Eigen::SVDBase< Derived > Class Template Reference

Base class of SVD algorithms. More...

Inheritance diagram for Eigen::SVDBase< Derived >:

Public Types
enum	{   RowsAtCompileTime ,   ColsAtCompileTime ,   DiagSizeAtCompileTime ,   MaxRowsAtCompileTime ,   MaxColsAtCompileTime ,   MaxDiagSizeAtCompileTime ,   MatrixOptions ,   MatrixUColsAtCompileTime ,   MatrixVColsAtCompileTime ,   MatrixUMaxColsAtCompileTime ,   MatrixVMaxColsAtCompileTime }
typedef Eigen::Index	Index
typedef internal::traits< Derived >::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< SVDBase< Derived > >
enum
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType
typedef EigenBase< SVDBase< Derived > >	Base
typedef Scalar	CoeffReturnType
typedef Transpose< const SVDBase< Derived > >	ConstTransposeReturnType
typedef internal::traits< SVDBase< 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
Index	cols () const
bool	computeU () const
bool	computeV () const
Derived &	derived ()
const Derived &	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
Derived &	setThreshold (const RealScalar &threshold)
Derived &	setThreshold (Default_t)
const SingularValuesType &	singularValues () const
template<typename Rhs >
const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const
RealScalar	threshold () const
Public Member Functions inherited from Eigen::SolverBase< SVDBase< Derived > >
const AdjointReturnType	adjoint () const
SVDBase< Derived > &	derived ()
const SVDBase< Derived > &	derived () const
const Solve< SVDBase< 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

Static Public Attributes
static constexpr bool	ShouldComputeFullU
static constexpr bool	ShouldComputeFullV
static constexpr bool	ShouldComputeThinU
static constexpr bool	ShouldComputeThinV

Protected Member Functions
void	_check_compute_assertions () const
template<bool Transpose_, typename Rhs >
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< SVDBase< Derived > >
void	_check_solve_assertion (const Rhs &b) const

Protected Attributes
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

Detailed Description

template<typename Derived>
class Eigen::SVDBase< Derived >

Base class of SVD algorithms.

Template Parameters

Derived

the type of the actual SVD decomposition

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.

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.

The status of the computation can be retrieved using the info() method. Unless info() returns Success, the results should be not considered well defined.

If the input matrix has inf or nan coefficients, the result of the computation is undefined, and info() will return InvalidInput, but the computation is guaranteed to terminate in finite (and reasonable) time.

See also

class BDCSVD, class JacobiSVD

Definition at line 118 of file SVDBase.h.

Member Typedef Documentation

Index

template<typename Derived >

typedef Eigen::Index Eigen::SVDBase< Derived >::Index

Deprecated:

since Eigen 3.3

Definition at line 130 of file SVDBase.h.

MatrixType

template<typename Derived >

typedef internal::traits<Derived>::MatrixType Eigen::SVDBase< Derived >::MatrixType

Definition at line 126 of file SVDBase.h.

MatrixUType

template<typename Derived >

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

Definition at line 156 of file SVDBase.h.

MatrixVType

template<typename Derived >

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

Definition at line 159 of file SVDBase.h.

RealScalar

template<typename Derived >

typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::SVDBase< Derived >::RealScalar

Definition at line 128 of file SVDBase.h.

Scalar

template<typename Derived >

typedef MatrixType::Scalar Eigen::SVDBase< Derived >::Scalar

Definition at line 127 of file SVDBase.h.

SingularValuesType

template<typename Derived >

typedef internal::plain_diag_type<MatrixType, RealScalar>::type Eigen::SVDBase< Derived >::SingularValuesType

Definition at line 161 of file SVDBase.h.

StorageIndex

template<typename Derived >

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

Definition at line 129 of file SVDBase.h.

Member Enumeration Documentation

anonymous enum

template<typename Derived >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions
MatrixUColsAtCompileTime
MatrixVColsAtCompileTime
MatrixUMaxColsAtCompileTime
MatrixVMaxColsAtCompileTime

Definition at line 137 of file SVDBase.h.

       {
    RowsAtCompileTime = MatrixType::RowsAtCompileTime,
    ColsAtCompileTime = MatrixType::ColsAtCompileTime,
    DiagSizeAtCompileTime = internal::min_size_prefer_dynamic(RowsAtCompileTime, ColsAtCompileTime),
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
    MaxDiagSizeAtCompileTime = internal::min_size_prefer_fixed(MaxRowsAtCompileTime, MaxColsAtCompileTime),
    MatrixOptions = MatrixType::Options,
    MatrixUColsAtCompileTime = internal::traits<Derived>::MatrixUColsAtCompileTime,
    MatrixVColsAtCompileTime = internal::traits<Derived>::MatrixVColsAtCompileTime,
    MatrixUMaxColsAtCompileTime = internal::traits<Derived>::MatrixUMaxColsAtCompileTime,
    MatrixVMaxColsAtCompileTime = internal::traits<Derived>::MatrixVMaxColsAtCompileTime
  };

Constructor & Destructor Documentation

SVDBase()

template<typename Derived >

Eigen::SVDBase< Derived >::SVDBase ( )

inlineprotected

Default Constructor.

Default constructor of SVDBase

Definition at line 358 of file SVDBase.h.

      : m_matrixU(MatrixUType()),
        m_matrixV(MatrixVType()),
        m_singularValues(SingularValuesType()),
        m_info(Success),
        m_isInitialized(false),
        m_isAllocated(false),
        m_usePrescribedThreshold(false),
        m_computeFullU(false),
        m_computeThinU(false),
        m_computeFullV(false),
        m_computeThinV(false),
        m_computationOptions(0),
        m_nonzeroSingularValues(0),
        m_rows(-1),
        m_cols(-1),
        m_diagSize(0),
        m_prescribedThreshold(0) {}

Member Function Documentation

_check_compute_assertions()

template<typename Derived >

void Eigen::SVDBase< Derived >::_check_compute_assertions ( ) const

inlineprotected

Definition at line 328 of file SVDBase.h.

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

_check_solve_assertion()

template<typename Derived >

template<bool Transpose_, typename Rhs >

void Eigen::SVDBase< Derived >::_check_solve_assertion ( const Rhs &  b ) const

inlineprotected

Definition at line 333 of file SVDBase.h.

                                                  {
      EIGEN_ONLY_USED_FOR_DEBUG(b);
      _check_compute_assertions();
      eigen_assert(computeU() && computeV() && "SVDBase::solve(): Both unitaries U and V are required to be computed (thin unitaries suffice).");
      eigen_assert((Transpose_?cols():rows())==b.rows() && "SVDBase::solve(): invalid number of rows of the right hand side matrix b");
  }

allocate()

template<typename Derived >

bool Eigen::SVDBase< Derived >::allocate ( Index  rows, Index  cols, unsigned int  computationOptions  )

protected

Definition at line 410 of file SVDBase.h.

                                                                                       {
  eigen_assert(rows >= 0 && cols >= 0);
 
  if (m_isAllocated &&
      rows == m_rows &&
      cols == m_cols &&
      computationOptions == m_computationOptions)
  {
    return true;
  }
 
  m_rows = rows;
  m_cols = cols;
  m_info = Success;
  m_isInitialized = false;
  m_isAllocated = true;
  m_computationOptions = computationOptions;
  m_computeFullU = ShouldComputeFullU || internal::should_svd_compute_full_u(computationOptions);
  m_computeThinU = ShouldComputeThinU || internal::should_svd_compute_thin_u(computationOptions);
  m_computeFullV = ShouldComputeFullV || internal::should_svd_compute_full_v(computationOptions);
  m_computeThinV = ShouldComputeThinV || internal::should_svd_compute_thin_v(computationOptions);
 
  eigen_assert(!(m_computeFullU && m_computeThinU) && "SVDBase: you can't ask for both full and thin U");
  eigen_assert(!(m_computeFullV && m_computeThinV) && "SVDBase: you can't ask for both full and thin V");
 
  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);
 
  return false;
}

cols()

template<typename Derived >

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

inline

Definition at line 287 of file SVDBase.h.

{ return m_cols; }

computeU()

template<typename Derived >

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

inline

Returns

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

Definition at line 282 of file SVDBase.h.

{ return m_computeFullU || m_computeThinU; }

computeV()

template<typename Derived >

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

inline

Returns

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

Definition at line 284 of file SVDBase.h.

{ return m_computeFullV || m_computeThinV; }

derived() [1/2]

template<typename Derived >

Derived& Eigen::SVDBase< Derived >::derived ( )

inline

Definition at line 163 of file SVDBase.h.

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

derived() [2/2]

template<typename Derived >

const Derived& Eigen::SVDBase< Derived >::derived ( ) const

inline

Definition at line 164 of file SVDBase.h.

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

info()

template<typename Derived >

ComputationInfo Eigen::SVDBase< Derived >::info ( ) const

inline

Reports whether previous computation was successful.

Returns

Success if computation was successful.

Definition at line 310 of file SVDBase.h.

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

matrixU()

template<typename Derived >

const MatrixUType& Eigen::SVDBase< Derived >::matrixU ( ) const

inline

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.

Definition at line 175 of file SVDBase.h.

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

matrixV()

template<typename Derived >

const MatrixVType& Eigen::SVDBase< Derived >::matrixV ( ) const

inline

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.

Definition at line 191 of file SVDBase.h.

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

nonzeroSingularValues()

template<typename Derived >

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

inline

Returns

the number of singular values that are not exactly 0

Definition at line 210 of file SVDBase.h.

  {
    _check_compute_assertions();
    return m_nonzeroSingularValues;
  }

rank()

template<typename Derived >

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&).

Definition at line 222 of file SVDBase.h.

  {
    using std::abs;
    _check_compute_assertions();
    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;
  }

rows()

template<typename Derived >

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

inline

Definition at line 286 of file SVDBase.h.

{ return m_rows; }

setThreshold() [1/2]

template<typename Derived >

Derived& Eigen::SVDBase< Derived >::setThreshold ( const RealScalar &  threshold )

inline

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)

Definition at line 247 of file SVDBase.h.

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

setThreshold() [2/2]

template<typename Derived >

Derived& Eigen::SVDBase< Derived >::setThreshold ( Default_t  )

inline

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); 

See the documentation of setThreshold(const RealScalar&).

Definition at line 262 of file SVDBase.h.

  {
    m_usePrescribedThreshold = false;
    return derived();
  }

singularValues()

template<typename Derived >

const SingularValuesType& Eigen::SVDBase< Derived >::singularValues ( ) const

inline

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.

Definition at line 203 of file SVDBase.h.

  {
    _check_compute_assertions();
    return m_singularValues;
  }

solve()

template<typename Derived >

template<typename Rhs >

const Solve<Derived, Rhs> Eigen::SVDBase< Derived >::solve ( const MatrixBase< Rhs > &  b ) const

inline

Returns

a (least squares) solution of \( A x = b \) 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 \( \Vert A x - b \Vert \).

threshold()

template<typename Derived >

RealScalar Eigen::SVDBase< Derived >::threshold ( ) const

inline

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

See the documentation of setThreshold(const RealScalar&).

Definition at line 272 of file SVDBase.h.

  {
    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
                                    : RealScalar(diagSize)*NumTraits<Scalar>::epsilon();
  }

Member Data Documentation

m_cols

template<typename Derived >

Index Eigen::SVDBase< Derived >::m_cols

protected

Definition at line 351 of file SVDBase.h.

m_computationOptions

template<typename Derived >

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

protected

Definition at line 350 of file SVDBase.h.

m_computeFullU

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_computeFullU

protected

Definition at line 348 of file SVDBase.h.

m_computeFullV

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_computeFullV

protected

Definition at line 349 of file SVDBase.h.

m_computeThinU

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_computeThinU

protected

Definition at line 348 of file SVDBase.h.

m_computeThinV

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_computeThinV

protected

Definition at line 349 of file SVDBase.h.

m_diagSize

template<typename Derived >

Index Eigen::SVDBase< Derived >::m_diagSize

protected

Definition at line 351 of file SVDBase.h.

m_info

template<typename Derived >

ComputationInfo Eigen::SVDBase< Derived >::m_info

protected

Definition at line 346 of file SVDBase.h.

m_isAllocated

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_isAllocated

protected

Definition at line 347 of file SVDBase.h.

m_isInitialized

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_isInitialized

protected

Definition at line 347 of file SVDBase.h.

m_matrixU

template<typename Derived >

MatrixUType Eigen::SVDBase< Derived >::m_matrixU

protected

Definition at line 343 of file SVDBase.h.

m_matrixV

template<typename Derived >

MatrixVType Eigen::SVDBase< Derived >::m_matrixV

protected

Definition at line 344 of file SVDBase.h.

m_nonzeroSingularValues

template<typename Derived >

Index Eigen::SVDBase< Derived >::m_nonzeroSingularValues

protected

Definition at line 351 of file SVDBase.h.

m_prescribedThreshold

template<typename Derived >

RealScalar Eigen::SVDBase< Derived >::m_prescribedThreshold

protected

Definition at line 352 of file SVDBase.h.

m_rows

template<typename Derived >

Index Eigen::SVDBase< Derived >::m_rows

protected

Definition at line 351 of file SVDBase.h.

m_singularValues

template<typename Derived >

SingularValuesType Eigen::SVDBase< Derived >::m_singularValues

protected

Definition at line 345 of file SVDBase.h.

m_usePrescribedThreshold

template<typename Derived >

bool Eigen::SVDBase< Derived >::m_usePrescribedThreshold

protected

Definition at line 347 of file SVDBase.h.

ShouldComputeFullU

template<typename Derived >

constexpr bool Eigen::SVDBase< Derived >::ShouldComputeFullU

staticconstexpr

Definition at line 132 of file SVDBase.h.

ShouldComputeFullV

template<typename Derived >

constexpr bool Eigen::SVDBase< Derived >::ShouldComputeFullV

staticconstexpr

Definition at line 134 of file SVDBase.h.

ShouldComputeThinU

template<typename Derived >

constexpr bool Eigen::SVDBase< Derived >::ShouldComputeThinU

staticconstexpr

Definition at line 133 of file SVDBase.h.

ShouldComputeThinV

template<typename Derived >

constexpr bool Eigen::SVDBase< Derived >::ShouldComputeThinV

staticconstexpr

Definition at line 135 of file SVDBase.h.

---

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

• ForwardDeclarations.h
• SVDBase.h