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JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference
Public Member Functions

JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference
[SVD module]

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

#include <JacobiSVD.h>

Public Member Functions

 JacobiSVD ()
 Default Constructor.
 JacobiSVD (Index rows, Index cols, unsigned int computationOptions=0)
 Default Constructor with memory preallocation.
 JacobiSVD (const MatrixType &matrix, unsigned int computationOptions=0)
 Constructor performing the decomposition of given matrix.
JacobiSVD & compute (const MatrixType &matrix, unsigned int computationOptions)
 Method performing the decomposition of given matrix using custom options.
JacobiSVD & compute (const MatrixType &matrix)
 Method performing the decomposition of given matrix using current options.
const MatrixUType & matrixU () const
const MatrixVType & matrixV () const
const SingularValuesType & singularValues () const
bool computeU () const
bool computeV () const
template<typename Rhs >
const internal::solve_retval
< JacobiSVD, Rhs > 		solve (const MatrixBase< Rhs > &b) const
Index nonzeroSingularValues () const
Index rank () const
JacobiSVD & setThreshold (const RealScalar &threshold)
 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.
JacobiSVD & setThreshold (Default_t)
 Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.
RealScalar threshold () const
 Returns the threshold that will be used by certain methods such as rank().

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: 


    

  • 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 500 of file JacobiSVD.h.


Constructor & Destructor Documentation






      

        

          JacobiSVD 
          (
           )
          
        

      







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 536 of file JacobiSVD.h.











      

        

          JacobiSVD 
          (
          Index 
           rows, 
        

        

          
          
          Index 
           cols, 
        

        

          
          
          unsigned int 
           computationOptions = 0 
        

        

          
          )
          
        

      







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 551 of file JacobiSVD.h.











      

        

          JacobiSVD 
          (
          const MatrixType & 
           matrix, 
        

        

          
          
          unsigned int 
           computationOptions = 0 
        

        

          
          )
          
        

      







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 571 of file JacobiSVD.h.







Member Function Documentation






      

        

          JacobiSVD& compute 
          (
          const MatrixType & 
           matrix )
          
        

      







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 599 of file JacobiSVD.h.











      

        

          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 824 of file JacobiSVD.h.











      

        

          bool computeU 
          (
           )
           const
        

      






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



Definition at line 648 of file JacobiSVD.h.











      

        

          bool computeV 
          (
           )
           const
        

      






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



Definition at line 650 of file JacobiSVD.h.











      

        

          const MatrixUType& matrixU 
          (
           )
           const
        

      






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 613 of file JacobiSVD.h.











      

        

          const MatrixVType& matrixV 
          (
           )
           const
        

      






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 629 of file JacobiSVD.h.











      

        

          Index nonzeroSingularValues 
          (
           )
           const
        

      






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



Definition at line 671 of file JacobiSVD.h.











      

        

          Index rank 
          (
           )
           const
        

      






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&). 



Definition at line 683 of file JacobiSVD.h.











      

        

          JacobiSVD& setThreshold 
          (
          Default_t 
           )
          
        

      







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&). 



Definition at line 723 of file JacobiSVD.h.











      

        

          JacobiSVD& setThreshold 
          (
          const RealScalar & 
           threshold )
          
        

      







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) 



Definition at line 708 of file JacobiSVD.h.











      

        

          const SingularValuesType& singularValues 
          (
           )
           const
        

      






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 641 of file JacobiSVD.h.











      

        

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

      






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 663 of file JacobiSVD.h.











      

        

          RealScalar threshold 
          (
           )
           const
        

      







Returns the threshold that will be used by certain methods such as rank(). 


See the documentation of setThreshold(const RealScalar&). 



Definition at line 733 of file JacobiSVD.h.









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