Eigen  3.2.90 (mercurial changeset f6878fc79b74)
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Eigen::JacobiSVD< MatrixType, QRPreconditioner > Class Template Reference

Detailed Description

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner>
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:

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()
+ Inheritance diagram for Eigen::JacobiSVD< MatrixType, QRPreconditioner >:

Public Member Functions

JacobiSVDcompute (const MatrixType &matrix, unsigned int computationOptions)
 Method performing the decomposition of given matrix using custom options. More...
 
JacobiSVDcompute (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 MatrixUTypematrixU () const
 
const MatrixVTypematrixV () const
 
Index nonzeroSingularValues () const
 
Index rank () const
 
JacobiSVD< _MatrixType,
QRPreconditioner > & 
setThreshold (const RealScalar &threshold)
 
JacobiSVD< _MatrixType,
QRPreconditioner > & 
setThreshold (Default_t)
 
const SingularValuesType & singularValues () const
 
const Solve< JacobiSVD
< _MatrixType,
QRPreconditioner >, Rhs > 
solve (const MatrixBase< Rhs > &b) const
 
RealScalar threshold () const
 

Constructor & Destructor Documentation

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner>
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&).

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner>
Eigen::JacobiSVD< MatrixType, QRPreconditioner >::JacobiSVD ( Index  rows,
Index  cols,
unsigned int  computationOptions = 0 
)
inlineexplicit

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

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.

Member Function Documentation

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.

References Eigen::JacobiRotation< Scalar >::transpose().

template<typename MatrixType, int QRPreconditioner = ColPivHouseholderQRPreconditioner>
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).

bool Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::computeU ( ) const
inlineinherited
Returns
true if U (full or thin) is asked for in this SVD decomposition
bool Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::computeV ( ) const
inlineinherited
Returns
true if V (full or thin) is asked for in this SVD decomposition
const MatrixUType& Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::matrixU ( ) const
inlineinherited
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.

References Eigen::SVDBase< Derived >::computeU().

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

const MatrixVType& Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::matrixV ( ) const
inlineinherited
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.

References Eigen::SVDBase< Derived >::computeV().

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

Index Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::nonzeroSingularValues ( ) const
inlineinherited
Returns
the number of singular values that are not exactly 0
Index Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::rank ( ) const
inlineinherited
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&).

References Eigen::SVDBase< Derived >::threshold().

JacobiSVD< _MatrixType, QRPreconditioner > & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::setThreshold ( const RealScalar &  threshold)
inlineinherited

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)

References Eigen::SVDBase< Derived >::threshold().

JacobiSVD< _MatrixType, QRPreconditioner > & Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::setThreshold ( Default_t  )
inlineinherited

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& Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::singularValues ( ) const
inlineinherited
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 Eigen::Transform< Scalar, Dim, Mode, _Options >::computeRotationScaling(), Eigen::Transform< Scalar, Dim, Mode, _Options >::computeScalingRotation(), and Eigen::umeyama().

const Solve<JacobiSVD< _MatrixType, QRPreconditioner > , Rhs> Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::solve ( const MatrixBase< Rhs > &  b) const
inlineinherited
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 $.

References Eigen::SVDBase< Derived >::computeU(), and Eigen::SVDBase< Derived >::computeV().

RealScalar Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >::threshold ( ) const
inlineinherited

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: