Public Types | Public Member Functions | Static Protected Member Functions | Protected Attributes | Friends | List of all members
Eigen::LLT< _MatrixType, _UpLo > Class Template Reference

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

#include <LLT.h>

Inheritance diagram for Eigen::LLT< _MatrixType, _UpLo >:
Inheritance graph
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Public Types

enum  { MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
 
enum  { PacketSize = internal::packet_traits<Scalar>::size, AlignmentMask = int(PacketSize)-1, UpLo = _UpLo }
 
typedef SolverBase< LLTBase
 
typedef _MatrixType MatrixType
 
typedef internal::LLT_Traits< MatrixType, UpLoTraits
 
- Public Types inherited from Eigen::SolverBase< LLT< _MatrixType, _UpLo > >
enum  
 
typedef internal::conditional< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, ConstTransposeReturnType >, ConstTransposeReturnType >::type AdjointReturnType
 
typedef EigenBase< LLT< _MatrixType, _UpLo > > Base
 
typedef Scalar CoeffReturnType
 
typedef internal::add_const< Transpose< const LLT< _MatrixType, _UpLo > > >::type ConstTransposeReturnType
 
typedef internal::traits< LLT< _MatrixType, _UpLo > >::Scalar Scalar
 
- Public Types inherited from Eigen::EigenBase< LLT< _MatrixType, _UpLo > >
typedef Eigen::Index Index
 The interface type of indices. More...
 
typedef internal::traits< LLT< _MatrixType, _UpLo > >::StorageKind StorageKind
 

Public Member Functions

template<typename RhsType , typename DstType >
void _solve_impl (const RhsType &rhs, DstType &dst) const
 
template<bool Conjugate, typename RhsType , typename DstType >
void _solve_impl_transposed (const RhsType &rhs, DstType &dst) const
 
const LLTadjoint () const EIGEN_NOEXCEPT
 
EIGEN_CONSTEXPR Index cols () const EIGEN_NOEXCEPT
 
template<typename InputType >
LLT< MatrixType, _UpLo > & compute (const EigenBase< InputType > &a)
 
template<typename InputType >
LLTcompute (const EigenBase< InputType > &matrix)
 
ComputationInfo info () const
 Reports whether previous computation was successful. More...
 
 LLT ()
 Default Constructor. More...
 
template<typename InputType >
 LLT (const EigenBase< InputType > &matrix)
 
template<typename InputType >
 LLT (EigenBase< InputType > &matrix)
 Constructs a LLT factorization from a given matrix. More...
 
 LLT (Index size)
 Default Constructor with memory preallocation. More...
 
Traits::MatrixL matrixL () const
 
const MatrixTypematrixLLT () const
 
Traits::MatrixU matrixU () const
 
template<typename VectorType >
LLT< _MatrixType, _UpLo > & rankUpdate (const VectorType &v, const RealScalar &sigma)
 
template<typename VectorType >
LLTrankUpdate (const VectorType &vec, const RealScalar &sigma=1)
 
RealScalar rcond () const
 
MatrixType reconstructedMatrix () const
 
EIGEN_CONSTEXPR Index rows () const EIGEN_NOEXCEPT
 
template<typename Derived >
void solveInPlace (const MatrixBase< Derived > &bAndX) const
 
- Public Member Functions inherited from Eigen::SolverBase< LLT< _MatrixType, _UpLo > >
AdjointReturnType adjoint () const
 
EIGEN_DEVICE_FUNC LLT< _MatrixType, _UpLo > & derived ()
 
const EIGEN_DEVICE_FUNC LLT< _MatrixType, _UpLo > & derived () const
 
const Solve< LLT< _MatrixType, _UpLo >, Rhs > solve (const MatrixBase< Rhs > &b) const
 
 SolverBase ()
 
ConstTransposeReturnType transpose () const
 
 ~SolverBase ()
 
- Public Member Functions inherited from Eigen::EigenBase< LLT< _MatrixType, _UpLo > >
EIGEN_DEVICE_FUNC void addTo (Dest &dst) const
 
EIGEN_DEVICE_FUNC void applyThisOnTheLeft (Dest &dst) const
 
EIGEN_DEVICE_FUNC void applyThisOnTheRight (Dest &dst) const
 
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index cols () const EIGEN_NOEXCEPT
 
EIGEN_DEVICE_FUNC LLT< _MatrixType, _UpLo > & const_cast_derived () const
 
const EIGEN_DEVICE_FUNC LLT< _MatrixType, _UpLo > & const_derived () const
 
EIGEN_DEVICE_FUNC LLT< _MatrixType, _UpLo > & derived ()
 
const EIGEN_DEVICE_FUNC LLT< _MatrixType, _UpLo > & derived () const
 
EIGEN_DEVICE_FUNC void evalTo (Dest &dst) const
 
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index rows () const EIGEN_NOEXCEPT
 
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index size () const EIGEN_NOEXCEPT
 
EIGEN_DEVICE_FUNC void subTo (Dest &dst) const
 

Static Protected Member Functions

static void check_template_parameters ()
 

Protected Attributes

ComputationInfo m_info
 
bool m_isInitialized
 
RealScalar m_l1_norm
 
MatrixType m_matrix
 

Friends

class SolverBase< LLT >
 

Additional Inherited Members

- Protected Member Functions inherited from Eigen::SolverBase< LLT< _MatrixType, _UpLo > >
void _check_solve_assertion (const Rhs &b) const
 

Detailed Description

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

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

Template 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:

MatrixXd A(3,3);
A << 4,-1,2, -1,6,0, 2,0,5;
cout << "The matrix A is" << endl << A << endl;
LLT<MatrixXd> lltOfA(A); // compute the Cholesky decomposition of A
MatrixXd L = lltOfA.matrixL(); // retrieve factor L in the decomposition
// The previous two lines can also be written as "L = A.llt().matrixL()"
cout << "The Cholesky factor L is" << endl << L << endl;
cout << "To check this, let us compute L * L.transpose()" << endl;
cout << L * L.transpose() << endl;
cout << "This should equal the matrix A" << endl;

Output:

Performance: for best performance, it is recommended to use a column-major storage format with the Lower triangular part (the default), or, equivalently, a row-major storage format with the Upper triangular part. Otherwise, you might get a 20% slowdown for the full factorization step, and rank-updates can be up to 3 times slower.

This class supports the inplace decomposition mechanism.

Note that during the decomposition, only the lower (or upper, as defined by _UpLo) triangular part of A is considered. Therefore, the strict lower part does not have to store correct values.

See also
MatrixBase::llt(), SelfAdjointView::llt(), class LDLT

Definition at line 66 of file LLT.h.

Member Typedef Documentation

◆ Base

template<typename _MatrixType , int _UpLo>
typedef SolverBase<LLT> Eigen::LLT< _MatrixType, _UpLo >::Base

Definition at line 71 of file LLT.h.

◆ MatrixType

template<typename _MatrixType , int _UpLo>
typedef _MatrixType Eigen::LLT< _MatrixType, _UpLo >::MatrixType

Definition at line 70 of file LLT.h.

◆ Traits

template<typename _MatrixType , int _UpLo>
typedef internal::LLT_Traits<MatrixType,UpLo> Eigen::LLT< _MatrixType, _UpLo >::Traits

Definition at line 85 of file LLT.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType , int _UpLo>
anonymous enum
Enumerator
MaxColsAtCompileTime 

Definition at line 75 of file LLT.h.

◆ anonymous enum

template<typename _MatrixType , int _UpLo>
anonymous enum
Enumerator
PacketSize 
AlignmentMask 
UpLo 

Definition at line 79 of file LLT.h.

Constructor & Destructor Documentation

◆ LLT() [1/4]

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&).

Definition at line 93 of file LLT.h.

◆ LLT() [2/4]

template<typename _MatrixType , int _UpLo>
Eigen::LLT< _MatrixType, _UpLo >::LLT ( Index  size)
inlineexplicit

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()

Definition at line 101 of file LLT.h.

◆ LLT() [3/4]

template<typename _MatrixType , int _UpLo>
template<typename InputType >
Eigen::LLT< _MatrixType, _UpLo >::LLT ( const EigenBase< InputType > &  matrix)
inlineexplicit

Definition at line 105 of file LLT.h.

◆ LLT() [4/4]

template<typename _MatrixType , int _UpLo>
template<typename InputType >
Eigen::LLT< _MatrixType, _UpLo >::LLT ( EigenBase< InputType > &  matrix)
inlineexplicit

Constructs a LLT factorization from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also
LLT(const EigenBase&)

Definition at line 120 of file LLT.h.

Member Function Documentation

◆ _solve_impl()

template<typename _MatrixType , int _UpLo>
template<typename RhsType , typename DstType >
void Eigen::LLT< _MatrixType, _UpLo >::_solve_impl ( const RhsType &  rhs,
DstType &  dst 
) const

Definition at line 485 of file LLT.h.

◆ _solve_impl_transposed()

template<typename _MatrixType , int _UpLo>
template<bool Conjugate, typename RhsType , typename DstType >
void Eigen::LLT< _MatrixType, _UpLo >::_solve_impl_transposed ( const RhsType &  rhs,
DstType &  dst 
) const

Definition at line 492 of file LLT.h.

◆ adjoint()

template<typename _MatrixType , int _UpLo>
const LLT& Eigen::LLT< _MatrixType, _UpLo >::adjoint ( ) const
inline
Returns
the adjoint of *this, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.

This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:

x = decomposition.adjoint().solve(b)

Definition at line 202 of file LLT.h.

◆ check_template_parameters()

template<typename _MatrixType , int _UpLo>
static void Eigen::LLT< _MatrixType, _UpLo >::check_template_parameters ( )
inlinestaticprotected

Definition at line 220 of file LLT.h.

◆ cols()

template<typename _MatrixType , int _UpLo>
EIGEN_CONSTEXPR Index Eigen::LLT< _MatrixType, _UpLo >::cols ( ) const
inline

Definition at line 205 of file LLT.h.

◆ compute() [1/2]

template<typename _MatrixType , int _UpLo>
template<typename InputType >
LLT<MatrixType,_UpLo>& Eigen::LLT< _MatrixType, _UpLo >::compute ( const EigenBase< InputType > &  a)

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

Returns
a reference to *this

Example:

#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main()
{
Matrix2f A, b;
A << 2, -1, -1, 3;
b << 1, 2, 3, 1;
cout << "Here is the matrix A:\n" << A << endl;
cout << "Here is the right hand side b:\n" << b << endl;
cout << "Computing LLT decomposition..." << endl;
llt.compute(A);
cout << "The solution is:\n" << llt.solve(b) << endl;
A(1,1)++;
cout << "The matrix A is now:\n" << A << endl;
cout << "Computing LLT decomposition..." << endl;
llt.compute(A);
cout << "The solution is now:\n" << llt.solve(b) << endl;
}

Output:

 

Definition at line 432 of file LLT.h.

◆ compute() [2/2]

template<typename _MatrixType , int _UpLo>
template<typename InputType >
LLT& Eigen::LLT< _MatrixType, _UpLo >::compute ( const EigenBase< InputType > &  matrix)

◆ 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 successful, NumericalIssue if the matrix.appears not to be positive definite.

Definition at line 191 of file LLT.h.

◆ matrixL()

template<typename _MatrixType , int _UpLo>
Traits::MatrixL Eigen::LLT< _MatrixType, _UpLo >::matrixL ( ) const
inline
Returns
a view of the lower triangular matrix L

Definition at line 135 of file LLT.h.

◆ 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

Definition at line 177 of file LLT.h.

◆ matrixU()

template<typename _MatrixType , int _UpLo>
Traits::MatrixU Eigen::LLT< _MatrixType, _UpLo >::matrixU ( ) const
inline
Returns
a view of the upper triangular matrix U

Definition at line 128 of file LLT.h.

◆ rankUpdate() [1/2]

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.

Definition at line 469 of file LLT.h.

◆ rankUpdate() [2/2]

template<typename _MatrixType , int _UpLo>
template<typename VectorType >
LLT& Eigen::LLT< _MatrixType, _UpLo >::rankUpdate ( const VectorType vec,
const RealScalar sigma = 1 
)

◆ rcond()

template<typename _MatrixType , int _UpLo>
RealScalar Eigen::LLT< _MatrixType, _UpLo >::rcond ( ) const
inline
Returns
an estimate of the reciprocal condition number of the matrix of which *this is the Cholesky decomposition.

Definition at line 166 of file LLT.h.

◆ reconstructedMatrix()

template<typename MatrixType , int _UpLo>
MatrixType Eigen::LLT< MatrixType, _UpLo >::reconstructedMatrix
Returns
the matrix represented by the decomposition, i.e., it returns the product: L L^*. This function is provided for debug purpose.

Definition at line 528 of file LLT.h.

◆ rows()

template<typename _MatrixType , int _UpLo>
EIGEN_CONSTEXPR Index Eigen::LLT< _MatrixType, _UpLo >::rows ( ) const
inline

Definition at line 204 of file LLT.h.

◆ solveInPlace()

template<typename MatrixType , int _UpLo>
template<typename Derived >
void Eigen::LLT< MatrixType, _UpLo >::solveInPlace ( const MatrixBase< Derived > &  bAndX) const

Definition at line 516 of file LLT.h.

Friends And Related Function Documentation

◆ SolverBase< LLT >

template<typename _MatrixType , int _UpLo>
friend class SolverBase< LLT >
friend

Definition at line 72 of file LLT.h.

Member Data Documentation

◆ m_info

template<typename _MatrixType , int _UpLo>
ComputationInfo Eigen::LLT< _MatrixType, _UpLo >::m_info
protected

Definition at line 232 of file LLT.h.

◆ m_isInitialized

template<typename _MatrixType , int _UpLo>
bool Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized
protected

Definition at line 231 of file LLT.h.

◆ m_l1_norm

template<typename _MatrixType , int _UpLo>
RealScalar Eigen::LLT< _MatrixType, _UpLo >::m_l1_norm
protected

Definition at line 230 of file LLT.h.

◆ m_matrix

template<typename _MatrixType , int _UpLo>
MatrixType Eigen::LLT< _MatrixType, _UpLo >::m_matrix
protected

Definition at line 229 of file LLT.h.


The documentation for this class was generated from the following file:
lltOfA
A<< 4,-1, 2, -1, 6, 0, 2, 0, 5;cout<< "The matrix A is"<< endl<< A<< endl;LLT< MatrixXd > lltOfA(A)
Eigen
Namespace containing all symbols from the Eigen library.
Definition: jet.h:637
llt
EIGEN_DONT_INLINE void llt(const Mat &A, const Mat &B, Mat &C)
Definition: llt.cpp:5
b
Scalar * b
Definition: benchVecAdd.cpp:17
x
set noclip points set clip one set noclip two set bar set border lt lw set xdata set ydata set zdata set x2data set y2data set boxwidth set dummy x
Definition: gnuplot_common_settings.hh:12
main
int main(int argc, char **argv)
Definition: cmake/example_cmake_find_gtsam/main.cpp:63
A
Matrix< SCALARA, Dynamic, Dynamic, opt_A > A
Definition: bench_gemm.cpp:48
A
Definition: test_numpy_dtypes.cpp:298
L
MatrixXd L
Definition: LLT_example.cpp:6
Eigen::LLT
Standard Cholesky decomposition (LL^T) of a matrix and associated features.
Definition: LLT.h:66
std
Definition: BFloat16.h:88


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autogenerated on Fri Nov 1 2024 03:44:56