Eigen  3.2.92
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Eigen::SVDBase< Derived > Class Template Reference
Dense linear problems and decompositions » Reference » SVD_Module

Detailed Description

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

Base class of SVD algorithms.

Template Parameters
Derivedthe 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.

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.

See also
MatrixBase::genericSvd()

Public Types

typedef Eigen::Index Index
 

Public Member Functions

bool computeU () const
 
bool computeV () const
 
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
 

Protected Member Functions

 SVDBase ()
 Default Constructor. More...
 

Member Typedef Documentation

template<typename Derived>
typedef Eigen::Index Eigen::SVDBase< Derived >::Index
Deprecated:
since Eigen 3.3

Constructor & Destructor Documentation

template<typename Derived>
Eigen::SVDBase< Derived >::SVDBase ( )
inlineprotected

Default Constructor.

Default constructor of SVDBase

Member Function Documentation

template<typename Derived>
bool Eigen::SVDBase< Derived >::computeU ( ) const
inline
Returns
true if U (full or thin) is asked for in this SVD decomposition
template<typename Derived>
bool Eigen::SVDBase< Derived >::computeV ( ) const
inline
Returns
true if V (full or thin) is asked for in this SVD decomposition
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.

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.

template<typename Derived>
Index Eigen::SVDBase< Derived >::nonzeroSingularValues ( ) const
inline
Returns
the number of singular values that are not exactly 0
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&).
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
thresholdThe 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)

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&).

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.

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
bthe 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 $.
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&).

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