Public Types | Public Member Functions | Protected Attributes

FullPivLU< _MatrixType > Class Template Reference

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

#include <FullPivLU.h>

List of all members.

Public Types

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

Public Member Functions

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

Protected Attributes

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

Detailed Description

template<typename _MatrixType>
class FullPivLU< _MatrixType >

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

Parameters:
MatrixType the 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() 


Member Typedef Documentation








template<typename _MatrixType> 

      

        

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

      

















template<typename _MatrixType> 

      

        

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

      







Reimplemented in LU< MatrixType >.













template<typename _MatrixType> 

      

        

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

      







Reimplemented in LU< MatrixType >.













template<typename _MatrixType> 

      

        

          typedef _MatrixType FullPivLU< _MatrixType >::MatrixType
        

      

















template<typename _MatrixType> 

      

        

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

      

















template<typename _MatrixType> 

      

        

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

      

















template<typename _MatrixType> 

      

        

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

      







Reimplemented in LU< MatrixType >.













template<typename _MatrixType> 

      

        

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

      







Reimplemented in LU< MatrixType >.













template<typename _MatrixType> 

      

        

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

      











Member Enumeration Documentation








template<typename _MatrixType> 

      

        

          anonymous enum
        

      






Enumerator: 


RowsAtCompileTime 



ColsAtCompileTime 



Options 



MaxRowsAtCompileTime 



MaxColsAtCompileTime 














Constructor & Destructor Documentation








template<typename _MatrixType> 

      

        

          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&). 













template<typename _MatrixType> 

      

        

          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() 













template<typename _MatrixType> 

      

        

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

      






Constructor.


Parameters:

  

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

  

  









Member Function Documentation








template<typename _MatrixType> 

      

        

          Index FullPivLU< _MatrixType >::cols 
          (
          void 
          
           ) 
           const [inline]
        

      

















template<typename _MatrixType> 

      

        

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

      






Computes the LU decomposition of the given matrix.


Parameters:

  

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

  

  




Returns:
a reference to *this 













template<typename _MatrixType> 

      

        

          internal::traits<MatrixType>::Scalar 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() 













template<typename _MatrixType> 

      

        

          Index 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&). 













template<typename _MatrixType> 

      

        

          const internal::image_retval<FullPivLU> 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:

  

    
originalMatrix the 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() 













template<typename _MatrixType> 

      

        

          const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> 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() 













template<typename _MatrixType> 

      

        

          bool 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&). 













template<typename _MatrixType> 

      

        

          bool 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&). 













template<typename _MatrixType> 

      

        

          bool 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&). 













template<typename _MatrixType> 

      

        

          const internal::kernel_retval<FullPivLU> 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() 













template<typename _MatrixType> 

      

        

          const MatrixType& 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() 













template<typename _MatrixType> 

      

        

          RealScalar FullPivLU< _MatrixType >::maxPivot 
          (
          
           ) 
           const [inline]
        

      






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













template<typename _MatrixType> 

      

        

          Index 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() 













template<typename _MatrixType> 

      

        

          const PermutationPType& FullPivLU< _MatrixType >::permutationP 
          (
          
           ) 
           const [inline]
        

      






Returns:
the permutation matrix P


See also:
permutationQ() 













template<typename _MatrixType> 

      

        

          const PermutationQType& FullPivLU< _MatrixType >::permutationQ 
          (
          
           ) 
           const [inline]
        

      






Returns:
the permutation matrix Q


See also:
permutationP() 













template<typename _MatrixType> 

      

        

          Index 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&). 













template<typename _MatrixType> 

      

        

          MatrixType FullPivLU< _MatrixType >::reconstructedMatrix 
          (
          
           ) 
           const
        

      

















template<typename _MatrixType> 

      

        

          Index FullPivLU< _MatrixType >::rows 
          (
          void 
          
           ) 
           const [inline]
        

      

















template<typename _MatrixType> 

      

        

          FullPivLU& 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&). 













template<typename _MatrixType> 

      

        

          FullPivLU& 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:

  

    
threshold The 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) 













template<typename _MatrixType> 



template<typename Rhs > 

      

        

          const internal::solve_retval<FullPivLU, Rhs> 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:

  

    
b the 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() 













template<typename _MatrixType> 

      

        

          RealScalar 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&). 







Member Data Documentation








template<typename _MatrixType> 

      

        

          IntRowVectorType FullPivLU< _MatrixType >::m_colsTranspositions [protected]
        

      

















template<typename _MatrixType> 

      

        

          Index FullPivLU< _MatrixType >::m_det_pq [protected]
        

      

















template<typename _MatrixType> 

      

        

          bool FullPivLU< _MatrixType >::m_isInitialized [protected]
        

      

















template<typename _MatrixType> 

      

        

          MatrixType FullPivLU< _MatrixType >::m_lu [protected]
        

      

















template<typename _MatrixType> 

      

        

          RealScalar FullPivLU< _MatrixType >::m_maxpivot [protected]
        

      

















template<typename _MatrixType> 

      

        

          Index FullPivLU< _MatrixType >::m_nonzero_pivots [protected]
        

      

















template<typename _MatrixType> 

      

        

          PermutationPType FullPivLU< _MatrixType >::m_p [protected]
        

      

















template<typename _MatrixType> 

      

        

          RealScalar FullPivLU< _MatrixType >::m_prescribedThreshold [protected]
        

      

















template<typename _MatrixType> 

      

        

          PermutationQType FullPivLU< _MatrixType >::m_q [protected]
        

      

















template<typename _MatrixType> 

      

        

          IntColVectorType FullPivLU< _MatrixType >::m_rowsTranspositions [protected]
        

      

















template<typename _MatrixType> 

      

        

          bool FullPivLU< _MatrixType >::m_usePrescribedThreshold [protected]
        

      











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



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