Eigen::SVDBase< Derived > Class Template Reference

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

Inheritance diagram for Eigen::SVDBase< Derived >:

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

Public Member Functions
bool	computeU () const
bool	computeV () 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
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
Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT
Derived &	derived ()
const Derived &	derived () const
EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT
EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

Protected Member Functions
	SVDBase ()
	Default Constructor. More...

Member Typedef Documentation

Index

template<typename Derived >

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

Deprecated:

since Eigen 3.3

Constructor & Destructor Documentation

SVDBase()

template<typename Derived >

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

inlineprotected

Default Constructor.

Default constructor of SVDBase

Member Function Documentation

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

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

info()

template<typename Derived >

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

inline

Reports whether previous computation was successful.

Returns

Success if computation was successful.

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.

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.

nonzeroSingularValues()

template<typename Derived >

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

inline

Returns

the number of singular values that are not exactly 0

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&).

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)

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&).

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.

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&).

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

• SVDBase.h