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Eigen  3.4.90 (git rev e3e74001f7c4bf95f0dde572e8a08c5b2918a3ab)
Eigen::LLT< MatrixType_, UpLo_ > Class Template Reference

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 decomposition: 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:

The matrix A is
 4 -1  2
-1  6  0
 2  0  5
The Cholesky factor L is
    2     0     0
 -0.5   2.4     0
    1 0.209  1.99
To check this, let us compute L * L.transpose()
 4 -1  2
-1  6  0
 2  0  5
This should equal the matrix A

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
+ Inheritance diagram for Eigen::LLT< MatrixType_, UpLo_ >:

Public Member Functions

const LLTadjoint () const EIGEN_NOEXCEPT
 
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...
 
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 MatrixType & matrixLLT () const
 
Traits::MatrixU matrixU () const
 
template<typename VectorType >
LLT< MatrixType_, UpLo_ > & rankUpdate (const VectorType &v, const RealScalar &sigma)
 
RealScalar rcond () const
 
MatrixType reconstructedMatrix () const
 
template<typename Rhs >
const Solve< LLT, Rhs > solve (const MatrixBase< Rhs > &b) const
 
- Public Member Functions inherited from Eigen::SolverBase< LLT< MatrixType_, UpLo_ > >
AdjointReturnType adjoint () const
 
LLT< MatrixType_, UpLo_ > & derived ()
 
const LLT< MatrixType_, UpLo_ > & derived () const
 
const Solve< LLT< MatrixType_, UpLo_ >, Rhs > solve (const MatrixBase< Rhs > &b) const
 
 SolverBase ()
 
ConstTransposeReturnType transpose () const
 
- Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_CONSTEXPR Index cols () const EIGEN_NOEXCEPT
 
Derived & derived ()
 
const Derived & derived () const
 
EIGEN_CONSTEXPR Index rows () const EIGEN_NOEXCEPT
 
EIGEN_CONSTEXPR Index size () const EIGEN_NOEXCEPT
 

Additional Inherited Members

- Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index Index
 The interface type of indices. More...
 

Constructor & Destructor Documentation

◆ LLT() [1/3]

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() [2/3]

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()

◆ LLT() [3/3]

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&)

Member Function Documentation

◆ 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)

◆ compute()

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;
}
Namespace containing all symbols from the Eigen library.
Definition: Core:141

Output:

Here is the matrix A:
 2 -1
-1  3
Here is the right hand side b:
1 2
3 1
Computing LLT decomposition...
The solution is:
1.2 1.4
1.4 0.8
The matrix A is now:
 2 -1
-1  4
Computing LLT decomposition...
The solution is now:
    1  1.29
    1 0.571

◆ 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.

◆ 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.

◆ 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.

◆ 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.

◆ solve()

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 \( A x = b \) 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:

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;
static const RandomReturnType Random()
Definition: Random.h:113

Output:

2.02
2.97
See also
solveInPlace(), MatrixBase::llt(), SelfAdjointView::llt()

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