Detailed Description

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner>
class Eigen::JacobiSVD< MatrixType, QRPreconditioner >

Two-sided Jacobi SVD decomposition of a rectangular 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:

MatrixXf m = MatrixXf::Random(3,2);
cout << "Here is the matrix m:" << endl << m << endl;
JacobiSVD<MatrixXf> svd(m, ComputeThinU | ComputeThinV);
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:
  0.68  0.597
-0.211  0.823
 0.566 -0.605
Its singular values are:
 1.19
0.899
Its left singular vectors are the columns of the thin U matrix:
  0.388   0.866
  0.712 -0.0634
 -0.586   0.496
Its right singular vectors are the columns of the thin V matrix:
-0.183  0.983
 0.983  0.183
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
0.888
0.496

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:

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.

See Also: MatrixBase::jacobiSvd()

Public Member Functions
JacobiSVD &	compute (const MatrixType &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix using custom options. More...

JacobiSVD &	compute (const MatrixType &matrix)
	Method performing the decomposition of given matrix using current options. More...

bool	computeU () const

bool	computeV () const

	JacobiSVD ()
	Default Constructor. More...

	JacobiSVD (Index rows, Index cols, unsigned int computationOptions=0)
	Default Constructor with memory preallocation. More...

	JacobiSVD (const MatrixType &matrix, unsigned int computationOptions=0)
	Constructor performing the decomposition of given matrix. More...

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

Index	nonzeroSingularValues () const

Index	rank () const

JacobiSVD &	setThreshold (const RealScalar &threshold)

JacobiSVD &	setThreshold (Default_t)

const SingularValuesType &	singularValues () const

template<typename Rhs >
const internal::solve_retval < JacobiSVD, Rhs >	solve (const MatrixBase< Rhs > &b) const

RealScalar	threshold () const

Constructor & Destructor Documentation

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	(	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()

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.

Member Function Documentation

JacobiSVD< MatrixType, QRPreconditioner > & compute	(	const MatrixType &	matrix,
		unsigned int	computationOptions
	)

Method performing the decomposition of given matrix using custom options.

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.

References JacobiRotation< Scalar >::transpose().

JacobiSVD& compute ( const MatrixType & matrix )

inline

Method performing the decomposition of given matrix using current options.

Parameters

matrix the matrix to decompose

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

bool computeU ( ) const

inline

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

bool computeV ( ) const

inline

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

const MatrixUType& 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.

Referenced by Transform< Scalar, Dim, Mode, _Options >::computeRotationScaling(), Transform< Scalar, Dim, Mode, _Options >::computeScalingRotation(), and Eigen::umeyama().

const MatrixVType& 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.

Referenced by Transform< Scalar, Dim, Mode, _Options >::computeRotationScaling(), Transform< Scalar, Dim, Mode, _Options >::computeScalingRotation(), QuaternionBase< Derived >::setFromTwoVectors(), Hyperplane< _Scalar, _AmbientDim, Options >::Through(), and Eigen::umeyama().

Index nonzeroSingularValues ( ) const

inline

Returns: the number of singular values that are not exactly 0

Index 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&).

JacobiSVD& 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)

JacobiSVD& 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&).

const SingularValuesType& 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.

Referenced by Transform< Scalar, Dim, Mode, _Options >::computeRotationScaling(), Transform< Scalar, Dim, Mode, _Options >::computeScalingRotation(), and Eigen::umeyama().

const internal::solve_retval<JacobiSVD, Rhs> solve ( const MatrixBase< Rhs > & b ) const

inline

Returns: a (least squares) solution of 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 $\Vert A x - b \Vert$ .

RealScalar 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 files:

Detailed Description

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner> class Eigen::JacobiSVD< MatrixType, QRPreconditioner >

Public Member Functions

Constructor & Destructor Documentation

Member Function Documentation

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner>
class Eigen::JacobiSVD< MatrixType, QRPreconditioner >