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

#include <LLT.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

enum	{ PacketSize = internal::packet_traits<Scalar>::size, AlignmentMask = int(PacketSize)-1, UpLo = _UpLo }

typedef Eigen::Index	Index

typedef _MatrixType	MatrixType

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

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

typedef internal::LLT_Traits< MatrixType, UpLo >	Traits

Public Member Functions
template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const

template<typename RhsType , typename DstType >
void	_solve_impl (const RhsType &rhs, DstType &dst) const

const LLT &	adjoint () const

Index	cols () const

template<typename InputType >
LLT &	compute (const EigenBase< InputType > &matrix)

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

ComputationInfo	info () const
	Reports whether previous computation was successful. More...

	LLT ()
	Default Constructor. More...

	LLT (Index size)
	Default Constructor with memory preallocation. More...

template<typename InputType >
	LLT (const EigenBase< InputType > &matrix)

template<typename InputType >
	LLT (EigenBase< InputType > &matrix)
	Constructs a LDLT factorization from a given matrix. More...

Traits::MatrixL	matrixL () const

const MatrixType &	matrixLLT () const

Traits::MatrixU	matrixU () 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)

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

Index	rows () const

template<typename Rhs >
const Solve< LLT, Rhs >	solve (const MatrixBase< Rhs > &b) const

template<typename Derived >
void	solveInPlace (const MatrixBase< Derived > &bAndX) 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

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

_MatrixType	the type of the matrix of which we are computing the LL^T Cholesky decomposition
_UpLo	the 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 56 of file LLT.h.

Member Typedef Documentation

template<typename _MatrixType, int _UpLo>

typedef Eigen::Index Eigen::LLT< _MatrixType, _UpLo >::Index

Deprecated:: since Eigen 3.3

Definition at line 67 of file LLT.h.

template<typename _MatrixType, int _UpLo>

typedef _MatrixType Eigen::LLT< _MatrixType, _UpLo >::MatrixType

Definition at line 59 of file LLT.h.

template<typename _MatrixType, int _UpLo>

typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::LLT< _MatrixType, _UpLo >::RealScalar

Definition at line 66 of file LLT.h.

template<typename _MatrixType, int _UpLo>

typedef MatrixType::Scalar Eigen::LLT< _MatrixType, _UpLo >::Scalar

Definition at line 65 of file LLT.h.

template<typename _MatrixType, int _UpLo>

typedef MatrixType::StorageIndex Eigen::LLT< _MatrixType, _UpLo >::StorageIndex

Definition at line 68 of file LLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 76 of file LLT.h.

Member Enumeration Documentation

template<typename _MatrixType, int _UpLo>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxColsAtCompileTime

Definition at line 60 of file LLT.h.

template<typename _MatrixType, int _UpLo>

anonymous enum

Enumerator
PacketSize
AlignmentMask
UpLo

Definition at line 70 of file LLT.h.

Constructor & Destructor Documentation

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 84 of file LLT.h.

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 92 of file LLT.h.

template<typename _MatrixType, int _UpLo>

template<typename InputType >

Eigen::LLT< _MatrixType, _UpLo >::LLT ( const EigenBase< InputType > & matrix )

inlineexplicit

Definition at line 96 of file LLT.h.

template<typename _MatrixType, int _UpLo>

template<typename InputType >

Eigen::LLT< _MatrixType, _UpLo >::LLT ( EigenBase< InputType > & matrix )

inlineexplicit

Constructs a LDLT 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 111 of file LLT.h.

Member Function Documentation

template<typename _MatrixType, int _UpLo>

template<typename RhsType , typename DstType >

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

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 478 of file LLT.h.

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 197 of file LLT.h.

template<typename _MatrixType, int _UpLo>

static void Eigen::LLT< _MatrixType, _UpLo >::check_template_parameters ( )

inlinestaticprotected

Definition at line 213 of file LLT.h.

template<typename _MatrixType, int _UpLo>

Index Eigen::LLT< _MatrixType, _UpLo >::cols ( ) const

inline

Definition at line 200 of file LLT.h.

template<typename _MatrixType, int _UpLo>

template<typename InputType >

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

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;
   LLT<Matrix2f> llt;
   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 425 of file LLT.h.

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 not to be positive definite.

Definition at line 186 of file LLT.h.

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 126 of file LLT.h.

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 172 of file LLT.h.

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 119 of file LLT.h.

template<typename _MatrixType, int _UpLo>

template<typename VectorType >

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

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 462 of file LLT.h.

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 161 of file LLT.h.

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.

Definition at line 512 of file LLT.h.

template<typename _MatrixType, int _UpLo>

Index Eigen::LLT< _MatrixType, _UpLo >::rows ( ) const

inline

Definition at line 199 of file LLT.h.

template<typename _MatrixType, int _UpLo>

template<typename Rhs >

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

inline

Returns: the solution x of 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:

typedef Matrix<float,Dynamic,2> DataMatrix;
// let's generate some samples on the 3D plane of equation z = 2x+3y (with some noise)
DataMatrix samples = DataMatrix::Random(12,2);
VectorXf elevations = 2*samples.col(0) + 3*samples.col(1) + VectorXf::Random(12)*0.1;
// and let's solve samples * [x y]^T = elevations in least square sense:
Matrix<float,2,1> xy
 = (samples.adjoint() * samples).llt().solve((samples.adjoint()*elevations));
cout << xy << endl;

Output:

See also: solveInPlace(), MatrixBase::llt(), SelfAdjointView::llt()

Definition at line 144 of file LLT.h.

template<typename MatrixType , int _UpLo>

template<typename Derived >

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

Definition at line 500 of file LLT.h.

Member Data Documentation

template<typename _MatrixType, int _UpLo>

ComputationInfo Eigen::LLT< _MatrixType, _UpLo >::m_info

protected

Definition at line 225 of file LLT.h.

template<typename _MatrixType, int _UpLo>

bool Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized

protected

Definition at line 224 of file LLT.h.

template<typename _MatrixType, int _UpLo>

RealScalar Eigen::LLT< _MatrixType, _UpLo >::m_l1_norm

protected

Definition at line 223 of file LLT.h.

template<typename _MatrixType, int _UpLo>

MatrixType Eigen::LLT< _MatrixType, _UpLo >::m_matrix

protected

Definition at line 222 of file LLT.h.

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

LLT.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

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

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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