Public Types | Public Member Functions | Protected Attributes | List of all members
Eigen::FullPivLU< _MatrixType > Class Template Reference

LU decomposition of a matrix with complete pivoting, and related features. More...

#include <ForwardDeclarations.h>

Public Types

enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
}
 
typedef MatrixType::Index Index
 
typedef internal::plain_col_type< MatrixType, Index >::type IntColVectorType
 
typedef internal::plain_row_type< MatrixType, Index >::type IntRowVectorType
 
typedef _MatrixType MatrixType
 
typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime > PermutationPType
 
typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime > PermutationQType
 
typedef NumTraits< typename MatrixType::Scalar >::Real RealScalar
 
typedef MatrixType::Scalar Scalar
 
typedef internal::traits< MatrixType >::StorageKind StorageKind
 

Public Member Functions

Index cols () const
 
FullPivLU & compute (const MatrixType &matrix)
 
internal::traits< MatrixType >::Scalar determinant () const
 
Index dimensionOfKernel () const
 
 FullPivLU ()
 Default Constructor. More...
 
 FullPivLU (Index rows, Index cols)
 Default Constructor with memory preallocation. More...
 
 FullPivLU (const MatrixType &matrix)
 
const internal::image_retval< FullPivLU > image (const MatrixType &originalMatrix) const
 
const internal::solve_retval< FullPivLU, typename MatrixType::IdentityReturnType > inverse () const
 
bool isInjective () const
 
bool isInvertible () const
 
bool isSurjective () const
 
const internal::kernel_retval< FullPivLU > kernel () const
 
const MatrixType & matrixLU () const
 
RealScalar maxPivot () const
 
Index nonzeroPivots () const
 
const PermutationPType & permutationP () const
 
const PermutationQType & permutationQ () const
 
Index rank () const
 
MatrixType reconstructedMatrix () const
 
Index rows () const
 
FullPivLU & setThreshold (const RealScalar &threshold)
 
FullPivLU & setThreshold (Default_t)
 
template<typename Rhs >
const internal::solve_retval< FullPivLU, Rhs > solve (const MatrixBase< Rhs > &b) const
 
RealScalar threshold () const
 

Protected Attributes

IntRowVectorType m_colsTranspositions
 
Index m_det_pq
 
bool m_isInitialized
 
MatrixType m_lu
 
RealScalar m_maxpivot
 
Index m_nonzero_pivots
 
PermutationPType m_p
 
RealScalar m_prescribedThreshold
 
PermutationQType m_q
 
IntColVectorType m_rowsTranspositions
 
bool m_usePrescribedThreshold
 

Detailed Description

template<typename _MatrixType>
class Eigen::FullPivLU< _MatrixType >

LU decomposition of a matrix with complete pivoting, and related features.

Parameters
MatrixTypethe type of the matrix of which we are computing the LU decomposition

This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is decomposed as A = PLUQ where L is unit-lower-triangular, U is upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any zeros are at the end.

This decomposition provides the generic approach to solving systems of linear equations, computing the rank, invertibility, inverse, kernel, and determinant.

This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, working with the SVD allows to select the smallest singular values of the matrix, something that the LU decomposition doesn't see.

The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP(), permutationQ().

As an exemple, here is how the original matrix can be retrieved:

Output: 

See also
MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse() 



Definition at line 216 of file ForwardDeclarations.h.


Member Typedef Documentation








template<typename _MatrixType> 

      

        

          typedef MatrixType::Index Eigen::FullPivLU< _MatrixType >::Index
        

      





Definition at line 59 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef internal::plain_col_type<MatrixType, Index>::type Eigen::FullPivLU< _MatrixType >::IntColVectorType
        

      





Definition at line 61 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef internal::plain_row_type<MatrixType, Index>::type Eigen::FullPivLU< _MatrixType >::IntRowVectorType
        

      





Definition at line 60 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef _MatrixType Eigen::FullPivLU< _MatrixType >::MatrixType
        

      





Definition at line 48 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::FullPivLU< _MatrixType >::PermutationPType
        

      





Definition at line 63 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::FullPivLU< _MatrixType >::PermutationQType
        

      





Definition at line 62 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::FullPivLU< _MatrixType >::RealScalar
        

      





Definition at line 57 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef MatrixType::Scalar Eigen::FullPivLU< _MatrixType >::Scalar
        

      





Definition at line 56 of file FullPivLU.h.













template<typename _MatrixType> 

      

        

          typedef internal::traits<MatrixType>::StorageKind Eigen::FullPivLU< _MatrixType >::StorageKind
        

      





Definition at line 58 of file FullPivLU.h.







Member Enumeration Documentation








template<typename _MatrixType> 

      

        

          anonymous enum
        

      






Enumerator
RowsAtCompileTime 



ColsAtCompileTime 



Options 



MaxRowsAtCompileTime 



MaxColsAtCompileTime 






Definition at line 49 of file FullPivLU.h.







Constructor & Destructor Documentation








template<typename MatrixType > 

      

        

          Eigen::FullPivLU< MatrixType >::FullPivLU 
          (
          )
          
        

      





Default Constructor. 


The default constructor is useful in cases in which the user intends to perform decompositions via LU::compute(const MatrixType&). 



Definition at line 387 of file FullPivLU.h.













template<typename MatrixType > 

      

        

          Eigen::FullPivLU< MatrixType >::FullPivLU 
          (
          Index 
          rows, 
        

        

          
          
          Index 
          cols 
        

        

          
          )
          
        

      





Default Constructor with memory preallocation. 


Like the default constructor but with preallocation of the internal data according to the specified problem size. 
See also
FullPivLU() 



Definition at line 393 of file FullPivLU.h.













template<typename MatrixType > 

      

        

          Eigen::FullPivLU< MatrixType >::FullPivLU 
          (
          const MatrixType & 
          matrix)
          
        

      




Constructor.


Parameters

  

    
matrixthe matrix of which to compute the LU decomposition. It is required to be nonzero. 

  

  





Definition at line 405 of file FullPivLU.h.







Member Function Documentation








template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::cols 
          (
          void 
          )
           const
        

      

  		
  
inline  
  







Definition at line 373 of file FullPivLU.h.













template<typename MatrixType > 

      

        

          FullPivLU< MatrixType > & Eigen::FullPivLU< MatrixType >::compute 
          (
          const MatrixType & 
          matrix)
          
        

      




Computes the LU decomposition of the given matrix.


Parameters

  

    
matrixthe matrix of which to compute the LU decomposition. It is required to be nonzero.

  

  




Returns
a reference to *this 



Definition at line 418 of file FullPivLU.h.













template<typename MatrixType > 

      

        

          internal::traits< MatrixType >::Scalar Eigen::FullPivLU< MatrixType >::determinant 
          (
          )
           const
        

      




Returns
the determinant of the matrix of which *this is the LU decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the LU decomposition has already been computed.


Note
This is only for square matrices.



For fixed-size matrices of size up to 4, MatrixBase::determinant() offers optimized paths.


Warning
a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow.


See also
MatrixBase::determinant() 



Definition at line 506 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::dimensionOfKernel 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the dimension of the kernel of the matrix of which *this is the LU decomposition.


Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&). 



Definition at line 311 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          const internal::image_retval<FullPivLU> Eigen::FullPivLU< _MatrixType >::image 
          (
          const MatrixType & 
          originalMatrix)
           const
        

      

  		
  
inline  
  






Returns
the image of the matrix, also called its column-space. The columns of the returned matrix will form a basis of the kernel.


Parameters

  

    
originalMatrixthe original matrix, of which *this is the LU decomposition. The reason why it is needed to pass it here, is that this allows a large optimization, as otherwise this method would need to reconstruct it from the LU decomposition.

  

  




Note
If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.



This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).


Example: 
 Output: 
See also
kernel() 



Definition at line 187 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> Eigen::FullPivLU< _MatrixType >::inverse 
          (
          void 
          )
           const
        

      

  		
  
inline  
  






Returns
the inverse of the matrix of which *this is the LU decomposition.


Note
If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.


See also
MatrixBase::inverse() 



Definition at line 362 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::FullPivLU< _MatrixType >::isInjective 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
true if the matrix of which *this is the LU decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.


Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&). 



Definition at line 324 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::FullPivLU< _MatrixType >::isInvertible 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
true if the matrix of which *this is the LU decomposition is invertible.


Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&). 



Definition at line 349 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::FullPivLU< _MatrixType >::isSurjective 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
true if the matrix of which *this is the LU decomposition represents a surjective linear map; false otherwise.


Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&). 



Definition at line 337 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          const internal::kernel_retval<FullPivLU> Eigen::FullPivLU< _MatrixType >::kernel 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the kernel of the matrix, also called its null-space. The columns of the returned matrix will form a basis of the kernel.


Note
If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.



This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).


Example: 
 Output: 
See also
image() 



Definition at line 161 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          const MatrixType& Eigen::FullPivLU< _MatrixType >::matrixLU 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the LU decomposition matrix: the upper-triangular part is U, the unit-lower-triangular part is L (at least for square matrices; in the non-square case, special care is needed, see the documentation of class FullPivLU).


See also
matrixL(), matrixU() 



Definition at line 103 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::FullPivLU< _MatrixType >::maxPivot 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U. 



Definition at line 125 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::nonzeroPivots 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the number of nonzero pivots in the LU decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.


See also
rank() 



Definition at line 116 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          const PermutationPType& Eigen::FullPivLU< _MatrixType >::permutationP 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the permutation matrix P


See also
permutationQ() 



Definition at line 131 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          const PermutationQType& Eigen::FullPivLU< _MatrixType >::permutationQ 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the permutation matrix Q


See also
permutationP() 



Definition at line 141 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::rank 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the rank of the matrix of which *this is the LU decomposition.


Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&). 



Definition at line 294 of file FullPivLU.h.













template<typename MatrixType > 

      

        

          MatrixType Eigen::FullPivLU< MatrixType >::reconstructedMatrix 
          (
          )
           const
        

      




Returns
the matrix represented by the decomposition, i.e., it returns the product: P^{-1} L U Q^{-1}. This function is provided for debug purpose. 



Definition at line 517 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::rows 
          (
          void 
          )
           const
        

      

  		
  
inline  
  







Definition at line 372 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          FullPivLU& Eigen::FullPivLU< _MatrixType >::setThreshold 
          (
          const RealScalar & 
          threshold)
          
        

      

  		
  
inline  
  






Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. This is not used for the LU decomposition itself.


When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.


Parameters

  

    
thresholdThe new value to use as the threshold.

  

  




A pivot will be considered nonzero if its absolute value is strictly greater than $ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $ where maxpivot is the biggest pivot.


If you want to come back to the default behavior, call setThreshold(Default_t) 



Definition at line 254 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          FullPivLU& Eigen::FullPivLU< _MatrixType >::setThreshold 
          (
          Default_t 
          )
          
        

      

  		
  
inline  
  






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. 
lu.setThreshold(Eigen::Default); 
See the documentation of setThreshold(const RealScalar&). 



Definition at line 269 of file FullPivLU.h.













template<typename _MatrixType> 



template<typename Rhs > 



  

  
      

        

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

      

  		
  
inline  
  






Returns
a solution x to the equation Ax=b, where A is the matrix of which *this is the LU decomposition.


Parameters

  

    
bthe right-hand-side of the equation to solve. Can be a vector or a matrix, the only requirement in order for the equation to make sense is that b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.

  

  




Returns
a solution.


Example: 
 Output: 
See also
TriangularView::solve(), kernel(), inverse() 



Definition at line 214 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::FullPivLU< _MatrixType >::threshold 
          (
          )
           const
        

      

  		
  
inline  
  






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


See the documentation of setThreshold(const RealScalar&). 



Definition at line 279 of file FullPivLU.h.







Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          IntRowVectorType Eigen::FullPivLU< _MatrixType >::m_colsTranspositions
        

      

  		
  
protected  
  







Definition at line 380 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::m_det_pq
        

      

  		
  
protected  
  







Definition at line 381 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::FullPivLU< _MatrixType >::m_isInitialized
        

      

  		
  
protected  
  







Definition at line 383 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::FullPivLU< _MatrixType >::m_lu
        

      

  		
  
protected  
  







Definition at line 376 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::FullPivLU< _MatrixType >::m_maxpivot
        

      

  		
  
protected  
  







Definition at line 382 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::FullPivLU< _MatrixType >::m_nonzero_pivots
        

      

  		
  
protected  
  







Definition at line 381 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          PermutationPType Eigen::FullPivLU< _MatrixType >::m_p
        

      

  		
  
protected  
  







Definition at line 377 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::FullPivLU< _MatrixType >::m_prescribedThreshold
        

      

  		
  
protected  
  







Definition at line 382 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          PermutationQType Eigen::FullPivLU< _MatrixType >::m_q
        

      

  		
  
protected  
  







Definition at line 378 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          IntColVectorType Eigen::FullPivLU< _MatrixType >::m_rowsTranspositions
        

      

  		
  
protected  
  







Definition at line 379 of file FullPivLU.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::FullPivLU< _MatrixType >::m_usePrescribedThreshold
        

      

  		
  
protected  
  







Definition at line 383 of file FullPivLU.h.







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







acado

Author(s): Milan Vukov, Rien Quirynen

autogenerated on Mon Jun 10 2019 12:35:34