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Eigen::LLT< _MatrixType, _UpLo > Class Template Reference
Dense linear problems and decompositions » Reference » Cholesky_Module

Standard Cholesky decomposition (LL^T) of a matrix and associated features. More...

#include <LLT.h>

Public Types

enum  { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
 
enum  { PacketSize = internal::packet_traits<Scalar>::size, AlignmentMask = int(PacketSize)-1, UpLo = _UpLo }
 
typedef _MatrixType MatrixType
 
typedef MatrixType::Scalar Scalar
 
typedef NumTraits< typename MatrixType::Scalar >::Real RealScalar
 
typedef MatrixType::Index Index
 
typedef internal::LLT_Traits< MatrixType, UpLo > Traits
 

Public Member Functions

 LLT ()
 Default Constructor. More...
 
 LLT (Index size)
 Default Constructor with memory preallocation. More...
 
 LLT (const MatrixType &matrix)
 
Traits::MatrixU matrixU () const
 
Traits::MatrixL matrixL () const
 
template<typename Rhs >
const internal::solve_retval< LLT, Rhs > solve (const MatrixBase< Rhs > &b) const
 
template<typename Derived >
void solveInPlace (MatrixBase< Derived > &bAndX) const
 
LLT & compute (const MatrixType &matrix)
 Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix. More...
 
const MatrixType & matrixLLT () const
 
MatrixType reconstructedMatrix () const
 
ComputationInfo info () const
 Reports whether previous computation was successful. More...
 
Index rows () const
 
Index cols () const
 
template<typename VectorType >
LLT rankUpdate (const VectorType &vec, const RealScalar &sigma=1)
 
template<typename VectorType >
LLT< _MatrixType, _UpLo > rankUpdate (const VectorType &v, const RealScalar &sigma)
 Performs a rank one update (or dowdate) of the current decomposition. More...
 

Static Protected Member Functions

static void check_template_parameters ()
 

Protected Attributes

MatrixType m_matrix
 
bool m_isInitialized
 
ComputationInfo m_info
 

Detailed Description

template<typename _MatrixType, int _UpLo>
class Eigen::LLT< _MatrixType, _UpLo >

Standard Cholesky decomposition (LL^T) of a matrix and associated features.

Parameters
MatrixTypethe type of the matrix of which we are computing the LL^T Cholesky decomposition
UpLothe triangular part that will be used for the decompositon: Lower (default) or Upper. The other triangular part won't be read.

This class performs a LL^T Cholesky decomposition of a symmetric, positive definite matrix A such that A = LL^* = U^*U, where L is lower triangular.

While the Cholesky decomposition is particularly useful to solve selfadjoint problems like D^*D x = b, for that purpose, we recommend the Cholesky decomposition without square root which is more stable and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other situations like generalised eigen problems with hermitian matrices.

Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices, use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations has a solution.

Example:

Output: 

See also
MatrixBase::llt(), class LDLT 


Constructor & Destructor Documentation



§ LLT() [1/2]








template<typename _MatrixType, int _UpLo> 



  

  
      

        

          Eigen::LLT< _MatrixType, _UpLo >::LLT 
          (
          )
          
        

      

  		
  
inline  
  







Default Constructor. 


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








§ LLT() [2/2]








template<typename _MatrixType, int _UpLo> 



  

  
      

        

          Eigen::LLT< _MatrixType, _UpLo >::LLT 
          (
          Index 
          size)
          
        

      

  		
  
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
LLT() 







Member Function Documentation



§ compute()








template<typename MatrixType , int _UpLo> 

      

        

          LLT< MatrixType, _UpLo > & Eigen::LLT< MatrixType, _UpLo >::compute 
          (
          const MatrixType & 
          a)
          
        

      





Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of matrix. 


Returns
a reference to *this


Example: 
 Output: 
 






§ info()








template<typename _MatrixType, int _UpLo> 



  

  
      

        

          ComputationInfo Eigen::LLT< _MatrixType, _UpLo >::info 
          (
          )
           const
        

      

  		
  
inline  
  







Reports whether previous computation was successful. 


Returns
Success if computation was succesful, NumericalIssue if the matrix.appears to be negative. 








§ matrixL()








template<typename _MatrixType, int _UpLo> 



  

  
      

        

          Traits::MatrixL Eigen::LLT< _MatrixType, _UpLo >::matrixL 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
a view of the lower triangular matrix L 








§ matrixLLT()








template<typename _MatrixType, int _UpLo> 



  

  
      

        

          const MatrixType& Eigen::LLT< _MatrixType, _UpLo >::matrixLLT 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
the LLT decomposition matrix


TODO: document the storage layout 








§ matrixU()








template<typename _MatrixType, int _UpLo> 



  

  
      

        

          Traits::MatrixU Eigen::LLT< _MatrixType, _UpLo >::matrixU 
          (
          )
           const
        

      

  		
  
inline  
  






Returns
a view of the upper triangular matrix U 








§ rankUpdate()








template<typename _MatrixType, int _UpLo> 



template<typename VectorType > 

      

        

          LLT<_MatrixType,_UpLo> Eigen::LLT< _MatrixType, _UpLo >::rankUpdate 
          (
          const VectorType & 
          v, 
        

        

          
          
          const RealScalar & 
          sigma 
        

        

          
          )
          
        

      





Performs a rank one update (or dowdate) of the current decomposition. 


If A = LL^* before the rank one update, then after it we have LL^* = A + sigma * v v^* where v must be a vector of same dimension. 








§ reconstructedMatrix()








template<typename MatrixType , int _UpLo> 

      

        

          MatrixType Eigen::LLT< MatrixType, _UpLo >::reconstructedMatrix 
          (
          )
           const
        

      




Returns
the matrix represented by the decomposition, i.e., it returns the product: L L^*. This function is provided for debug purpose. 








§ solve()








template<typename _MatrixType, int _UpLo> 



template<typename Rhs > 



  

  
      

        

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

      

  		
  
inline  
  






Returns
the solution x of \( A x = b \) using the current decomposition of A.


Since this LLT class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.


Example: 
 Output: 
See also
solveInPlace(), MatrixBase::llt() 







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





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