#include <Eigen/src/SVD/JacobiSVD.h>

Detailed Description

template<typename MatrixType_, int Options_>
class Eigen::JacobiSVD< MatrixType_, Options_ >

Two-sided Jacobi SVD decomposition of a rectangular matrix.

Template Parameters

MatrixType_	the type of the matrix of which we are computing the SVD decomposition
Options	this optional parameter allows one to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. Additionally, it allows one to specify whether to compute thin or full unitaries U and V. 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:

MatrixXf m = MatrixXf::Random(3, 2);
cout << "Here is the matrix m:" << endl << m << endl;
JacobiSVD<MatrixXf, ComputeThinU | ComputeThinV> svd(m);
cout << "Its singular values are:" << endl << svd.singularValues() << endl;
cout << "Its left singular vectors are the columns of the thin U matrix:" << endl << svd.matrixU() << endl;
cout << "Its right singular vectors are the columns of the thin V matrix:" << endl << svd.matrixV() << endl;
Vector3f rhs(1, 0, 0);
cout << "Now consider this rhs vector:" << endl << rhs << endl;
cout << "A least-squares solution of m*x = rhs is:" << endl << svd.solve(rhs) << endl;

Output:

Here is the matrix m:
 -0.824  -0.199
  0.924  -0.237
-0.0532   0.146
Its singular values are:
 1.24
0.338
Its left singular vectors are the columns of the thin U matrix:
-0.657 -0.695
 0.753 -0.579
-0.048  0.426
Its right singular vectors are the columns of the thin V matrix:
 0.999  0.044
-0.044  0.999
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
-0.62
-2.03

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 QR preconditioners that can be set with Options template parameter are:

ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. Contrary to other QRs, it doesn't allow computing thin unitaries.
HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive process is more reliable than the optimized bidiagonal SVD iterations.
NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking if QR preconditioning is needed before applying it anyway.

One may also use the Options template parameter to specify how the unitaries should be computed. The options are ComputeThinU, ComputeThinV, ComputeFullU, ComputeFullV. It is not possible to request both the thin and full versions of a unitary. By default, unitaries will not be computed.

You can set the QRPreconditioner and unitary options together: JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner | ComputeThinU | ComputeFullV>

See also: MatrixBase::jacobiSvd()

Inheritance diagram for Eigen::JacobiSVD< MatrixType_, Options_ >:

Public Member Functions
template<typename Derived >
JacobiSVD &	compute (const MatrixBase< Derived > &matrix)
	Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified using the Options template parameter or the class constructor. More...

template<typename Derived >
JacobiSVD &	compute (const MatrixBase< Derived > &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix, as specified by the `computationOptions` parameter. More...

bool	computeU () const

bool	computeV () const

	JacobiSVD ()
	Default Constructor. More...

	JacobiSVD (Index rows, Index cols)
	Default Constructor with memory preallocation. More...

	JacobiSVD (Index rows, Index cols, unsigned int computationOptions)
	Default Constructor with memory preallocation. More...

template<typename Derived >
	JacobiSVD (const MatrixBase< Derived > &matrix)
	Constructor performing the decomposition of given matrix, using the custom options specified with the Options template parameter. More...

template<typename Derived >
	JacobiSVD (const MatrixBase< Derived > &matrix, unsigned int computationOptions)
	Constructor performing the decomposition of given matrix using specified options for computing unitaries. More...

Index	rank () const

Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
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

JacobiSVD< MatrixType_, Options_ > &	setThreshold (const RealScalar &threshold)

JacobiSVD< MatrixType_, Options_ > &	setThreshold (Default_t)

const SingularValuesType &	singularValues () const

const Solve< JacobiSVD< MatrixType_, Options_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const

RealScalar	threshold () const

Public Member Functions inherited from Eigen::SolverBase< SVDBase< JacobiSVD< MatrixType_, Options_ > > >
const AdjointReturnType	adjoint () const

constexpr const SVDBase< JacobiSVD< MatrixType_, Options_ > > &	derived () const

constexpr SVDBase< JacobiSVD< MatrixType_, Options_ > > &	derived ()

const Solve< SVDBase< JacobiSVD< MatrixType_, Options_ > >, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()

const ConstTransposeReturnType	transpose () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
constexpr Index	cols () const noexcept

constexpr Derived &	derived ()

constexpr const Derived &	derived () const

constexpr Index	rows () const noexcept

constexpr Index	size () const noexcept

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

Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, Options_ > >
	SVDBase ()
	Default Constructor. More...

Constructor & Destructor Documentation

◆ JacobiSVD() [1/5]

template<typename MatrixType_, int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD ( )

inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via JacobiSVD::compute(const MatrixType&).

◆ JacobiSVD() [2/5]

template<typename MatrixType_, int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD	(	Index	rows,
		Index	cols
	)

inline

Default Constructor with memory preallocation.

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

See also: JacobiSVD()

◆ JacobiSVD() [3/5]

template<typename MatrixType_, int Options_>

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions
	)

inline

Default Constructor with memory preallocation.

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

One cannot request unitaries using both the Options template parameter and the constructor. If possible, prefer using the Options template parameter.

Parameters

rows	number of rows for the input matrix
cols	number of columns for the input matrix
computationOptions	specify whether to compute Thin/Full unitaries U/V

See also: JacobiSVD()

◆ JacobiSVD() [4/5]

template<typename MatrixType_, int Options_>

template<typename Derived >

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD ( const MatrixBase< Derived > & matrix )

inlineexplicit

Constructor performing the decomposition of given matrix, using the custom options specified with the Options template parameter.

Parameters

matrix the matrix to decompose

◆ JacobiSVD() [5/5]

template<typename MatrixType_, int Options_>

template<typename Derived >

Eigen::JacobiSVD< MatrixType_, Options_ >::JacobiSVD	(	const MatrixBase< Derived > &	matrix,
		unsigned int	computationOptions
	)

inline

Constructor performing the decomposition of given matrix using specified options for computing unitaries.

One cannot request unitiaries using both the Options template parameter and the constructor. If possible, prefer using the Options template parameter.

Parameters

matrix	the matrix to decompose
computationOptions	specify whether to compute Thin/Full unitaries U/V

Member Function Documentation

◆ compute() [1/2]

template<typename MatrixType_, int Options_>

template<typename Derived >

JacobiSVD& Eigen::JacobiSVD< MatrixType_, Options_ >::compute ( const MatrixBase< Derived > & matrix )

inline

Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified using the Options template parameter or the class constructor.

Parameters

matrix the matrix to decompose

◆ compute() [2/2]

template<typename MatrixType_, int Options_>

template<typename Derived >

JacobiSVD& Eigen::JacobiSVD< MatrixType_, Options_ >::compute	(	const MatrixBase< Derived > &	matrix,
		unsigned int	computationOptions
	)

inline

Method performing the decomposition of given matrix, as specified by the computationOptions parameter.

Parameters

matrix	the matrix to decompose
computationOptions	specify whether to compute Thin/Full unitaries U/V

◆ computeU()

template<typename MatrixType_, int Options_>

bool Eigen::SVDBase< Derived >::computeU

inline

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

◆ computeV()

template<typename MatrixType_, int Options_>

bool Eigen::SVDBase< Derived >::computeV

inline

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

◆ rank()

template<typename MatrixType_, int Options_>

Index Eigen::SVDBase< Derived >::rank

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&).

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

/builddir/build/BUILD/eigen-5.0.1/Eigen/src/Core/util/ForwardDeclarations.h
/builddir/build/BUILD/eigen-5.0.1/Eigen/src/SVD/JacobiSVD.h

	Eigen 5.0.1-dev

Detailed Description

template<typename MatrixType_, int Options_> class Eigen::JacobiSVD< MatrixType_, Options_ >

Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ JacobiSVD() [1/5]

◆ JacobiSVD() [2/5]

◆ JacobiSVD() [3/5]

◆ JacobiSVD() [4/5]

◆ JacobiSVD() [5/5]

Member Function Documentation

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeU()

◆ computeV()

◆ rank()

template<typename MatrixType_, int Options_>
class Eigen::JacobiSVD< MatrixType_, Options_ >