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

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

#include <ForwardDeclarations.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 internal::plain_col_type< MatrixType >::type ColType
 
typedef MatrixType::Index Index
 
typedef _MatrixType MatrixType
 
typedef Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime > MatrixUType
 
typedef Matrix< Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime > MatrixVType
 
typedef NumTraits< typename MatrixType::Scalar >::Real RealScalar
 
typedef internal::plain_row_type< MatrixType >::type RowType
 
typedef MatrixType::Scalar Scalar
 
typedef internal::plain_diag_type< MatrixType, RealScalar >::type SingularValuesType
 
typedef Matrix< Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime > WorkMatrixType
 

Public Member Functions

Index cols () const
 
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 rows () const
 
const SingularValuesType & singularValues () const
 
template<typename Rhs >
const internal::solve_retval< JacobiSVD, Rhs > solve (const MatrixBase< Rhs > &b) const
 

Protected Attributes

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
 
internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows > m_qr_precond_morecols
 
internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols > m_qr_precond_morerows
 
Index m_rows
 
SingularValuesType m_singularValues
 
WorkMatrixType m_workMatrix
 

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
 

Detailed Description

template<typename _MatrixType, int QRPreconditioner>
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:

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() 



Definition at line 224 of file ForwardDeclarations.h.


Member Typedef Documentation








template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef internal::plain_col_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::ColType
        

      





Definition at line 518 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef MatrixType::Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Index
        

      





Definition at line 499 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef _MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType
        

      





Definition at line 496 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixUType
        

      





Definition at line 512 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixVType
        

      





Definition at line 515 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RealScalar
        

      





Definition at line 498 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef internal::plain_row_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RowType
        

      





Definition at line 517 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef MatrixType::Scalar Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Scalar
        

      





Definition at line 497 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef internal::plain_diag_type<MatrixType, RealScalar>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::SingularValuesType
        

      





Definition at line 516 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::WorkMatrixType
        

      





Definition at line 521 of file JacobiSVD.h.







Member Enumeration Documentation








template<typename _MatrixType, int QRPreconditioner> 

      

        

          anonymous enum
        

      






Enumerator
RowsAtCompileTime 



ColsAtCompileTime 



DiagSizeAtCompileTime 



MaxRowsAtCompileTime 



MaxColsAtCompileTime 



MaxDiagSizeAtCompileTime 



MatrixOptions 






Definition at line 500 of file JacobiSVD.h.







Constructor & Destructor Documentation








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&). 



Definition at line 528 of file JacobiSVD.h.













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() 



Definition at line 542 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD 
          (
          const MatrixType & 
          matrix, 
        

        

          
          
          unsigned int 
          computationOptions = 0 
        

        

          
          )
          
        

      

  		
  
inline  
  







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. 



Definition at line 561 of file JacobiSVD.h.







Member Function Documentation








template<typename MatrixType , int QRPreconditioner> 



  

  
      

        

          void Eigen::JacobiSVD< MatrixType, QRPreconditioner >::allocate 
          (
          Index 
          rows, 
        

        

          
          
          Index 
          cols, 
        

        

          
          
          unsigned int 
          computationOptions 
        

        

          
          )
          
        

      

  		
  
private  
  







Definition at line 693 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::cols 
          (
          void 
          )
           const
        

      

  		
  
inline  
  







Definition at line 667 of file JacobiSVD.h.













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. 



Definition at line 742 of file JacobiSVD.h.













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). 



Definition at line 588 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::computeU 
          (
          )
           const
        

      

  		
  
inline  
  






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



Definition at line 637 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::computeV 
          (
          )
           const
        

      

  		
  
inline  
  






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



Definition at line 639 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          const MatrixUType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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. 



Definition at line 602 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          const MatrixVType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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. 



Definition at line 618 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::nonzeroSingularValues 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the number of singular values that are not exactly 0 



Definition at line 660 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::rows 
          (
          void 
          )
           const
        

      

  		
  
inline  
  







Definition at line 666 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          const SingularValuesType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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. 



Definition at line 630 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



template<typename Rhs > 



  

  
      

        

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

      

  		
  
inline  
  






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 $. 



Definition at line 652 of file JacobiSVD.h.







Friends And Related Function Documentation








template<typename _MatrixType, int QRPreconditioner> 



template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything> 



  

  
      

        

          friend struct internal::qr_preconditioner_impl
        

      

  		
  
friend  
  







Definition at line 686 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex> 



  

  
      

        

          friend struct internal::svd_precondition_2x2_block_to_be_real
        

      

  		
  
friend  
  







Definition at line 684 of file JacobiSVD.h.







Member Data Documentation








template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_cols
        

      

  		
  
protected  
  







Definition at line 681 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          unsigned int Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computationOptions
        

      

  		
  
protected  
  







Definition at line 680 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullU
        

      

  		
  
protected  
  







Definition at line 678 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullV
        

      

  		
  
protected  
  







Definition at line 679 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinU
        

      

  		
  
protected  
  







Definition at line 678 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinV
        

      

  		
  
protected  
  







Definition at line 679 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_diagSize
        

      

  		
  
protected  
  







Definition at line 681 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_isAllocated
        

      

  		
  
protected  
  







Definition at line 677 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized
        

      

  		
  
protected  
  







Definition at line 677 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          MatrixUType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU
        

      

  		
  
protected  
  







Definition at line 673 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          MatrixVType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV
        

      

  		
  
protected  
  







Definition at line 674 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues
        

      

  		
  
protected  
  







Definition at line 681 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morecols
        

      

  		
  
protected  
  







Definition at line 688 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morerows
        

      

  		
  
protected  
  







Definition at line 689 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_rows
        

      

  		
  
protected  
  







Definition at line 681 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          SingularValuesType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues
        

      

  		
  
protected  
  







Definition at line 675 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



  

  
      

        

          WorkMatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix
        

      

  		
  
protected  
  







Definition at line 676 of file JacobiSVD.h.







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







tuw_aruco

Author(s): Lukas Pfeifhofer

autogenerated on Mon Jun 10 2019 15:41:07