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JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference

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

#include <JacobiSVD.h>

List of all members.

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.
JacobiSVD & compute (const MatrixType &matrix)
 Method performing the decomposition of given matrix using current options.
bool computeU () const
bool computeV () const
 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.
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
Index m_rows
SingularValuesType m_singularValues
WorkMatrixType m_workMatrix

Private Member Functions

void allocate (Index rows, Index cols, unsigned int computationOptions)

Friends

struct internal::qr_preconditioner_impl
struct internal::svd_precondition_2x2_block_to_be_real

Detailed Description

template<typename _MatrixType, int QRPreconditioner>
class 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: 

MatrixXf m = MatrixXf::Random(3,2);
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: 

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 343 of file JacobiSVD.h.


Member Typedef Documentation








template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 369 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 350 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          typedef _MatrixType JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType
        

      







Definition at line 347 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 363 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 366 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 349 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 368 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 348 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 367 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

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

      







Definition at line 372 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 351 of file JacobiSVD.h.







Constructor & Destructor Documentation








template<typename _MatrixType, int QRPreconditioner> 

      

        

          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 379 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          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 393 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          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 412 of file JacobiSVD.h.







Member Function Documentation








template<typename MatrixType , int QRPreconditioner> 

      

        

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

        

          
          
          Index 
          cols, 
        

        

          
          
          unsigned int 
          computationOptions 
        

        

          
          )
           [private]
        

      







Definition at line 541 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::cols 
          (
          void 
          )
           const [inline]
        

      







Definition at line 518 of file JacobiSVD.h.













template<typename MatrixType , int QRPreconditioner> 

      

        

          JacobiSVD< MatrixType, QRPreconditioner > & 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 585 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          JacobiSVD& 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 439 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::computeU 
          (
          )
           const [inline]
        

      






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



Definition at line 488 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::computeV 
          (
          )
           const [inline]
        

      






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



Definition at line 490 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          const MatrixUType& 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 453 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          const MatrixVType& 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 469 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::nonzeroSingularValues 
          (
          )
           const [inline]
        

      






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



Definition at line 511 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::rows 
          (
          void 
          )
           const [inline]
        

      







Definition at line 517 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          const SingularValuesType& 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 481 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 



template<typename Rhs > 

      

        

          const internal::solve_retval<JacobiSVD, Rhs> 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 503 of file JacobiSVD.h.







Friends And Related Function Documentation








template<typename _MatrixType, int QRPreconditioner> 

      

        

          friend struct internal::qr_preconditioner_impl [friend]
        

      







Definition at line 537 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          friend struct internal::svd_precondition_2x2_block_to_be_real [friend]
        

      







Definition at line 535 of file JacobiSVD.h.







Member Data Documentation








template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::m_cols [protected]
        

      







Definition at line 532 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          unsigned int JacobiSVD< _MatrixType, QRPreconditioner >::m_computationOptions [protected]
        

      







Definition at line 531 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullU [protected]
        

      







Definition at line 529 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeFullV [protected]
        

      







Definition at line 530 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinU [protected]
        

      







Definition at line 529 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::m_computeThinV [protected]
        

      







Definition at line 530 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::m_diagSize [protected]
        

      







Definition at line 532 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::m_isAllocated [protected]
        

      







Definition at line 528 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          bool JacobiSVD< _MatrixType, QRPreconditioner >::m_isInitialized [protected]
        

      







Definition at line 528 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          MatrixUType JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixU [protected]
        

      







Definition at line 524 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          MatrixVType JacobiSVD< _MatrixType, QRPreconditioner >::m_matrixV [protected]
        

      







Definition at line 525 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::m_nonzeroSingularValues [protected]
        

      







Definition at line 532 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          Index JacobiSVD< _MatrixType, QRPreconditioner >::m_rows [protected]
        

      







Definition at line 532 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          SingularValuesType JacobiSVD< _MatrixType, QRPreconditioner >::m_singularValues [protected]
        

      







Definition at line 526 of file JacobiSVD.h.













template<typename _MatrixType, int QRPreconditioner> 

      

        

          WorkMatrixType JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix [protected]
        

      







Definition at line 527 of file JacobiSVD.h.







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







libicr

Author(s): Robert Krug

autogenerated on Mon Jan 6 2014 11:34:21