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

Inheritance diagram for Eigen::JacobiSVD< MatrixType_, Options_ >:

Public Types
enum	{ Options , QRPreconditioner , RowsAtCompileTime , ColsAtCompileTime , DiagSizeAtCompileTime , MaxRowsAtCompileTime , MaxColsAtCompileTime , MaxDiagSizeAtCompileTime , MatrixOptions }

typedef Base::Index	Index

typedef MatrixType_	MatrixType

typedef Base::MatrixUType	MatrixUType

typedef Base::MatrixVType	MatrixVType

typedef Base::RealScalar	RealScalar

typedef Base::Scalar	Scalar

typedef Base::SingularValuesType	SingularValuesType

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

Public Types inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
enum

typedef Eigen::Index	Index

typedef internal::traits< JacobiSVD< MatrixType_, Options_ > >::MatrixType	MatrixType

typedef internal::make_proper_matrix_type< Scalar, RowsAtCompileTime, MatrixUColsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MatrixUMaxColsAtCompileTime >::type	MatrixUType

typedef internal::make_proper_matrix_type< Scalar, ColsAtCompileTime, MatrixVColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MatrixVMaxColsAtCompileTime >::type	MatrixVType

typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar

typedef internal::plain_diag_type< MatrixType, RealScalar >::type	SingularValuesType

typedef Eigen::internal::traits< SVDBase >::StorageIndex	StorageIndex

Public Types inherited from Eigen::SolverBase< Derived >
enum	{ RowsAtCompileTime , ColsAtCompileTime , SizeAtCompileTime , MaxRowsAtCompileTime , MaxColsAtCompileTime , MaxSizeAtCompileTime , IsVectorAtCompileTime , NumDimensions }

typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType

typedef EigenBase< Derived >	Base

typedef Scalar	CoeffReturnType

typedef Transpose< const Derived >	ConstTransposeReturnType

typedef internal::traits< Derived >::Scalar	Scalar

Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index	Index
	The interface type of indices. More...

typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

JacobiSVD &	compute (const MatrixType &matrix)
	Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified using the Options template parameter or the class constructor. More...

EIGEN_DEPRECATED JacobiSVD &	compute (const MatrixType &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix, as specified by the `computationOptions` parameter. More...

	JacobiSVD ()
	Default Constructor. More...

	JacobiSVD (const MatrixType &matrix)
	Constructor performing the decomposition of given matrix, using the custom options specified with the Options template paramter. More...

	JacobiSVD (const MatrixType &matrix, unsigned int computationOptions)
	Constructor performing the decomposition of given matrix using specified options for computing unitaries. More...

	JacobiSVD (Index rows, Index cols)
	Default Constructor with memory preallocation. More...

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

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
Index	cols () const

bool	computeU () const

bool	computeV () const

JacobiSVD< MatrixType_, Options_ > &	derived ()

const JacobiSVD< MatrixType_, Options_ > &	derived () const

ComputationInfo	info () const
	Reports whether previous computation was successful. More...

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

Index	nonzeroSingularValues () const

Index	rank () const

Index	rows () const

JacobiSVD< MatrixType_, Options_ > &	setThreshold (const RealScalar &threshold)

JacobiSVD< MatrixType_, Options_ > &	setThreshold (Default_t)

const SingularValuesType &	singularValues () const

const Solve< JacobiSVD< MatrixType_, Options_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const

RealScalar	threshold () const

Public Member Functions inherited from Eigen::SolverBase< Derived >
const AdjointReturnType	adjoint () const

Derived &	derived ()

const Derived &	derived () const

template<typename Rhs >
const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()

const ConstTransposeReturnType	transpose () const

	~SolverBase ()

Public Member Functions inherited from Eigen::EigenBase< Derived >
template<typename Dest >
void	addTo (Dest &dst) const

template<typename Dest >
void	applyThisOnTheLeft (Dest &dst) const

template<typename Dest >
void	applyThisOnTheRight (Dest &dst) const

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

Derived &	const_cast_derived () const

const Derived &	const_derived () const

Derived &	derived ()

const Derived &	derived () const

template<typename Dest >
void	evalTo (Dest &dst) const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

template<typename Dest >
void	subTo (Dest &dst) const

Protected Member Functions
	EIGEN_STATIC_ASSERT (!(ShouldComputeThinU &&int(QRPreconditioner)==int(FullPivHouseholderQRPreconditioner)) &&!(ShouldComputeThinU &&int(QRPreconditioner)==int(FullPivHouseholderQRPreconditioner)), "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. " "Use the ColPivHouseholderQR preconditioner instead.") template< typename MatrixType__

Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
void	_check_compute_assertions () const

void	_check_solve_assertion (const Rhs &b) const

bool	allocate (Index rows, Index cols, unsigned int computationOptions)

	SVDBase ()
	Default Constructor. More...

Protected Member Functions inherited from Eigen::SolverBase< Derived >
template<bool Transpose_, typename Rhs >
void	_check_solve_assertion (const Rhs &b) const

Protected Attributes
internal::qr_preconditioner_impl< MatrixType, Options, QRPreconditioner, internal::PreconditionIfMoreColsThanRows >	m_qr_precond_morecols

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

MatrixType	m_scaledMatrix

WorkMatrixType	m_workMatrix

int	Options__

Protected Attributes inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
Index	m_cols

unsigned int	m_computationOptions

bool	m_computeFullU

bool	m_computeFullV

bool	m_computeThinU

bool	m_computeThinV

Index	m_diagSize

ComputationInfo	m_info

bool	m_isAllocated

bool	m_isInitialized

MatrixUType	m_matrixU

MatrixVType	m_matrixV

Index	m_nonzeroSingularValues

RealScalar	m_prescribedThreshold

Index	m_rows

SingularValuesType	m_singularValues

bool	m_usePrescribedThreshold

Private Types
typedef SVDBase< JacobiSVD >	Base

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

JacobiSVD &	compute_impl (const MatrixType &matrix, unsigned int computationOptions)

Additional Inherited Members
Static Public Attributes inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
static constexpr bool	ShouldComputeFullU

static constexpr bool	ShouldComputeFullV

static constexpr bool	ShouldComputeThinU

static constexpr bool	ShouldComputeThinV

Detailed Description

template<typename MatrixType_, int Options_>
class Eigen::JacobiSVD< MatrixType_, Options_ >

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
Options	this optional parameter allows one to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. Additionally, it allows one to specify whether to compute thin or full unitaries U and V. 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:

MatrixXf m = MatrixXf::Random(3,2);
cout << "Here is the matrix m:" << endl << m << endl;
JacobiSVD<MatrixXf, ComputeThinU | ComputeThinV> svd(m);
cout << "Its singular values are:" << endl << svd.singularValues() << endl;
cout << "Its left singular vectors are the columns of the thin U matrix:" << endl << svd.matrixU() << endl;
cout << "Its right singular vectors are the columns of the thin V matrix:" << endl << svd.matrixV() << endl;
Vector3f rhs(1, 0, 0);
cout << "Now consider this rhs vector:" << endl << rhs << endl;
cout << "A least-squares solution of m*x = rhs is:" << endl << svd.solve(rhs) << endl;

Output:

Here is the matrix m:
  0.68  0.597
-0.211  0.823
 0.566 -0.605
Its singular values are:
 1.19
0.899
Its left singular vectors are the columns of the thin U matrix:
  0.388   0.866
  0.712 -0.0634
 -0.586   0.496
Its right singular vectors are the columns of the thin V matrix:
-0.183  0.983
 0.983  0.183
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
0.888
0.496

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 QR preconditioners that can be set with Options template parameter 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.

One may also use the Options template parameter to specify how the unitaries should be computed. The options are ComputeThinU, ComputeThinV, ComputeFullU, ComputeFullV. It is not possible to request both the thin and full versions of a unitary. By default, unitaries will not be computed.

You can set the QRPreconditioner and unitary options together: JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner | ComputeThinU | ComputeFullV>

See also: MatrixBase::jacobiSvd()

Definition at line 514 of file JacobiSVD.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , int Options_>

typedef SVDBase<JacobiSVD> Eigen::JacobiSVD< MatrixType_, Options_ >::Base

private

Definition at line 515 of file JacobiSVD.h.

◆ Index

template<typename MatrixType_ , int Options_>

typedef Base::Index Eigen::JacobiSVD< MatrixType_, Options_ >::Index

Definition at line 521 of file JacobiSVD.h.

◆ MatrixType

template<typename MatrixType_ , int Options_>

typedef MatrixType_ Eigen::JacobiSVD< MatrixType_, Options_ >::MatrixType

Definition at line 518 of file JacobiSVD.h.

◆ MatrixUType

template<typename MatrixType_ , int Options_>

typedef Base::MatrixUType Eigen::JacobiSVD< MatrixType_, Options_ >::MatrixUType

Definition at line 534 of file JacobiSVD.h.

◆ MatrixVType

template<typename MatrixType_ , int Options_>

typedef Base::MatrixVType Eigen::JacobiSVD< MatrixType_, Options_ >::MatrixVType

Definition at line 535 of file JacobiSVD.h.

◆ RealScalar

template<typename MatrixType_ , int Options_>

typedef Base::RealScalar Eigen::JacobiSVD< MatrixType_, Options_ >::RealScalar

Definition at line 520 of file JacobiSVD.h.

◆ Scalar

template<typename MatrixType_ , int Options_>

typedef Base::Scalar Eigen::JacobiSVD< MatrixType_, Options_ >::Scalar

Definition at line 519 of file JacobiSVD.h.

◆ SingularValuesType

template<typename MatrixType_ , int Options_>

typedef Base::SingularValuesType Eigen::JacobiSVD< MatrixType_, Options_ >::SingularValuesType

Definition at line 536 of file JacobiSVD.h.

◆ WorkMatrixType

template<typename MatrixType_ , int Options_>

typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< MatrixType_, Options_ >::WorkMatrixType

Definition at line 539 of file JacobiSVD.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , int Options_>

anonymous enum

Enumerator
Options
QRPreconditioner
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

Definition at line 522 of file JacobiSVD.h.

        {
     Options = Options_,
     QRPreconditioner = internal::get_qr_preconditioner(Options),
     RowsAtCompileTime = Base::RowsAtCompileTime,
     ColsAtCompileTime = Base::ColsAtCompileTime,
     DiagSizeAtCompileTime = Base::DiagSizeAtCompileTime,
     MaxRowsAtCompileTime = Base::MaxRowsAtCompileTime,
     MaxColsAtCompileTime = Base::MaxColsAtCompileTime,
     MaxDiagSizeAtCompileTime = Base::MaxDiagSizeAtCompileTime,
     MatrixOptions = Base::MatrixOptions
   };

Constructor & Destructor Documentation

◆ JacobiSVD() [1/5]

template<typename MatrixType_ , int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD ( )

inline

Default Constructor.

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

Definition at line 546 of file JacobiSVD.h.

546 {}

◆ JacobiSVD() [2/5]

template<typename MatrixType_ , int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD	(	Index	rows,
		Index	cols
	)

inline

Default Constructor with memory preallocation.

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

See also: JacobiSVD()

Definition at line 555 of file JacobiSVD.h.

555 { allocate(rows, cols, internal::get_computation_options(Options)); }

Eigen::JacobiSVD::cols

EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT

Definition: EigenBase.h:65

Eigen::JacobiSVD::allocate

void allocate(Index rows, Index cols, unsigned int computationOptions)

Definition: JacobiSVD.h:674

Eigen::JacobiSVD::rows

EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT

Definition: EigenBase.h:62

Eigen::internal::get_computation_options

constexpr int get_computation_options(int options)

Definition: SVDBase.h:33

◆ JacobiSVD() [3/5]

template<typename MatrixType_ , int Options_>

EIGEN_DEPRECATED Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions
	)

inline

Default Constructor with memory preallocation.

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

One cannot request unitaries using both the Options template parameter and the constructor. If possible, prefer using the Options template parameter.

Parameters

computationOptions specify whether to compute Thin/Full unitaries U/V

See also: JacobiSVD()

Deprecated:: Will be removed in the next major Eigen version. Options should be specified in the Options template parameter.

Definition at line 572 of file JacobiSVD.h.

                                                                      {
     internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, rows, cols);
     allocate(rows, cols, computationOptions);
   }

◆ JacobiSVD() [4/5]

template<typename MatrixType_ , int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD ( const MatrixType & matrix )

inlineexplicit

Constructor performing the decomposition of given matrix, using the custom options specified with the Options template paramter.

Parameters

matrix the matrix to decompose

Definition at line 582 of file JacobiSVD.h.

582 { compute_impl(matrix, internal::get_computation_options(Options)); }

Eigen::JacobiSVD::compute_impl

JacobiSVD & compute_impl(const MatrixType &matrix, unsigned int computationOptions)

Definition: JacobiSVD.h:689

◆ JacobiSVD() [5/5]

template<typename MatrixType_ , int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD	(	const MatrixType &	matrix,
		unsigned int	computationOptions
	)

inline

Constructor performing the decomposition of given matrix using specified options for computing unitaries.

One cannot request unitiaries using both the Options template parameter and the constructor. If possible, prefer using the Options template parameter.

Parameters

matrix	the matrix to decompose
computationOptions	specify whether to compute Thin/Full unitaries U/V

Deprecated:: Will be removed in the next major Eigen version. Options should be specified in the Options template parameter.

Definition at line 597 of file JacobiSVD.h.

                                                                        {
     internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, matrix.rows(), matrix.cols());
     compute_impl(matrix, computationOptions);
   }

Member Function Documentation

◆ allocate()

template<typename MatrixType , int Options>

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

private

Definition at line 674 of file JacobiSVD.h.

                                                                                                      {
   if (Base::allocate(rows, cols, computationOptions)) return;
  
   eigen_assert(!(ShouldComputeThinU && int(QRPreconditioner) == int(FullPivHouseholderQRPreconditioner)) &&
                !(ShouldComputeThinU && int(QRPreconditioner) == int(FullPivHouseholderQRPreconditioner)) &&
                "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
                "Use the ColPivHouseholderQR preconditioner instead.");
  
   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);
 }

◆ cols()

template<typename MatrixType_ , int Options_>

EIGEN_CONSTEXPR Index Eigen::EigenBase< Derived >::cols ( void )

inline

Returns: the number of columns.

See also: rows(), ColsAtCompileTime

Definition at line 65 of file EigenBase.h.

65 { return derived().cols(); }

Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::derived

JacobiSVD< MatrixType_, Options_ > & derived()

Definition: SVDBase.h:163

◆ compute() [1/2]

template<typename MatrixType_ , int Options_>

JacobiSVD& Eigen::JacobiSVD< MatrixType_, Options_ >::compute ( const MatrixType & matrix )

inline

Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified using the Options template parameter or the class constructor.

Parameters

matrix the matrix to decompose

Definition at line 607 of file JacobiSVD.h.

607 { return compute_impl(matrix, m_computationOptions); }

Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computationOptions

unsigned int m_computationOptions

Definition: SVDBase.h:350

◆ compute() [2/2]

template<typename MatrixType_ , int Options_>

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

inline

Method performing the decomposition of given matrix, as specified by the computationOptions parameter.

Parameters

matrix	the matrix to decompose
computationOptions	specify whether to compute Thin/Full unitaries U/V

Deprecated:: Will be removed in the next major Eigen version. Options should be specified in the Options template parameter.

Definition at line 619 of file JacobiSVD.h.

                                                                                 {
     internal::check_svd_options_assertions<MatrixType, Options>(m_computationOptions, matrix.rows(), matrix.cols());
     return compute_impl(matrix, computationOptions);
   }

◆ compute_impl()

template<typename MatrixType , int Options>

JacobiSVD< MatrixType, Options > & Eigen::JacobiSVD< MatrixType, Options >::compute_impl	(	const MatrixType &	matrix,
		unsigned int	computationOptions
	)

private

Definition at line 689 of file JacobiSVD.h.

                                                                                                               {
   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().template maxCoeff<PropagateNaN>();
   if (!(numext::isfinite)(scale)) {
     m_isInitialized = true;
     m_info = InvalidInput;
     m_nonzeroSingularValues = 0;
     return *this;
   }
   if(numext::is_exactly_zero(scale)) scale = RealScalar(1);
   
  
   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);
   }
  
  
   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, Options>::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))));
           }
         }
       }
     }
   }
  
  
   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;
  
  
   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(numext::is_exactly_zero(maxRemainingSingularValue))
     {
       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;
 }

◆ EIGEN_STATIC_ASSERT()

template<typename MatrixType_ , int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::EIGEN_STATIC_ASSERT	(	!	ShouldComputeThinU &&int(QRPreconditioner)==int(FullPivHouseholderQRPreconditioner)) &&!(ShouldComputeThinU &&int(QRPreconditioner)==int(FullPivHouseholderQRPreconditioner),
		"JacobiSVD< MatrixType_, Options_ >: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. " "Use the ColPivHouseholderQR preconditioner instead."
	)

protected

◆ rows()

template<typename MatrixType_ , int Options_>

EIGEN_CONSTEXPR Index Eigen::EigenBase< Derived >::rows ( void )

inline

Returns: the number of rows.

See also: cols(), RowsAtCompileTime

Definition at line 62 of file EigenBase.h.

62 { return derived().rows(); }

Member Data Documentation

◆ m_qr_precond_morecols

template<typename MatrixType_ , int Options_>

internal::qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> Eigen::JacobiSVD< MatrixType_, Options_ >::m_qr_precond_morecols

protected

Definition at line 666 of file JacobiSVD.h.

◆ m_qr_precond_morerows

template<typename MatrixType_ , int Options_>

internal::qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> Eigen::JacobiSVD< MatrixType_, Options_ >::m_qr_precond_morerows

protected

Definition at line 668 of file JacobiSVD.h.

◆ m_scaledMatrix

template<typename MatrixType_ , int Options_>

MatrixType Eigen::JacobiSVD< MatrixType_, Options_ >::m_scaledMatrix

protected

Definition at line 670 of file JacobiSVD.h.

◆ m_workMatrix

template<typename MatrixType_ , int Options_>

WorkMatrixType Eigen::JacobiSVD< MatrixType_, Options_ >::m_workMatrix

protected

Definition at line 669 of file JacobiSVD.h.

◆ Options__

template<typename MatrixType_ , int Options_>

int Eigen::JacobiSVD< MatrixType_, Options_ >::Options__

protected

Definition at line 660 of file JacobiSVD.h.

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Private Types

Private Member Functions

Additional Inherited Members

Detailed Description

template<typename MatrixType_, int Options_> class Eigen::JacobiSVD< MatrixType_, Options_ >

Member Typedef Documentation

◆ Base

◆ Index

◆ MatrixType

◆ MatrixUType

◆ MatrixVType

◆ RealScalar

◆ Scalar

◆ SingularValuesType

◆ WorkMatrixType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ JacobiSVD() [1/5]

◆ JacobiSVD() [2/5]

◆ JacobiSVD() [3/5]

◆ JacobiSVD() [4/5]

◆ JacobiSVD() [5/5]

Member Function Documentation

◆ allocate()

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ compute_impl()

◆ EIGEN_STATIC_ASSERT()

◆ rows()

Member Data Documentation

◆ m_qr_precond_morecols

◆ m_qr_precond_morerows

◆ m_scaledMatrix

◆ m_workMatrix

◆ Options__

template<typename MatrixType_, int Options_>
class Eigen::JacobiSVD< MatrixType_, Options_ >