValuesType & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::singularValues ( ) const

inlineinherited

Returns: the vector of singular values.

For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the returned vector has size m. Singular values are always sorted in decreasing order.

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    return m_singularValues;
  }

◆ solve()

const Solve< JacobiSVD< _MatrixType, QRPreconditioner > , Rhs > Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::solve ( const MatrixBase< Rhs > & b ) const

inlineinherited

Returns: a (least squares) solution of using the current SVD decomposition of A.

Parameters

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.; 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 $\Vert A x - b \Vert$ .

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    eigen_assert(computeU() && computeV() && "SVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
    return Solve<Derived, Rhs>(derived(), b.derived());
  }

◆ threshold()

RealScalar Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::threshold ( ) const

inlineinherited

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

  {
    eigen_assert(m_isInitialized || m_usePrescribedThreshold);
    // this temporary is needed to workaround a MSVC issue
    Index diagSize = (std::max<Index>)(1,m_diagSize);
    return m_usePrescribedThreshold ? m_prescribedThreshold
                                    : diagSize*NumTraits<Scalar>::epsilon();
  }

Friends And Related Symbol Documentation

◆ internal::qr_preconditioner_impl

template<typename _MatrixType , int QRPreconditioner>

template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything>

friend struct internal::qr_preconditioner_impl

friend

◆ internal::svd_precondition_2x2_block_to_be_real

template<typename _MatrixType , int QRPreconditioner>

template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex>

friend struct internal::svd_precondition_2x2_block_to_be_real

friend

Member Data Documentation

◆ m_cols

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::m_cols

protected

◆ m_computationOptions

template<typename _MatrixType , int QRPreconditioner>

unsigned int Eigen::SVDBase< Derived >::m_computationOptions

protected

◆ m_computeFullU

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_computeFullU

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run().

◆ m_computeFullV

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_computeFullV

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run().

◆ m_computeThinU

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_computeThinU

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run().

◆ m_computeThinV

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_computeThinV

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::allocate(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run().

◆ m_diagSize

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::m_diagSize

protected

◆ m_isAllocated

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_isAllocated

protected

◆ m_isInitialized

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_isInitialized

protected

◆ m_matrixU

template<typename _MatrixType , int QRPreconditioner>

MatrixUType Eigen::SVDBase< Derived >::m_matrixU

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), and Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run().

◆ m_matrixV

template<typename _MatrixType , int QRPreconditioner>

MatrixVType Eigen::SVDBase< Derived >::m_matrixV

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), and Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run().

◆ m_nonzeroSingularValues

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::m_nonzeroSingularValues

protected

◆ m_prescribedThreshold

template<typename _MatrixType , int QRPreconditioner>

RealScalar Eigen::SVDBase< Derived >::m_prescribedThreshold

protected

◆ m_qr_precond_morecols

template<typename _MatrixType , int QRPreconditioner>

internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morecols

protected

◆ m_qr_precond_morerows

template<typename _MatrixType , int QRPreconditioner>

internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morerows

protected

◆ m_rows

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::m_rows

protected

◆ m_scaledMatrix

template<typename _MatrixType , int QRPreconditioner>

MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_scaledMatrix

protected

◆ m_singularValues

template<typename _MatrixType , int QRPreconditioner>

SingularValuesType Eigen::SVDBase< Derived >::m_singularValues

protected

◆ m_usePrescribedThreshold

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::m_usePrescribedThreshold

protected

◆ m_workMatrix

template<typename _MatrixType , int QRPreconditioner>

WorkMatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix

protected

Referenced by Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run(), Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run(), and Eigen::internal::qr_preconditioner_impl< MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true >::run().

The documentation for this class was generated from the following files:

src/eigen/Eigen/src/Core/util/ForwardDeclarations.h
src/eigen/Eigen/src/SVD/JacobiSVD.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Private Types

Private Member Functions

Friends

Detailed Description

Member Typedef Documentation

◆ Base

◆ ColType

◆ Index

◆ MatrixType

◆ MatrixUType

◆ MatrixVType

◆ RealScalar

◆ RowType

◆ Scalar

◆ SingularValuesType

◆ StorageIndex

◆ WorkMatrixType

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Constructor & Destructor Documentation

◆ JacobiSVD() [1/3]

◆ JacobiSVD() [2/3]

◆ JacobiSVD() [3/3]

Member Function Documentation

◆ _solve_impl() [1/2]

◆ _solve_impl() [2/2]

◆ allocate()

◆ check_template_parameters()

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeU()

◆ computeV()

◆ derived() [1/2]

◆ derived() [2/2]

◆ matrixU()

◆ matrixV()

◆ nonzeroSingularValues()

◆ rank()

◆ rows()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ singularValues()

◆ solve()

◆ threshold()

Friends And Related Symbol Documentation

◆ internal::qr_preconditioner_impl

◆ internal::svd_precondition_2x2_block_to_be_real

Member Data Documentation

◆ m_cols

◆ m_computationOptions

◆ m_computeFullU

◆ m_computeFullV

◆ m_computeThinU

◆ m_computeThinV

◆ m_diagSize

◆ m_isAllocated

◆ m_isInitialized

◆ m_matrixU

◆ m_matrixV

◆ m_nonzeroSingularValues

◆ m_prescribedThreshold

◆ m_qr_precond_morecols

◆ m_qr_precond_morerows

◆ m_rows

◆ m_scaledMatrix

◆ m_singularValues

◆ m_usePrescribedThreshold

◆ m_workMatrix