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

#include <src/eigen/Eigen/src/Cholesky/LLT.h>

Collaboration diagram for Eigen::LLT< _MatrixType, _UpLo >:

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 _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef NumTraits< typenameMatrixType::Scalar >::Real	RealScalar

typedef Eigen::Index	Index

typedef MatrixType::StorageIndex	StorageIndex

typedef internal::LLT_Traits< MatrixType, UpLo >	Traits

Public Member Functions
	LLT ()
	Default Constructor.

	LLT (Index size)
	Default Constructor with memory preallocation.

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

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

Traits::MatrixU	matrixU () const

Traits::MatrixL	matrixL () const

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

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

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

RealScalar	rcond () const

const MatrixType &	matrixLLT () const

MatrixType	reconstructedMatrix () const

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

const LLT &	adjoint () const

Index	rows () const

Index	cols () const

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

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

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

template<typename VectorType >
LLT< _MatrixType, _UpLo >	rankUpdate (const VectorType &v, const RealScalar &sigma)

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

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
MatrixType	m_matrix

RealScalar	m_l1_norm

bool	m_isInitialized

ComputationInfo	m_info

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:

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

Member Typedef Documentation

◆ Index

template<typename _MatrixType , int _UpLo>

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

◆ MatrixType

template<typename _MatrixType , int _UpLo>

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

◆ RealScalar

template<typename _MatrixType , int _UpLo>

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

◆ Scalar

template<typename _MatrixType , int _UpLo>

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

◆ StorageIndex

template<typename _MatrixType , int _UpLo>

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

◆ Traits

template<typename _MatrixType , int _UpLo>

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

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType , int _UpLo>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxColsAtCompileTime

         {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

◆ anonymous enum

template<typename _MatrixType , int _UpLo>

anonymous enum

Enumerator
PacketSize
AlignmentMask
UpLo

         {
      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&).

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

Eigen::LLT::m_isInitialized

bool m_isInitialized

Definition LLT.h:224

Eigen::LLT::m_matrix

MatrixType m_matrix

Definition LLT.h:222

◆ 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()

92 : m_matrix(size, size),

93 m_isInitialized(false) {}

◆ LLT() [3/4]

template<typename _MatrixType , int _UpLo>

template<typename InputType >

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

inlineexplicit

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

References Eigen::LLT< _MatrixType, _UpLo >::compute(), and Eigen::EigenBase< Derived >::derived().

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◆ LLT() [4/4]

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&)

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

References Eigen::LLT< _MatrixType, _UpLo >::compute(), and Eigen::EigenBase< Derived >::derived().

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Member Function Documentation

◆ _solve_impl() [1/2]

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

◆ _solve_impl() [2/2]

template<typename _MatrixType , int _UpLo>

template<typename RhsType , typename DstType >

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

{
  dst = rhs;
  solveInPlace(dst);
}

◆ 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)

197{ return *this; };

◆ check_template_parameters()

template<typename _MatrixType , int _UpLo>

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

inlinestaticprotected

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ cols()

template<typename _MatrixType , int _UpLo>

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

inline

200{ return m_matrix.cols(); }

References Eigen::LLT< _MatrixType, _UpLo >::m_matrix.

◆ 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:

Output:

{
  check_template_parameters();
 
  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;
}

References col(), eigen_assert, Eigen::internal::is_same_dense(), Eigen::Lower, Eigen::NumericalIssue, and Eigen::Success.

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◆ compute() [2/2]

template<typename _MatrixType , int _UpLo>

template<typename InputType >

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

Referenced by Eigen::LLT< _MatrixType, _UpLo >::LLT(), Eigen::LLT< _MatrixType, _UpLo >::LLT(), and igl::copyleft::quadprog().

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◆ 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.

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

References eigen_assert, Eigen::LLT< _MatrixType, _UpLo >::m_info, and Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized.

◆ matrixL()

template<typename _MatrixType , int _UpLo>

Traits::MatrixL Eigen::LLT< _MatrixType, _UpLo >::matrixL ( ) const

inline

Returns: a view of the lower triangular matrix L

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

References eigen_assert, Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized, and Eigen::LLT< _MatrixType, _UpLo >::m_matrix.

Referenced by Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::compute().

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◆ 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

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

References eigen_assert, Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized, and Eigen::LLT< _MatrixType, _UpLo >::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

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

References eigen_assert, Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized, and Eigen::LLT< _MatrixType, _UpLo >::m_matrix.

Referenced by Eigen::GeneralizedSelfAdjointEigenSolver< _MatrixType >::compute(), and igl::copyleft::quadprog().

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◆ 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.

{
  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;
}

References eigen_assert, EIGEN_STATIC_ASSERT_VECTOR_ONLY, Eigen::NumericalIssue, and Eigen::Success.

◆ 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.

    {
      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);
    }

References eigen_assert, Eigen::LLT< _MatrixType, _UpLo >::m_info, Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized, Eigen::LLT< _MatrixType, _UpLo >::m_l1_norm, Eigen::internal::rcond_estimate_helper(), and Eigen::Success.

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◆ 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.

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

References eigen_assert.

◆ rows()

template<typename _MatrixType , int _UpLo>

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

inline

199{ return m_matrix.rows(); }

References Eigen::LLT< _MatrixType, _UpLo >::m_matrix.

◆ 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 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:

Output:

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

    {
      eigen_assert(m_isInitialized && "LLT is not initialized.");
      eigen_assert(m_matrix.rows()==b.rows()
                && "LLT::solve(): invalid number of rows of the right hand side matrix b");
      return Solve<LLT, Rhs>(*this, b.derived());
    }

References eigen_assert, Eigen::LLT< _MatrixType, _UpLo >::m_isInitialized, and Eigen::LLT< _MatrixType, _UpLo >::m_matrix.

Referenced by igl::copyleft::quadprog().

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◆ solveInPlace()

template<typename MatrixType , int _UpLo>

template<typename Derived >

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

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

References eigen_assert.

Member Data Documentation

◆ m_info

template<typename _MatrixType , int _UpLo>

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

protected

Referenced by Eigen::LLT< _MatrixType, _UpLo >::info(), and Eigen::LLT< _MatrixType, _UpLo >::rcond().

◆ m_isInitialized

template<typename _MatrixType , int _UpLo>

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

protected

Referenced by Eigen::LLT< _MatrixType, _UpLo >::info(), Eigen::LLT< _MatrixType, _UpLo >::matrixL(), Eigen::LLT< _MatrixType, _UpLo >::matrixLLT(), Eigen::LLT< _MatrixType, _UpLo >::matrixU(), Eigen::LLT< _MatrixType, _UpLo >::rcond(), and Eigen::LLT< _MatrixType, _UpLo >::solve().

◆ m_l1_norm

template<typename _MatrixType , int _UpLo>

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

protected

Referenced by Eigen::LLT< _MatrixType, _UpLo >::rcond().

◆ m_matrix

template<typename _MatrixType , int _UpLo>

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

protected

Referenced by Eigen::LLT< _MatrixType, _UpLo >::cols(), Eigen::LLT< _MatrixType, _UpLo >::matrixL(), Eigen::LLT< _MatrixType, _UpLo >::matrixLLT(), Eigen::LLT< _MatrixType, _UpLo >::matrixU(), Eigen::LLT< _MatrixType, _UpLo >::rows(), and Eigen::LLT< _MatrixType, _UpLo >::solve().

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

src/eigen/Eigen/src/Cholesky/LLT.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ Index

◆ MatrixType

◆ RealScalar

◆ Scalar

◆ StorageIndex

◆ 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

◆ _solve_impl() [1/2]

◆ _solve_impl() [2/2]

◆ adjoint()

◆ check_template_parameters()

◆ 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