Main MRPT website > C++ reference
MRPT logo
Public Types | Public Member Functions | Protected Attributes | Private Member Functions | Friends

JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference


Detailed Description

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

Two-sided Jacobi SVD decomposition of a square matrix.

Parameters:
MatrixType the type of the matrix of which we are computing the SVD decomposition
QRPreconditioner this 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 343 of file JacobiSVD.h.

#include <src/SVD/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 _MatrixType MatrixType
typedef MatrixType::Scalar Scalar
typedef NumTraits< typename
MatrixType::Scalar >::Real 
RealScalar
typedef MatrixType::Index Index
typedef Matrix< Scalar,
RowsAtCompileTime,
RowsAtCompileTime,
MatrixOptions,
MaxRowsAtCompileTime,
MaxRowsAtCompileTime > 
MatrixUType
typedef Matrix< Scalar,
ColsAtCompileTime,
ColsAtCompileTime,
MatrixOptions,
MaxColsAtCompileTime,
MaxColsAtCompileTime > 
MatrixVType
typedef
internal::plain_diag_type
< MatrixType, RealScalar >
::type 
SingularValuesType
typedef
internal::plain_row_type
< MatrixType >::type 
RowType
typedef
internal::plain_col_type
< MatrixType >::type 
ColType
typedef Matrix< Scalar,
DiagSizeAtCompileTime,
DiagSizeAtCompileTime,
MatrixOptions,
MaxDiagSizeAtCompileTime,
MaxDiagSizeAtCompileTime > 
WorkMatrixType

Public Member Functions

 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.
JacobiSVDcompute (const MatrixType &matrix, unsigned int computationOptions=0)
 Method performing the decomposition of given matrix.
const MatrixUTypematrixU () const
const MatrixVTypematrixV () const
const SingularValuesTypesingularValues () const
bool computeU () const
bool computeV () const
template<typename Rhs >
const internal::solve_retval
< JacobiSVD, Rhs > 
solve (const MatrixBase< Rhs > &b) const
Index nonzeroSingularValues () const
Index rows () const
Index cols () const

Protected Attributes

MatrixUType m_matrixU
MatrixVType m_matrixV
SingularValuesType m_singularValues
WorkMatrixType m_workMatrix
bool m_isInitialized
bool m_computeFullU
bool m_computeThinU
bool m_computeFullV
bool m_computeThinV
Index m_nonzeroSingularValues
Index m_rows
Index m_cols
Index m_diagSize

Private Member Functions

void allocate (Index rows, Index cols, unsigned int computationOptions=0)

Friends

struct internal::svd_precondition_2x2_block_to_be_real
struct internal::qr_preconditioner_impl

Member Typedef Documentation

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

Definition at line 369 of file JacobiSVD.h.

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

Definition at line 350 of file JacobiSVD.h.

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

Definition at line 347 of file JacobiSVD.h.

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

Definition at line 363 of file JacobiSVD.h.

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

Definition at line 366 of file JacobiSVD.h.

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

Definition at line 349 of file JacobiSVD.h.

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

Definition at line 368 of file JacobiSVD.h.

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

Definition at line 348 of file JacobiSVD.h.

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

Definition at line 367 of file JacobiSVD.h.

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

Definition at line 372 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 351 of file JacobiSVD.h.


Constructor & Destructor Documentation

template<typename _MatrixType, int QRPreconditioner>
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 379 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>
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 388 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::allocate().

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

Constructor performing the decomposition of given matrix.

Parameters:
matrix the matrix to decompose
computationOptions optional 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 403 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::compute().


Member Function Documentation

template<typename MatrixType , int QRPreconditioner>
void JacobiSVD< MatrixType, QRPreconditioner >::allocate ( Index  rows,
Index  cols,
unsigned int  computationOptions = 0 
) [private]
template<typename _MatrixType, int QRPreconditioner>
Index JacobiSVD< _MatrixType, QRPreconditioner >::cols ( void   )  const [inline]

Definition at line 494 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::m_cols.

template<typename MatrixType , int QRPreconditioner>
JacobiSVD< MatrixType, QRPreconditioner > & JacobiSVD< MatrixType, QRPreconditioner >::compute ( const MatrixType matrix,
unsigned int  computationOptions = 0 
)

Method performing the decomposition of given matrix.

Parameters:
matrix the matrix to decompose
computationOptions optional 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 548 of file JacobiSVD.h.

References abs(), JacobiSVD< _MatrixType, QRPreconditioner >::allocate(), PlainObjectBase< Derived >::coeff(), JacobiSVD< _MatrixType, QRPreconditioner >::computeU(), JacobiSVD< _MatrixType, QRPreconditioner >::computeV(), JacobiSVD< _MatrixType, QRPreconditioner >::m_cols, JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullU, JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullV, JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinU, JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinV, JacobiSVD< _MatrixType, QRPreconditioner >::m_diagSize, JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU, JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV, JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues, JacobiSVD< _MatrixType, QRPreconditioner >::m_rows, JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues, JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix, internal::real_2x2_jacobi_svd(), Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols >::swap(), and JacobiRotation< Scalar >::transpose().

Referenced by JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD().

template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::computeU (  )  const [inline]
template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::computeV (  )  const [inline]
template<typename _MatrixType, int QRPreconditioner>
const MatrixUType& 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 429 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::computeU(), eigen_assert, JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU.

Referenced by Transform< _Scalar, _Dim, _Mode >::computeRotationScaling(), Transform< _Scalar, _Dim, _Mode >::computeScalingRotation(), and umeyama().

template<typename _MatrixType, int QRPreconditioner>
const MatrixVType& 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 445 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::computeV(), eigen_assert, JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV.

Referenced by Transform< _Scalar, _Dim, _Mode >::computeRotationScaling(), Transform< _Scalar, _Dim, _Mode >::computeScalingRotation(), QuaternionBase< Derived >::setFromTwoVectors(), and umeyama().

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

Definition at line 487 of file JacobiSVD.h.

References eigen_assert, JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues.

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

Definition at line 493 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::m_rows.

template<typename _MatrixType, int QRPreconditioner>
const SingularValuesType& 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.

Definition at line 457 of file JacobiSVD.h.

References eigen_assert, JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized, and JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues.

Referenced by Transform< _Scalar, _Dim, _Mode >::computeRotationScaling(), Transform< _Scalar, _Dim, _Mode >::computeScalingRotation(), and umeyama().

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

Definition at line 479 of file JacobiSVD.h.

References JacobiSVD< _MatrixType, QRPreconditioner >::computeU(), JacobiSVD< _MatrixType, QRPreconditioner >::computeV(), eigen_assert, and JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized.


Friends And Related Function Documentation

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

Definition at line 512 of file JacobiSVD.h.

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

Definition at line 510 of file JacobiSVD.h.


Member Data Documentation

template<typename _MatrixType, int QRPreconditioner>
Index JacobiSVD< _MatrixType, QRPreconditioner >::m_cols [protected]
template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullU [protected]
template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullV [protected]
template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinU [protected]
template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinV [protected]
template<typename _MatrixType, int QRPreconditioner>
Index JacobiSVD< _MatrixType, QRPreconditioner >::m_diagSize [protected]
template<typename _MatrixType, int QRPreconditioner>
bool JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized [protected]
template<typename _MatrixType, int QRPreconditioner>
MatrixUType JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU [protected]
template<typename _MatrixType, int QRPreconditioner>
MatrixVType JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV [protected]
template<typename _MatrixType, int QRPreconditioner>
Index JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues [protected]
template<typename _MatrixType, int QRPreconditioner>
Index JacobiSVD< _MatrixType, QRPreconditioner >::m_rows [protected]
template<typename _MatrixType, int QRPreconditioner>
SingularValuesType JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues [protected]
template<typename _MatrixType, int QRPreconditioner>
WorkMatrixType JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix [protected]



Page generated by Doxygen 1.7.1 for MRPT 0.9.4 SVN: at Mon Jan 10 23:33:19 UTC 2011