Eigen::LDLT< MatrixType_, UpLo_ > Class Template Reference

Robust Cholesky decomposition of a matrix with pivoting. More...

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

Public Types
enum	{   MaxRowsAtCompileTime ,   MaxColsAtCompileTime ,   UpLo }
typedef SolverBase< LDLT >	Base
typedef MatrixType_	MatrixType
typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType
typedef Matrix< Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1 >	TmpMatrixType
typedef internal::LDLT_Traits< MatrixType, UpLo >	Traits
typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType
Public Types inherited from Eigen::SolverBase< LDLT< MatrixType_, UpLo_ > >
enum
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType
typedef EigenBase< LDLT< MatrixType_, UpLo_ > >	Base
typedef Scalar	CoeffReturnType
typedef Transpose< const LDLT< MatrixType_, UpLo_ > >	ConstTransposeReturnType
typedef internal::traits< LDLT< 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 LDLT &	adjoint () const
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT
template<typename InputType >
LDLT< MatrixType, UpLo_ > &	compute (const EigenBase< InputType > &a)
template<typename InputType >
LDLT &	compute (const EigenBase< InputType > &matrix)
ComputationInfo	info () const
	Reports whether previous computation was successful. More...
bool	isNegative (void) const
bool	isPositive () const
	LDLT ()
	Default Constructor. More...
template<typename InputType >
	LDLT (const EigenBase< InputType > &matrix)
	Constructor with decomposition. More...
template<typename InputType >
	LDLT (EigenBase< InputType > &matrix)
	Constructs a LDLT factorization from a given matrix. More...
	LDLT (Index size)
	Default Constructor with memory preallocation. More...
Traits::MatrixL	matrixL () const
const MatrixType &	matrixLDLT () const
Traits::MatrixU	matrixU () const
template<typename Derived >
LDLT &	rankUpdate (const MatrixBase< Derived > &w, const RealScalar &alpha=1)
template<typename Derived >
LDLT< MatrixType, UpLo_ > &	rankUpdate (const MatrixBase< Derived > &w, const typename LDLT< MatrixType, UpLo_ >::RealScalar &sigma)
RealScalar	rcond () const
MatrixType	reconstructedMatrix () const
EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT
void	setZero ()
template<typename Rhs >
const Solve< LDLT, Rhs >	solve (const MatrixBase< Rhs > &b) const
template<typename Derived >
bool	solveInPlace (MatrixBase< Derived > &bAndX) const
const TranspositionType &	transpositionsP () const
Diagonal< const MatrixType >	vectorD () const
Public Member Functions inherited from Eigen::SolverBase< LDLT< MatrixType_, UpLo_ > >
const AdjointReturnType	adjoint () const
LDLT< MatrixType_, UpLo_ > &	derived ()
const LDLT< MatrixType_, UpLo_ > &	derived () const
const Solve< LDLT< 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
internal::SignMatrix	m_sign
TmpMatrixType	m_temporary
TranspositionType	m_transpositions

Additional Inherited Members
Protected Member Functions inherited from Eigen::SolverBase< LDLT< MatrixType_, UpLo_ > >
void	_check_solve_assertion (const Rhs &b) const

Detailed Description

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

Robust Cholesky decomposition of a matrix with pivoting.

Template Parameters

MatrixType_	the type of the matrix of which to compute the LDL^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.

Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite matrix \( A \) such that \( A = P^TLDL^*P \), where P is a permutation matrix, L is lower triangular with a unit diagonal and D is a diagonal matrix.

The decomposition uses pivoting to ensure stability, so that D will have zeros in the bottom right rank(A) - n submatrix. Avoiding the square root on D also stabilizes the computation.

Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky decomposition to determine whether a system of equations has a solution.

This class supports the inplace decomposition mechanism.

See also

MatrixBase::ldlt(), SelfAdjointView::ldlt(), class LLT

Definition at line 61 of file LDLT.h.

Member Typedef Documentation

Base

template<typename MatrixType_ , int UpLo_>

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

Definition at line 66 of file LDLT.h.

MatrixType

template<typename MatrixType_ , int UpLo_>

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

Definition at line 65 of file LDLT.h.

PermutationType

template<typename MatrixType_ , int UpLo_>

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::LDLT< MatrixType_, UpLo_ >::PermutationType

Definition at line 78 of file LDLT.h.

TmpMatrixType

template<typename MatrixType_ , int UpLo_>

typedef Matrix<Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1> Eigen::LDLT< MatrixType_, UpLo_ >::TmpMatrixType

Definition at line 75 of file LDLT.h.

Traits

template<typename MatrixType_ , int UpLo_>

typedef internal::LDLT_Traits<MatrixType,UpLo> Eigen::LDLT< MatrixType_, UpLo_ >::Traits

Definition at line 80 of file LDLT.h.

TranspositionType

template<typename MatrixType_ , int UpLo_>

typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::LDLT< MatrixType_, UpLo_ >::TranspositionType

Definition at line 77 of file LDLT.h.

Member Enumeration Documentation

anonymous enum

template<typename MatrixType_ , int UpLo_>

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime
UpLo

Definition at line 70 of file LDLT.h.

         {
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
      UpLo = UpLo_
    };

Constructor & Destructor Documentation

LDLT() [1/4]

template<typename MatrixType_ , int UpLo_>

Eigen::LDLT< MatrixType_, UpLo_ >::LDLT ( )

inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via LDLT::compute(const MatrixType&).

Definition at line 87 of file LDLT.h.

      : m_matrix(),
        m_transpositions(),
        m_sign(internal::ZeroSign),
        m_isInitialized(false)
    {}

LDLT() [2/4]

template<typename MatrixType_ , int UpLo_>

Eigen::LDLT< MatrixType_, UpLo_ >::LDLT ( 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

LDLT()

Definition at line 100 of file LDLT.h.

      : m_matrix(size, size),
        m_transpositions(size),
        m_temporary(size),
        m_sign(internal::ZeroSign),
        m_isInitialized(false)
    {}

LDLT() [3/4]

template<typename MatrixType_ , int UpLo_>

template<typename InputType >

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

inlineexplicit

Constructor with decomposition.

This calculates the decomposition for the input matrix.

See also

LDLT(Index size)

Definition at line 115 of file LDLT.h.

      : m_matrix(matrix.rows(), matrix.cols()),
        m_transpositions(matrix.rows()),
        m_temporary(matrix.rows()),
        m_sign(internal::ZeroSign),
        m_isInitialized(false)
    {
      compute(matrix.derived());
    }

LDLT() [4/4]

template<typename MatrixType_ , int UpLo_>

template<typename InputType >

Eigen::LDLT< MatrixType_, UpLo_ >::LDLT ( 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

LDLT(const EigenBase&)

Definition at line 132 of file LDLT.h.

      : m_matrix(matrix.derived()),
        m_transpositions(matrix.rows()),
        m_temporary(matrix.rows()),
        m_sign(internal::ZeroSign),
        m_isInitialized(false)
    {
      compute(matrix.derived());
    }

Member Function Documentation

adjoint()

template<typename MatrixType_ , int UpLo_>

const LDLT& Eigen::LDLT< 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 249 of file LDLT.h.

{ return *this; }

cols()

template<typename MatrixType_ , int UpLo_>

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

inline

Definition at line 252 of file LDLT.h.

{ return m_matrix.cols(); }

compute() [1/2]

template<typename MatrixType_ , int UpLo_>

template<typename InputType >

LDLT<MatrixType,UpLo_>& Eigen::LDLT< MatrixType_, UpLo_ >::compute

(

const EigenBase< InputType > &

a

)

Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of matrix

Definition at line 498 of file LDLT.h.

{
  eigen_assert(a.rows()==a.cols());
  const Index size = a.rows();
 
  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_transpositions.resize(size);
  m_isInitialized = false;
  m_temporary.resize(size);
  m_sign = internal::ZeroSign;
 
  m_info = internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign) ? Success : NumericalIssue;
 
  m_isInitialized = true;
  return *this;
}

compute() [2/2]

template<typename MatrixType_ , int UpLo_>

template<typename InputType >

LDLT& Eigen::LDLT< MatrixType_, UpLo_ >::compute

(

const EigenBase< InputType > &

matrix

)

info()

template<typename MatrixType_ , int UpLo_>

ComputationInfo Eigen::LDLT< MatrixType_, UpLo_ >::info ( ) const

inline

Reports whether previous computation was successful.

Returns

Success if computation was successful, NumericalIssue if the factorization failed because of a zero pivot.

Definition at line 259 of file LDLT.h.

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

isNegative()

template<typename MatrixType_ , int UpLo_>

bool Eigen::LDLT< MatrixType_, UpLo_ >::isNegative ( void  ) const

inline

Returns

true if the matrix is negative (semidefinite)

Definition at line 187 of file LDLT.h.

    {
      eigen_assert(m_isInitialized && "LDLT is not initialized.");
      return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
    }

isPositive()

template<typename MatrixType_ , int UpLo_>

bool Eigen::LDLT< MatrixType_, UpLo_ >::isPositive ( ) const

inline

Returns

true if the matrix is positive (semidefinite)

Definition at line 180 of file LDLT.h.

    {
      eigen_assert(m_isInitialized && "LDLT is not initialized.");
      return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
    }

matrixL()

template<typename MatrixType_ , int UpLo_>

Traits::MatrixL Eigen::LDLT< MatrixType_, UpLo_ >::matrixL ( ) const

inline

Returns

a view of the lower triangular matrix L

Definition at line 158 of file LDLT.h.

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

matrixLDLT()

template<typename MatrixType_ , int UpLo_>

const MatrixType& Eigen::LDLT< MatrixType_, UpLo_ >::matrixLDLT ( ) const

inline

Returns

the internal LDLT decomposition matrix

TODO: document the storage layout

Definition at line 236 of file LDLT.h.

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

matrixU()

template<typename MatrixType_ , int UpLo_>

Traits::MatrixU Eigen::LDLT< MatrixType_, UpLo_ >::matrixU ( ) const

inline

Returns

a view of the upper triangular matrix U

Definition at line 151 of file LDLT.h.

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

rankUpdate() [1/2]

template<typename MatrixType_ , int UpLo_>

template<typename Derived >

LDLT& Eigen::LDLT< MatrixType_, UpLo_ >::rankUpdate	(	const MatrixBase< Derived > &	w,
		const RealScalar &	alpha = 1
	)

rankUpdate() [2/2]

template<typename MatrixType_ , int UpLo_>

template<typename Derived >

LDLT<MatrixType,UpLo_>& Eigen::LDLT< MatrixType_, UpLo_ >::rankUpdate	(	const MatrixBase< Derived > &	w,
		const typename LDLT< MatrixType, UpLo_ >::RealScalar &	sigma
	)

Update the LDLT decomposition: given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.

Parameters

w	a vector to be incorporated into the decomposition.
sigma	a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.

See also

setZero()

Definition at line 536 of file LDLT.h.

{
  typedef typename TranspositionType::StorageIndex IndexType;
  const Index size = w.rows();
  if (m_isInitialized)
  {
    eigen_assert(m_matrix.rows()==size);
  }
  else
  {
    m_matrix.resize(size,size);
    m_matrix.setZero();
    m_transpositions.resize(size);
    for (Index i = 0; i < size; i++)
      m_transpositions.coeffRef(i) = IndexType(i);
    m_temporary.resize(size);
    m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
    m_isInitialized = true;
  }
 
  internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
 
  return *this;
}

rcond()

template<typename MatrixType_ , int UpLo_>

RealScalar Eigen::LDLT< MatrixType_, UpLo_ >::rcond ( ) const

inline

Returns

an estimate of the reciprocal condition number of the matrix of which *this is the LDLT decomposition.

Definition at line 223 of file LDLT.h.

    {
      eigen_assert(m_isInitialized && "LDLT is not initialized.");
      return internal::rcond_estimate_helper(m_l1_norm, *this);
    }

reconstructedMatrix()

template<typename MatrixType , int UpLo_>

MatrixType Eigen::LDLT< MatrixType, UpLo_ >::reconstructedMatrix

Returns

the matrix represented by the decomposition, i.e., it returns the product: P^T L D L^* P. This function is provided for debug purpose.

Definition at line 640 of file LDLT.h.

{
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  const Index size = m_matrix.rows();
  MatrixType res(size,size);
 
  // P
  res.setIdentity();
  res = transpositionsP() * res;
  // L^* P
  res = matrixU() * res;
  // D(L^*P)
  res = vectorD().real().asDiagonal() * res;
  // L(DL^*P)
  res = matrixL() * res;
  // P^T (LDL^*P)
  res = transpositionsP().transpose() * res;
 
  return res;
}

rows()

template<typename MatrixType_ , int UpLo_>

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

inline

Definition at line 251 of file LDLT.h.

{ return m_matrix.rows(); }

setZero()

template<typename MatrixType_ , int UpLo_>

void Eigen::LDLT< MatrixType_, UpLo_ >::setZero ( )

inline

Clear any existing decomposition

See also

rankUpdate(w,sigma)

Definition at line 145 of file LDLT.h.

    {
      m_isInitialized = false;
    }

solve()

template<typename MatrixType_ , int UpLo_>

template<typename Rhs >

const Solve<LDLT, Rhs> Eigen::LDLT< MatrixType_, UpLo_ >::solve ( const MatrixBase< Rhs > &  b ) const

inline

Returns

a solution x of \( A x = b \) using the current decomposition of A.

This function also supports in-place solves using the syntax x = decompositionObject.solve(x) .

This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:

bool a_solution_exists = (A*result).isApprox(b, precision); 

This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get inf or nan values.

More precisely, this method solves \( A x = b \) using the decomposition \( A = P^T L D L^* P \) by solving the systems \( P^T y_1 = b \), \( L y_2 = y_1 \), \( D y_3 = y_2 \), \( L^* y_4 = y_3 \) and \( P x = y_4 \) in succession. If the matrix \( A \) is singular, then \( D \) will also be singular (all the other matrices are invertible). In that case, the least-square solution of \( D y_3 = y_2 \) is computed. This does not mean that this function computes the least-square solution of \( A x = b \) if \( A \) is singular.

See also

MatrixBase::ldlt(), SelfAdjointView::ldlt()

solveInPlace()

template<typename MatrixType , int UpLo_>

template<typename Derived >

bool Eigen::LDLT< MatrixType, UpLo_ >::solveInPlace

(

MatrixBase< Derived > &

bAndX

)

const

Definition at line 626 of file LDLT.h.

{
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  eigen_assert(m_matrix.rows() == bAndX.rows());
 
  bAndX = this->solve(bAndX);
 
  return true;
}

transpositionsP()

template<typename MatrixType_ , int UpLo_>

const TranspositionType& Eigen::LDLT< MatrixType_, UpLo_ >::transpositionsP ( ) const

inline

Returns

the permutation matrix P as a transposition sequence.

Definition at line 166 of file LDLT.h.

    {
      eigen_assert(m_isInitialized && "LDLT is not initialized.");
      return m_transpositions;
    }

vectorD()

template<typename MatrixType_ , int UpLo_>

Diagonal<const MatrixType> Eigen::LDLT< MatrixType_, UpLo_ >::vectorD ( ) const

inline

Returns

the coefficients of the diagonal matrix D

Definition at line 173 of file LDLT.h.

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

Member Data Documentation

m_info

template<typename MatrixType_ , int UpLo_>

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

protected

Definition at line 289 of file LDLT.h.

m_isInitialized

template<typename MatrixType_ , int UpLo_>

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

protected

Definition at line 288 of file LDLT.h.

m_l1_norm

template<typename MatrixType_ , int UpLo_>

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

protected

Definition at line 284 of file LDLT.h.

m_matrix

template<typename MatrixType_ , int UpLo_>

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

protected

Definition at line 283 of file LDLT.h.

m_sign

template<typename MatrixType_ , int UpLo_>

internal::SignMatrix Eigen::LDLT< MatrixType_, UpLo_ >::m_sign

protected

Definition at line 287 of file LDLT.h.

m_temporary

template<typename MatrixType_ , int UpLo_>

TmpMatrixType Eigen::LDLT< MatrixType_, UpLo_ >::m_temporary

protected

Definition at line 286 of file LDLT.h.

m_transpositions

template<typename MatrixType_ , int UpLo_>

TranspositionType Eigen::LDLT< MatrixType_, UpLo_ >::m_transpositions

protected

Definition at line 285 of file LDLT.h.

---

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

• LDLT.h