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

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

Public Types
enum	{ MaxColsAtCompileTime }

enum	{ PacketSize , AlignmentMask , UpLo }

typedef SolverBase< LLT >	Base

typedef MatrixType_	MatrixType

typedef internal::LLT_Traits< MatrixType, UpLo >	Traits

Public Types inherited from Eigen::SolverBase< LLT< MatrixType_, UpLo_ > >
enum

typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType

typedef EigenBase< LLT< MatrixType_, UpLo_ > >	Base

typedef Scalar	CoeffReturnType

typedef Transpose< const LLT< MatrixType_, UpLo_ > >	ConstTransposeReturnType

typedef internal::traits< LLT< MatrixType_, UpLo_ > >::Scalar	Scalar

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

typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
const LLT &	adjoint () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

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

template<typename InputType >
LLT &	compute (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 MatrixType &	matrixLLT () const

Traits::MatrixU	matrixU () const

template<typename VectorType >
LLT< MatrixType_, UpLo_ > &	rankUpdate (const VectorType &v, const RealScalar &sigma)

template<typename VectorType >
LLT &	rankUpdate (const VectorType &vec, const RealScalar &sigma=1)

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

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

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

Public Member Functions inherited from Eigen::SolverBase< LLT< MatrixType_, UpLo_ > >
const 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 ()

const ConstTransposeReturnType	transpose () const

	~SolverBase ()

Public Member Functions inherited from Eigen::EigenBase< Derived >
template<typename Dest >
void	addTo (Dest &dst) const

template<typename Dest >
void	applyThisOnTheLeft (Dest &dst) const

template<typename Dest >
void	applyThisOnTheRight (Dest &dst) const

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

Derived &	const_cast_derived () const

const Derived &	const_derived () const

Derived &	derived ()

const Derived &	derived () const

template<typename Dest >
void	evalTo (Dest &dst) const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

template<typename Dest >
void	subTo (Dest &dst) const

Protected Attributes
ComputationInfo	m_info

bool	m_isInitialized

RealScalar	m_l1_norm

MatrixType	m_matrix

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

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

Definition at line 68 of file LLT.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , int UpLo_>

typedef SolverBase<LLT> Eigen::LLT< MatrixType_, UpLo_ >::Base

Definition at line 73 of file LLT.h.

◆ MatrixType

template<typename MatrixType_ , int UpLo_>

typedef MatrixType_ Eigen::LLT< MatrixType_, UpLo_ >::MatrixType

Definition at line 72 of file LLT.h.

◆ Traits

template<typename MatrixType_ , int UpLo_>

typedef internal::LLT_Traits<MatrixType,UpLo> Eigen::LLT< MatrixType_, UpLo_ >::Traits

Definition at line 87 of file LLT.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , int UpLo_>

anonymous enum

Enumerator
MaxColsAtCompileTime

Definition at line 77 of file LLT.h.

          {
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     };

◆ anonymous enum

template<typename MatrixType_ , int UpLo_>

anonymous enum

Enumerator
PacketSize
AlignmentMask
UpLo

Definition at line 81 of file LLT.h.

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

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 95 of file LLT.h.

95 : m_matrix(), m_isInitialized(false) {}

Eigen::LLT::m_matrix

MatrixType m_matrix

Definition: LLT.h:228

Eigen::LLT::m_isInitialized

bool m_isInitialized

Definition: LLT.h:230

◆ 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 103 of file LLT.h.

103 : m_matrix(size, size),

104 m_isInitialized(false) {}

Eigen::EigenBase::size

EIGEN_CONSTEXPR Index size() const EIGEN_NOEXCEPT

Definition: EigenBase.h:69

◆ LLT() [3/4]

template<typename MatrixType_ , int UpLo_>

template<typename InputType >

Eigen::LLT< MatrixType_, UpLo_ >::LLT ( const EigenBase< InputType > & matrix )

inlineexplicit

Definition at line 107 of file LLT.h.

       : m_matrix(matrix.rows(), matrix.cols()),
         m_isInitialized(false)
     {
       compute(matrix.derived());
     }

◆ 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 122 of file LLT.h.

       : m_matrix(matrix.derived()),
         m_isInitialized(false)
     {
       compute(matrix.derived());
     }

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)

b

Array< int, 3, 1 > b

Definition: Array_variadic_ctor_cxx11.cpp:2

x

Definition: BiCGSTAB_simple.cpp:7

Definition at line 204 of file LLT.h.

204 { return *this; }

◆ cols()

template<typename MatrixType_ , int UpLo_>

EIGEN_CONSTEXPR Index Eigen::LLT< MatrixType_, UpLo_ >::cols ( ) const

inline

Definition at line 207 of file LLT.h.

207 { return m_matrix.cols(); }

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

Definition at line 431 of file LLT.h.

 {
   eigen_assert(a.rows()==a.cols());
   const Index size = a.rows();
   m_matrix.resize(size, size);
   if (!internal::is_same_dense(m_matrix, a.derived()))
     m_matrix = a.derived();
  
   // Compute matrix L1 norm = max abs column sum.
   m_l1_norm = RealScalar(0);
   // TODO move this code to SelfAdjointView
   for (Index col = 0; col < size; ++col) {
     RealScalar abs_col_sum;
     if (UpLo_ == Lower)
       abs_col_sum = m_matrix.col(col).tail(size - col).template lpNorm<1>() + m_matrix.row(col).head(col).template lpNorm<1>();
     else
       abs_col_sum = m_matrix.col(col).head(col).template lpNorm<1>() + m_matrix.row(col).tail(size - col).template lpNorm<1>();
     if (abs_col_sum > m_l1_norm)
       m_l1_norm = abs_col_sum;
   }
  
   m_isInitialized = true;
   bool ok = Traits::inplace_decomposition(m_matrix);
   m_info = ok ? Success : NumericalIssue;
  
   return *this;
 }

◆ 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 193 of file LLT.h.

     {
       eigen_assert(m_isInitialized && "LLT is not initialized.");
       return m_info;
     }

◆ 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 137 of file LLT.h.

     {
       eigen_assert(m_isInitialized && "LLT is not initialized.");
       return Traits::getL(m_matrix);
     }

◆ 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 179 of file LLT.h.

     {
       eigen_assert(m_isInitialized && "LLT is not initialized.");
       return m_matrix;
     }

◆ 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 130 of file LLT.h.

     {
       eigen_assert(m_isInitialized && "LLT is not initialized.");
       return Traits::getU(m_matrix);
     }

◆ 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 466 of file LLT.h.

 {
   EIGEN_STATIC_ASSERT_VECTOR_ONLY(VectorType);
   eigen_assert(v.size()==m_matrix.cols());
   eigen_assert(m_isInitialized);
   if(internal::llt_inplace<typename MatrixType::Scalar, UpLo>::rankUpdate(m_matrix,v,sigma)>=0)
     m_info = NumericalIssue;
   else
     m_info = Success;
  
   return *this;
 }

◆ 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 168 of file LLT.h.

     {
       eigen_assert(m_isInitialized && "LLT is not initialized.");
       eigen_assert(m_info == Success && "LLT failed because matrix appears to be negative");
       return internal::rcond_estimate_helper(m_l1_norm, *this);
     }

◆ 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 525 of file LLT.h.

 {
   eigen_assert(m_isInitialized && "LLT is not initialized.");
   return matrixL() * matrixL().adjoint().toDenseMatrix();
 }

◆ rows()

template<typename MatrixType_ , int UpLo_>

EIGEN_CONSTEXPR Index Eigen::LLT< MatrixType_, UpLo_ >::rows ( ) const

inline

Definition at line 206 of file LLT.h.

206 { return m_matrix.rows(); }

◆ 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;

Output:

2.02
2.97

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

◆ solveInPlace()

template<typename MatrixType , int UpLo_>

template<typename Derived >

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

Definition at line 513 of file LLT.h.

 {
   eigen_assert(m_isInitialized && "LLT is not initialized.");
   eigen_assert(m_matrix.rows()==bAndX.rows());
   matrixL().solveInPlace(bAndX);
   matrixU().solveInPlace(bAndX);
 }

Member Data Documentation

◆ m_info

template<typename MatrixType_ , int UpLo_>

ComputationInfo Eigen::LLT< MatrixType_, UpLo_ >::m_info

protected

Definition at line 231 of file LLT.h.

◆ m_isInitialized

template<typename MatrixType_ , int UpLo_>

bool Eigen::LLT< MatrixType_, UpLo_ >::m_isInitialized

protected

Definition at line 230 of file LLT.h.

◆ m_l1_norm

template<typename MatrixType_ , int UpLo_>

RealScalar Eigen::LLT< MatrixType_, UpLo_ >::m_l1_norm

protected

Definition at line 229 of file LLT.h.

◆ m_matrix

template<typename MatrixType_ , int UpLo_>

MatrixType Eigen::LLT< MatrixType_, UpLo_ >::m_matrix

protected

Definition at line 228 of file LLT.h.

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

LLT.h

Public Types

Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

template<typename MatrixType_, int UpLo_> class Eigen::LLT< MatrixType_, UpLo_ >

Member Typedef Documentation

◆ Base

◆ MatrixType

◆ Traits

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Constructor & Destructor Documentation

◆ LLT() [1/4]

◆ LLT() [2/4]

◆ LLT() [3/4]

◆ LLT() [4/4]

Member Function Documentation

◆ adjoint()

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ info()

◆ matrixL()

◆ matrixLLT()

◆ matrixU()

◆ rankUpdate() [1/2]

◆ rankUpdate() [2/2]

◆ rcond()

◆ reconstructedMatrix()

◆ rows()

◆ solve()

◆ solveInPlace()

Member Data Documentation

◆ m_info

◆ m_isInitialized

◆ m_l1_norm

◆ m_matrix

template<typename MatrixType_, int UpLo_>
class Eigen::LLT< MatrixType_, UpLo_ >