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Eigen::JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference

Two-sided Jacobi SVD decomposition of a rectangular matrix. More...

#include <JacobiSVD.h>

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 Base::MatrixUType MatrixUType
 
typedef Base::MatrixVType MatrixVType
 
typedef Base::SingularValuesType 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. 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...
 
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
 
Index rows () const
 
Index cols () const
 
Index rank () const
 

Protected Attributes

WorkMatrixType m_workMatrix
 
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
 
MatrixUType m_matrixU
 
MatrixVType m_matrixV
 
SingularValuesType m_singularValues
 
bool m_isInitialized
 
bool m_isAllocated
 
bool m_usePrescribedThreshold
 
bool m_computeFullU
 
bool m_computeThinU
 
bool m_computeFullV
 
bool m_computeThinV
 
unsigned int m_computationOptions
 
Index m_nonzeroSingularValues
 
Index m_rows
 
Index m_cols
 
Index m_diagSize
 
RealScalar m_prescribedThreshold
 

Friends

template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex>
struct internal::svd_precondition_2x2_block_to_be_real
 
template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything>
struct internal::qr_preconditioner_impl
 

Detailed Description

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

Two-sided Jacobi SVD decomposition of a rectangular matrix.

Template 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:

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() 


Constructor & Destructor Documentation



◆ JacobiSVD() [1/3]








template<typename _MatrixType , int QRPreconditioner> 



  

  
      

        

          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&). 








◆ JacobiSVD() [2/3]








template<typename _MatrixType , int QRPreconditioner> 



  

  
      

        

          Eigen::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() 








◆ JacobiSVD() [3/3]








template<typename _MatrixType , int QRPreconditioner> 



  

  
      

        

          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



◆ compute() [1/2]








template<typename _MatrixType , int QRPreconditioner> 



  

  
      

        

          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). 








◆ compute() [2/2]








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. 








◆ computeU()








template<typename _MatrixType , int QRPreconditioner> 



  

  
      

        

          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 QRPreconditioner> 



  

  
      

        

          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 QRPreconditioner> 



  

  
      

        

          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:





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