 Eigen  3.3.7 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 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:

```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.

## Public Types

typedef Eigen::Index Index

## Public Member Functions

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 LDLT 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

## ◆ Index

template<typename _MatrixType , int _UpLo>
 typedef Eigen::Index Eigen::LLT< _MatrixType, _UpLo >::Index
Deprecated:
since Eigen 3.3

## ◆ 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.

LLT()

## ◆ LLT() [3/3]

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.

LLT(const EigenBase&)

## Member Function Documentation

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:

## ◆ compute()

template<typename _MatrixType , int _UpLo>
template<typename InputType >
 LLT& 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:

```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 succesful, `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 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
cout << xy << endl;

Output:

```2.02
2.97
```

The documentation for this class was generated from the following file:
Eigen
Namespace containing all symbols from the Eigen library.
Definition: Core:306
Eigen::DenseBase::Random
static const RandomReturnType Random()
Definition: Random.h:113
Eigen::LLT
Standard Cholesky decomposition (LL^T) of a matrix and associated features.
Definition: LLT.h:56
Eigen::Matrix
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:178
Eigen::LLT::solve
const Solve< LLT, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: LLT.h:144