Two-sided Jacobi SVD decomposition of a rectangular matrix. More...
#include <ForwardDeclarations.h>
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... | |
| 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... | |
 Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > | |
| EIGEN_DEVICE_FUNC void | _solve_impl (const RhsType &rhs, DstType &dst) const |
| Index | cols () const |
| bool | computeU () const |
| bool | computeV () const |
| JacobiSVD< _MatrixType, QRPreconditioner > & | derived () |
| const JacobiSVD< _MatrixType, QRPreconditioner > & | derived () const |
| const MatrixUType & | matrixU () const |
| const MatrixVType & | matrixV () const |
| Index | nonzeroSingularValues () const |
| Index | rank () const |
| Index | rows () 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 |
Protected Attributes | |
| internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows > | m_qr_precond_morecols |
| internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols > | m_qr_precond_morerows |
| MatrixType | m_scaledMatrix |
| WorkMatrixType | m_workMatrix |
| Protected Attributes inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > | |
| 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 |
| RealScalar | m_prescribedThreshold |
| Index | m_rows |
| SingularValuesType | m_singularValues |
| bool | m_usePrescribedThreshold |
Private Types | |
| typedef SVDBase< JacobiSVD > | Base |
Private Member Functions | |
| void | allocate (Index rows, Index cols, unsigned int computationOptions) |
Friends | |
| template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything> | |
| struct | internal::qr_preconditioner_impl |
| template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex> | |
| struct | internal::svd_precondition_2x2_block_to_be_real |
Additional Inherited Members | |
| Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > | |
| bool | allocate (Index rows, Index cols, unsigned int computationOptions) |
| SVDBase () | |
| Default Constructor. More... | |
| Static Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > | |
| static void | check_template_parameters () |
Detailed Description
template<typename _MatrixType, int QRPreconditioner>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >
Two-sided Jacobi SVD decomposition of a rectangular matrix.
- Template Parameters
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_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
![\[ A = U S V^* \]](form_174.png)
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 
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()
Definition at line 258 of file ForwardDeclarations.h.
Member Typedef Documentation
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private |
Definition at line 490 of file JacobiSVD.h.
| typedef internal::plain_col_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::ColType |
Definition at line 511 of file JacobiSVD.h.
| typedef _MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType |
Definition at line 493 of file JacobiSVD.h.
| typedef Base::MatrixUType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixUType |
Definition at line 506 of file JacobiSVD.h.
| typedef Base::MatrixVType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixVType |
Definition at line 507 of file JacobiSVD.h.
| typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RealScalar |
Definition at line 495 of file JacobiSVD.h.
| typedef internal::plain_row_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RowType |
Definition at line 510 of file JacobiSVD.h.
| typedef MatrixType::Scalar Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Scalar |
Definition at line 494 of file JacobiSVD.h.
| typedef Base::SingularValuesType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::SingularValuesType |
Definition at line 508 of file JacobiSVD.h.
| typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::WorkMatrixType |
Definition at line 514 of file JacobiSVD.h.
Member Enumeration Documentation
| anonymous enum |
| Enumerator | |
|---|---|
| RowsAtCompileTime | |
| ColsAtCompileTime | |
| DiagSizeAtCompileTime | |
| MaxRowsAtCompileTime | |
| MaxColsAtCompileTime | |
| MaxDiagSizeAtCompileTime | |
| MatrixOptions | |
Definition at line 496 of file JacobiSVD.h.
Constructor & Destructor Documentation
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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 521 of file JacobiSVD.h.
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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 531 of file JacobiSVD.h.
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inlineexplicit |
Constructor performing the decomposition of given matrix.
- Parameters
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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 546 of file JacobiSVD.h.
Member Function Documentation
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private |
Definition at line 613 of file JacobiSVD.h.
| 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
-
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 663 of file JacobiSVD.h.
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inline |
Method performing the decomposition of given matrix using current options.
- Parameters
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matrix the 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 569 of file JacobiSVD.h.
Friends And Related Function Documentation
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friend |
Definition at line 605 of file JacobiSVD.h.
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friend |
Definition at line 603 of file JacobiSVD.h.
Member Data Documentation
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protected |
Definition at line 607 of file JacobiSVD.h.
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protected |
Definition at line 608 of file JacobiSVD.h.
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protected |
Definition at line 609 of file JacobiSVD.h.
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Definition at line 600 of file JacobiSVD.h.
The documentation for this class was generated from the following files: