Eigen::JacobiSVD< MatrixType_, Options_ > Class Template Reference

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

#include <JacobiSVD.h>

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

Public Types
enum	: int {   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 }
typedef MatrixType_	MatrixType
typedef Base::Scalar	Scalar
typedef Base::RealScalar	RealScalar
typedef Base::MatrixUType	MatrixUType
typedef Base::MatrixVType	MatrixVType
typedef Base::SingularValuesType	SingularValuesType
typedef Matrix< Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime >	WorkMatrixType
Public Types inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
enum
typedef internal::traits< JacobiSVD< MatrixType_, Options_ > >::MatrixType	MatrixType
typedef MatrixType::Scalar	Scalar
typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar
typedef Eigen::internal::traits< SVDBase >::StorageIndex	StorageIndex
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 internal::plain_diag_type< MatrixType, RealScalar >::type	SingularValuesType
Public Types inherited from Eigen::SolverBase< Derived >
enum	{   RowsAtCompileTime = internal::traits<Derived>::RowsAtCompileTime , ColsAtCompileTime = internal::traits<Derived>::ColsAtCompileTime , SizeAtCompileTime = (internal::size_of_xpr_at_compile_time<Derived>::ret) , MaxRowsAtCompileTime = internal::traits<Derived>::MaxRowsAtCompileTime ,   MaxColsAtCompileTime = internal::traits<Derived>::MaxColsAtCompileTime , MaxSizeAtCompileTime , IsVectorAtCompileTime , NumDimensions }
typedef EigenBase< Derived >	Base
typedef internal::traits< Derived >::Scalar	Scalar
typedef Scalar	CoeffReturnType
typedef Transpose< const Derived >	ConstTransposeReturnType
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType
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
	JacobiSVD ()
	Default Constructor. 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...
	JacobiSVD (const MatrixType &matrix)
	Constructor performing the decomposition of given matrix, using the custom options specified with the Options template parameter. More...
	JacobiSVD (const MatrixType &matrix, unsigned int computationOptions)
	Constructor performing the decomposition of given matrix using specified options for computing unitaries. More...
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...
void	allocate (Index rows_, Index cols_, unsigned int computationOptions)
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT
Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
JacobiSVD< MatrixType_, Options_ > &	derived ()
const JacobiSVD< MatrixType_, Options_ > &	derived () const
const MatrixUType &	matrixU () const
const MatrixVType &	matrixV () const
const SingularValuesType &	singularValues () const
Index	nonzeroSingularValues () const
Index	rank () const
JacobiSVD< MatrixType_, Options_ > &	setThreshold (const RealScalar &threshold)
JacobiSVD< MatrixType_, Options_ > &	setThreshold (Default_t)
RealScalar	threshold () const
bool	computeU () const
bool	computeV () const
Index	rows () const
Index	cols () const
Index	diagSize () const
EIGEN_DEVICE_FUNC ComputationInfo	info () const
	Reports whether previous computation was successful. More...
void	_solve_impl (const RhsType &rhs, DstType &dst) const
void	_solve_impl_transposed (const RhsType &rhs, DstType &dst) const
Public Member Functions inherited from Eigen::SolverBase< Derived >
	SolverBase ()
	~SolverBase ()
template<typename Rhs >
const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const
const ConstTransposeReturnType	transpose () const
const AdjointReturnType	adjoint () const
constexpr EIGEN_DEVICE_FUNC Derived &	derived ()
constexpr EIGEN_DEVICE_FUNC const Derived &	derived () const
Public Member Functions inherited from Eigen::EigenBase< Derived >
constexpr EIGEN_DEVICE_FUNC Derived &	derived ()
constexpr EIGEN_DEVICE_FUNC const Derived &	derived () const
EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const
EIGEN_DEVICE_FUNC const Derived &	const_derived () const
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT
template<typename Dest >
EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const
template<typename Dest >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const
template<typename Dest >
EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const
template<typename Dest >
EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const
template<typename Dest >
EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const
template<typename Device >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE DeviceWrapper< Derived, Device >	device (Device &device)
template<typename Device >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE DeviceWrapper< const Derived, Device >	device (Device &device) 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
int	Options__
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
WorkMatrixType	m_workMatrix
Protected Attributes inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
MatrixUType	m_matrixU
MatrixVType	m_matrixV
SingularValuesType	m_singularValues
ComputationInfo	m_info
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
internal::variable_if_dynamic< Index, RowsAtCompileTime >	m_rows
internal::variable_if_dynamic< Index, ColsAtCompileTime >	m_cols
internal::variable_if_dynamic< Index, DiagSizeAtCompileTime >	m_diagSize
RealScalar	m_prescribedThreshold

Private Types
typedef SVDBase< JacobiSVD >	Base

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

Friends
template<typename MatrixType__ , int Options__, int QRPreconditioner_, int Case_, bool DoAnything_>
struct	internal::qr_preconditioner_impl

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

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()

Member Typedef Documentation

Base

template<typename MatrixType_ , int Options_>

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

private

MatrixType

template<typename MatrixType_ , int Options_>

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

MatrixUType

template<typename MatrixType_ , int Options_>

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

MatrixVType

template<typename MatrixType_ , int Options_>

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

RealScalar

template<typename MatrixType_ , int Options_>

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

Scalar

template<typename MatrixType_ , int Options_>

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

SingularValuesType

template<typename MatrixType_ , int Options_>

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

WorkMatrixType

template<typename MatrixType_ , int Options_>

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

Member Enumeration Documentation

anonymous enum

template<typename MatrixType_ , int Options_>

anonymous enum : int

Enumerator
Options
QRPreconditioner
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

       : int {
    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&).

{}

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()

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

References Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), Eigen::JacobiSVD< MatrixType_, Options_ >::cols(), Eigen::internal::get_computation_options(), Eigen::JacobiSVD< MatrixType_, Options_ >::Options, and Eigen::JacobiSVD< MatrixType_, Options_ >::rows().

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.

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

References Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), Eigen::JacobiSVD< MatrixType_, Options_ >::cols(), and Eigen::JacobiSVD< MatrixType_, Options_ >::rows().

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 parameter.

Parameters

matrix

the matrix to decompose

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

References Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), Eigen::internal::get_computation_options(), matrix(), and Eigen::JacobiSVD< MatrixType_, Options_ >::Options.

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.

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

References Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), and matrix().

Member Function Documentation

allocate()

template<typename MatrixType_ , int Options_>

void Eigen::JacobiSVD< MatrixType_, Options_ >::allocate ( Index  rows_, Index  cols_, unsigned int  computationOptions  )

inline

                                                                           {
    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(diagSize(), diagSize());
    if (cols() > rows()) m_qr_precond_morecols.allocate(*this);
    if (rows() > cols()) m_qr_precond_morerows.allocate(*this);
  }

References Eigen::SVDBase< Derived >::allocate(), Eigen::JacobiSVD< MatrixType_, Options_ >::cols(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::diagSize(), eigen_assert, Eigen::FullPivHouseholderQRPreconditioner, Eigen::JacobiSVD< MatrixType_, Options_ >::m_qr_precond_morecols, Eigen::JacobiSVD< MatrixType_, Options_ >::m_qr_precond_morerows, Eigen::JacobiSVD< MatrixType_, Options_ >::m_workMatrix, Eigen::JacobiSVD< MatrixType_, Options_ >::QRPreconditioner, Eigen::PlainObjectBase< Derived >::resize(), Eigen::JacobiSVD< MatrixType_, Options_ >::rows(), and Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::ShouldComputeThinU.

Referenced by Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), and Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD().

cols()

template<typename MatrixType_ , int Options_>

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::EigenBase< Derived >::cols

inline

Returns

the number of columns.

See also

rows(), ColsAtCompileTime

{ return derived().cols(); }

Referenced by gdb.printers._MatrixEntryIterator::__next__(), Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

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

{ return compute_impl(matrix, m_computationOptions); }

References Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computationOptions, and matrix().

Referenced by bench(), and ctms_decompositions().

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.

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

References Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computationOptions, and matrix().

compute_impl()

template<typename MatrixType , int Options>

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

private

                                                                                                              {
  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);
 
  /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
 
  if (rows() != cols()) {
    m_qr_precond_morecols.run(*this, matrix / scale);
    m_qr_precond_morerows.run(*this, matrix / scale);
  } else {
    m_workMatrix =
        matrix.template topLeftCorner<DiagSizeAtCompileTime, DiagSizeAtCompileTime>(diagSize(), diagSize()) / scale;
    if (m_computeFullU) m_matrixU.setIdentity(rows(), rows());
    if (m_computeThinU) m_matrixU.setIdentity(rows(), diagSize());
    if (m_computeFullV) m_matrixV.setIdentity(cols(), cols());
    if (m_computeThinV) m_matrixV.setIdentity(cols(), 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 < 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))));
          }
        }
      }
    }
  }
 
  /*** 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 < 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 = diagSize();
  for (Index i = 0; i < diagSize(); i++) {
    Index pos;
    RealScalar maxRemainingSingularValue = m_singularValues.tail(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;
}

References a, abs(), Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), Eigen::PlainObjectBase< Derived >::coeff(), Eigen::JacobiSVD< MatrixType_, Options_ >::cols(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::computeU(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::computeV(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::diagSize(), oomph::SarahBL::epsilon, MergeRestartFiles::finished, i, imag(), Eigen::InvalidInput, Eigen::numext::is_exactly_zero(), Eigen::numext::isfinite(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computeFullU, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computeFullV, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computeThinU, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_computeThinV, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_info, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_isInitialized, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_matrixU, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_matrixV, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_nonzeroSingularValues, Eigen::JacobiSVD< MatrixType_, Options_ >::m_qr_precond_morecols, Eigen::JacobiSVD< MatrixType_, Options_ >::m_qr_precond_morerows, Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::m_singularValues, Eigen::JacobiSVD< MatrixType_, Options_ >::m_workMatrix, matrix(), min, p, Eigen::numext::q, Eigen::internal::real_2x2_jacobi_svd(), Eigen::JacobiSVD< MatrixType_, Options_ >::rows(), Eigen::PlainObjectBase< Derived >::swap(), swap(), Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >::threshold(), and Eigen::JacobiRotation< Scalar >::transpose().

Referenced by Eigen::JacobiSVD< MatrixType_, Options_ >::compute(), and Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD().

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_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::EigenBase< Derived >::rows

inline

Returns

the number of rows.

See also

cols(), RowsAtCompileTime

{ return derived().rows(); }

Referenced by gdb.printers._MatrixEntryIterator::__next__(), Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl(), Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

Friends And Related Function Documentation

internal::qr_preconditioner_impl

template<typename MatrixType_ , int Options_>

template<typename MatrixType__ , int Options__, int QRPreconditioner_, int Case_, bool DoAnything_>

friend struct internal::qr_preconditioner_impl

friend

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

Referenced by Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), and Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl().

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

Referenced by Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), and Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl().

m_workMatrix

template<typename MatrixType_ , int Options_>

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

protected

Referenced by Eigen::JacobiSVD< MatrixType_, Options_ >::allocate(), and Eigen::JacobiSVD< MatrixType_, Options_ >::compute_impl().

Options__

template<typename MatrixType_ , int Options_>

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

protected

---

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

• ForwardDeclarations.h
• JacobiSVD.h