Public Types | Public Member Functions | Protected Attributes | Private Member Functions | Friends
Eigen::JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference

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

#include <JacobiSVD.h>

List of all members.

Public Types

enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime), MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime), MatrixOptions = MatrixType::Options
}
typedef
internal::plain_col_type
< MatrixType >::type 
ColType
typedef MatrixType::Index Index
typedef _MatrixType MatrixType
typedef Matrix< Scalar,
RowsAtCompileTime,
RowsAtCompileTime,
MatrixOptions,
MaxRowsAtCompileTime,
MaxRowsAtCompileTime
MatrixUType
typedef Matrix< Scalar,
ColsAtCompileTime,
ColsAtCompileTime,
MatrixOptions,
MaxColsAtCompileTime,
MaxColsAtCompileTime
MatrixVType
typedef NumTraits< typename
MatrixType::Scalar >::Real 
RealScalar
typedef
internal::plain_row_type
< MatrixType >::type 
RowType
typedef MatrixType::Scalar Scalar
typedef
internal::plain_diag_type
< MatrixType, RealScalar >
::type 
SingularValuesType
typedef Matrix< Scalar,
DiagSizeAtCompileTime,
DiagSizeAtCompileTime,
MatrixOptions,
MaxDiagSizeAtCompileTime,
MaxDiagSizeAtCompileTime
WorkMatrixType

Public Member Functions

Index cols () const
JacobiSVDcompute (const MatrixType &matrix, unsigned int computationOptions)
 Method performing the decomposition of given matrix using custom options.
JacobiSVDcompute (const MatrixType &matrix)
 Method performing the decomposition of given matrix using current options.
bool computeU () const
bool computeV () const
 JacobiSVD ()
 Default Constructor.
 JacobiSVD (Index rows, Index cols, unsigned int computationOptions=0)
 Default Constructor with memory preallocation.
 JacobiSVD (const MatrixType &matrix, unsigned int computationOptions=0)
 Constructor performing the decomposition of given matrix.
const MatrixUTypematrixU () const
const MatrixVTypematrixV () const
Index nonzeroSingularValues () const
Index rows () const
const SingularValuesTypesingularValues () const
template<typename Rhs >
const internal::solve_retval
< JacobiSVD, Rhs > 
solve (const MatrixBase< 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
bool m_isAllocated
bool m_isInitialized
MatrixUType m_matrixU
MatrixVType m_matrixV
Index m_nonzeroSingularValues
internal::qr_preconditioner_impl
< MatrixType, QRPreconditioner,
internal::PreconditionIfMoreColsThanRows
m_qr_precond_morecols
internal::qr_preconditioner_impl
< MatrixType, QRPreconditioner,
internal::PreconditionIfMoreRowsThanCols
m_qr_precond_morerows
Index m_rows
SingularValuesType m_singularValues
WorkMatrixType m_workMatrix

Private Member Functions

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

Friends

struct internal::qr_preconditioner_impl
struct internal::svd_precondition_2x2_block_to_be_real

Detailed Description

template<typename _MatrixType, int QRPreconditioner>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >

Two-sided Jacobi SVD decomposition of a rectangular matrix.

Parameters:
MatrixTypethe type of the matrix of which we are computing the SVD decomposition
QRPreconditionerthis optional parameter allows to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. 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:

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 values for QRPreconditioner are:

See also:
MatrixBase::jacobiSvd()

Definition at line 492 of file JacobiSVD.h.


Member Typedef Documentation

template<typename _MatrixType, int QRPreconditioner>
typedef internal::plain_col_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::ColType

Definition at line 518 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef MatrixType::Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Index

Definition at line 499 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef _MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType

Definition at line 496 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixUType

Definition at line 512 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixVType

Definition at line 515 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RealScalar

Definition at line 498 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef internal::plain_row_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RowType

Definition at line 517 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef MatrixType::Scalar Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Scalar

Definition at line 497 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef internal::plain_diag_type<MatrixType, RealScalar>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::SingularValuesType

Definition at line 516 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::WorkMatrixType

Definition at line 521 of file JacobiSVD.h.


Member Enumeration Documentation

template<typename _MatrixType, int QRPreconditioner>
anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
DiagSizeAtCompileTime 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 
MaxDiagSizeAtCompileTime 
MatrixOptions 

Definition at line 500 of file JacobiSVD.h.


Constructor & Destructor Documentation

template<typename _MatrixType, int QRPreconditioner>
Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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 528 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD ( Index  rows,
Index  cols,
unsigned int  computationOptions = 0 
) [inline]

Default Constructor with memory preallocation.

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

See also:
JacobiSVD()

Definition at line 542 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD ( const MatrixType matrix,
unsigned int  computationOptions = 0 
) [inline]

Constructor performing the decomposition of given matrix.

Parameters:
matrixthe matrix to decompose
computationOptionsoptional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are ComputeFullU, ComputeThinU, ComputeFullV, ComputeThinV.

Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner.

Definition at line 561 of file JacobiSVD.h.


Member Function Documentation

template<typename MatrixType , int QRPreconditioner>
void Eigen::JacobiSVD< MatrixType, QRPreconditioner >::allocate ( Index  rows,
Index  cols,
unsigned int  computationOptions 
) [private]

Definition at line 693 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::cols ( void  ) const [inline]

Definition at line 667 of file JacobiSVD.h.

template<typename MatrixType , int QRPreconditioner>
JacobiSVD< MatrixType, QRPreconditioner > & Eigen::JacobiSVD< MatrixType, QRPreconditioner >::compute ( const MatrixType matrix,
unsigned int  computationOptions 
)

Method performing the decomposition of given matrix using custom options.

Parameters:
matrixthe matrix to decompose
computationOptionsoptional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are ComputeFullU, ComputeThinU, ComputeFullV, ComputeThinV.

Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner.

Definition at line 742 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
JacobiSVD& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::compute ( const MatrixType matrix) [inline]

Method performing the decomposition of given matrix using current options.

Parameters:
matrixthe matrix to decompose

This method uses the current computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).

Definition at line 588 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::computeU ( ) const [inline]
Returns:
true if U (full or thin) is asked for in this SVD decomposition

Definition at line 637 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::computeV ( ) const [inline]
Returns:
true if V (full or thin) is asked for in this SVD decomposition

Definition at line 639 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
const MatrixUType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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 602 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
const MatrixVType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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 618 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::nonzeroSingularValues ( ) const [inline]
Returns:
the number of singular values that are not exactly 0

Definition at line 660 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::rows ( void  ) const [inline]

Definition at line 666 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
const SingularValuesType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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 630 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
template<typename Rhs >
const internal::solve_retval<JacobiSVD, Rhs> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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 $.

Definition at line 652 of file JacobiSVD.h.


Friends And Related Function Documentation

template<typename _MatrixType, int QRPreconditioner>
friend struct internal::qr_preconditioner_impl [friend]

Definition at line 686 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
friend struct internal::svd_precondition_2x2_block_to_be_real [friend]

Definition at line 684 of file JacobiSVD.h.


Member Data Documentation

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_cols [protected]

Definition at line 681 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
unsigned int Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computationOptions [protected]

Definition at line 680 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullU [protected]

Definition at line 678 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullV [protected]

Definition at line 679 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinU [protected]

Definition at line 678 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinV [protected]

Definition at line 679 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_diagSize [protected]

Definition at line 681 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_isAllocated [protected]

Definition at line 677 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized [protected]

Definition at line 677 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
MatrixUType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU [protected]

Definition at line 673 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
MatrixVType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV [protected]

Definition at line 674 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues [protected]

Definition at line 681 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morecols [protected]

Definition at line 688 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morerows [protected]

Definition at line 689 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_rows [protected]

Definition at line 681 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
SingularValuesType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues [protected]

Definition at line 675 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
WorkMatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix [protected]

Definition at line 676 of file JacobiSVD.h.


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


acado
Author(s): Milan Vukov, Rien Quirynen
autogenerated on Sat Jun 8 2019 19:40:46