LU decomposition of a matrix with partial pivoting, and related features. More...

Inheritance diagram for Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >:

Public Types
enum	{ MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef SolverBase< PartialPivLU >	Base

typedef MatrixType_	MatrixType

using	PermutationIndex = PermutationIndex_

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex >	PermutationType

typedef MatrixType::PlainObject	PlainObject

typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex >	TranspositionType

Public Types inherited from Eigen::SolverBase< PartialPivLU< MatrixType_, PermutationIndex_ > >
enum

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

typedef EigenBase< PartialPivLU< MatrixType_, PermutationIndex_ > >	Base

typedef Scalar	CoeffReturnType

typedef Transpose< const PartialPivLU< MatrixType_, PermutationIndex_ > >	ConstTransposeReturnType

typedef internal::traits< PartialPivLU< MatrixType_, PermutationIndex_ > >::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
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

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

Scalar	determinant () const

const Inverse< PartialPivLU >	inverse () const

const MatrixType &	matrixLU () const

	PartialPivLU ()
	Default Constructor. More...

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

template<typename InputType >
	PartialPivLU (EigenBase< InputType > &matrix)

	PartialPivLU (Index size)
	Default Constructor with memory preallocation. More...

const PermutationType &	permutationP () const

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

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

Public Member Functions inherited from Eigen::SolverBase< PartialPivLU< MatrixType_, PermutationIndex_ > >
const AdjointReturnType	adjoint () const

PartialPivLU< MatrixType_, PermutationIndex_ > &	derived ()

const PartialPivLU< MatrixType_, PermutationIndex_ > &	derived () const

const Solve< PartialPivLU< MatrixType_, PermutationIndex_ >, 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 Member Functions
void	compute ()

Protected Member Functions inherited from Eigen::SolverBase< PartialPivLU< MatrixType_, PermutationIndex_ > >
void	_check_solve_assertion (const Rhs &b) const

Protected Attributes
signed char	m_det_p

bool	m_isInitialized

RealScalar	m_l1_norm

MatrixType	m_lu

PermutationType	m_p

TranspositionType	m_rowsTranspositions

Detailed Description

template<typename MatrixType_, typename PermutationIndex_>
class Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >

LU decomposition of a matrix with partial pivoting, and related features.

Template Parameters

MatrixType_ the type of the matrix of which we are computing the LU decomposition

This class represents a LU decomposition of a square invertible matrix, with partial pivoting: the matrix A is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P is a permutation matrix.

Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.

The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided by class FullPivLU.

This is not a rank-revealing LU decomposition. Many features are intentionally absent from this class, such as rank computation. If you need these features, use class FullPivLU.

This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses in the general case. On the other hand, it is not suitable to determine whether a given matrix is invertible.

The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().

This class supports the inplace decomposition mechanism.

See also: MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class FullPivLU

Definition at line 78 of file PartialPivLU.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

typedef SolverBase<PartialPivLU> Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::Base

Definition at line 84 of file PartialPivLU.h.

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef MatrixType_ Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::MatrixType

Definition at line 83 of file PartialPivLU.h.

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

using Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::PermutationIndex = PermutationIndex_

Definition at line 92 of file PartialPivLU.h.

◆ PermutationType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex> Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::PermutationType

Definition at line 93 of file PartialPivLU.h.

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

typedef MatrixType::PlainObject Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::PlainObject

Definition at line 95 of file PartialPivLU.h.

◆ TranspositionType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex> Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::TranspositionType

Definition at line 94 of file PartialPivLU.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 88 of file PartialPivLU.h.

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

Constructor & Destructor Documentation

◆ PartialPivLU() [1/4]

template<typename MatrixType , typename PermutationIndex >

Eigen::PartialPivLU< MatrixType, PermutationIndex >::PartialPivLU

Default Constructor.

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

Definition at line 284 of file PartialPivLU.h.

   : m_lu(),
     m_p(),
     m_rowsTranspositions(),
     m_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
 }

◆ PartialPivLU() [2/4]

template<typename MatrixType , typename PermutationIndex >

Eigen::PartialPivLU< MatrixType, PermutationIndex >::PartialPivLU ( Index size )

explicit

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: PartialPivLU()

Definition at line 295 of file PartialPivLU.h.

   : m_lu(size, size),
     m_p(size),
     m_rowsTranspositions(size),
     m_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
 }

◆ PartialPivLU() [3/4]

template<typename MatrixType , typename PermutationIndex >

template<typename InputType >

Eigen::PartialPivLU< MatrixType, PermutationIndex >::PartialPivLU ( const EigenBase< InputType > & matrix )

explicit

Constructor.

Parameters

matrix the matrix of which to compute the LU decomposition.

Warning: The matrix should have full rank (e.g. if it's square, it should be invertible). If you need to deal with non-full rank, use class FullPivLU instead.

Definition at line 307 of file PartialPivLU.h.

   : m_lu(matrix.rows(),matrix.cols()),
     m_p(matrix.rows()),
     m_rowsTranspositions(matrix.rows()),
     m_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
   compute(matrix.derived());
 }

◆ PartialPivLU() [4/4]

template<typename MatrixType , typename PermutationIndex >

template<typename InputType >

Eigen::PartialPivLU< MatrixType, PermutationIndex >::PartialPivLU ( EigenBase< InputType > & matrix )

explicit

Constructor for inplace decomposition

Parameters

matrix the matrix of which to compute the LU decomposition.

Warning: The matrix should have full rank (e.g. if it's square, it should be invertible). If you need to deal with non-full rank, use class FullPivLU instead.

Definition at line 320 of file PartialPivLU.h.

   : m_lu(matrix.derived()),
     m_p(matrix.rows()),
     m_rowsTranspositions(matrix.rows()),
     m_l1_norm(0),
     m_det_p(0),
     m_isInitialized(false)
 {
   compute();
 }

Member Function Documentation

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

EIGEN_CONSTEXPR Index Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::cols ( void ) const

inline

Definition at line 223 of file PartialPivLU.h.

223 { return m_lu.cols(); }

◆ compute() [1/2]

template<typename MatrixType , typename PermutationIndex >

void Eigen::PartialPivLU< MatrixType, PermutationIndex >::compute

protected

Definition at line 525 of file PartialPivLU.h.

 {
   eigen_assert(m_lu.rows()<NumTraits<PermutationIndex>::highest());
  
   if(m_lu.cols()>0)
     m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
   else
     m_l1_norm = RealScalar(0);
  
   eigen_assert(m_lu.rows() == m_lu.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
   const Index size = m_lu.rows();
  
   m_rowsTranspositions.resize(size);
  
   typename TranspositionType::StorageIndex nb_transpositions;
   internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
   m_det_p = (nb_transpositions%2) ? -1 : 1;
  
   m_p = m_rowsTranspositions;
  
   m_isInitialized = true;
 }

◆ compute() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

PartialPivLU& Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::compute ( const EigenBase< InputType > & matrix )

inline

Definition at line 134 of file PartialPivLU.h.

                                                               {
       m_lu = matrix.derived();
       compute();
       return *this;
     }

◆ determinant()

template<typename MatrixType , typename PermutationIndex >

PartialPivLU< MatrixType, PermutationIndex >::Scalar Eigen::PartialPivLU< MatrixType, PermutationIndex >::determinant

Returns: the determinant of the matrix of which *this is the LU decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the LU decomposition has already been computed.

Note: For fixed-size matrices of size up to 4, MatrixBase::determinant() offers optimized paths.

Warning: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow.

See also: MatrixBase::determinant()

Definition at line 549 of file PartialPivLU.h.

 {
   eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
   return Scalar(m_det_p) * m_lu.diagonal().prod();
 }

◆ inverse()

template<typename MatrixType_ , typename PermutationIndex_ >

const Inverse<PartialPivLU> Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::inverse ( ) const

inline

Returns: the inverse of the matrix of which *this is the LU decomposition.

Warning: The matrix being decomposed here is assumed to be invertible. If you need to check for invertibility, use class FullPivLU instead.

See also: MatrixBase::inverse(), LU::inverse()

Definition at line 199 of file PartialPivLU.h.

     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return Inverse<PartialPivLU>(*this);
     }

◆ matrixLU()

template<typename MatrixType_ , typename PermutationIndex_ >

const MatrixType& Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::matrixLU ( ) const

inline

Returns: the LU decomposition matrix: the upper-triangular part is U, the unit-lower-triangular part is L (at least for square matrices; in the non-square case, special care is needed, see the documentation of class FullPivLU).

See also: matrixL(), matrixU()

Definition at line 146 of file PartialPivLU.h.

     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return m_lu;
     }

◆ permutationP()

template<typename MatrixType_ , typename PermutationIndex_ >

const PermutationType& Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::permutationP ( ) const

inline

Returns: the permutation matrix P.

Definition at line 154 of file PartialPivLU.h.

     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return m_p;
     }

◆ rcond()

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::rcond ( ) const

inline

Returns: an estimate of the reciprocal condition number of the matrix of which *this is the LU decomposition.

Definition at line 186 of file PartialPivLU.h.

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

◆ reconstructedMatrix()

template<typename MatrixType , typename PermutationIndex >

MatrixType Eigen::PartialPivLU< MatrixType, PermutationIndex >::reconstructedMatrix

Returns: the matrix represented by the decomposition, i.e., it returns the product: P^{-1} L U. This function is provided for debug purpose.

Definition at line 559 of file PartialPivLU.h.

 {
   eigen_assert(m_isInitialized && "LU is not initialized.");
   // LU
   MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix()
                  * m_lu.template triangularView<Upper>();
  
   // P^{-1}(LU)
   res = m_p.inverse() * res;
  
   return res;
 }

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

EIGEN_CONSTEXPR Index Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::rows ( void ) const

inline

Definition at line 222 of file PartialPivLU.h.

222 { return m_lu.rows(); }

◆ solve()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename Rhs >

const Solve<PartialPivLU, Rhs> Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::solve ( const MatrixBase< Rhs > & b ) const

inline

This method returns the solution x to the equation Ax=b, where A is the matrix of which *this is the LU decomposition.

Parameters

b	the right-hand-side of the equation to solve. Can be a vector or a matrix, the only requirement in order for the equation to make sense is that b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.

Returns: the solution.

Example:

MatrixXd A = MatrixXd::Random(3,3);
MatrixXd B = MatrixXd::Random(3,2);
cout << "Here is the invertible matrix A:" << endl << A << endl;
cout << "Here is the matrix B:" << endl << B << endl;
MatrixXd X = A.lu().solve(B);
cout << "Here is the (unique) solution X to the equation AX=B:" << endl << X << endl;
cout << "Relative error: " << (A*X-B).norm() / B.norm() << endl;

Output:

Here is the invertible matrix A:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Here is the matrix B:
  0.108   -0.27
-0.0452  0.0268
  0.258   0.904
Here is the (unique) solution X to the equation AX=B:
 0.609   2.68
-0.231  -1.57
  0.51   3.51
Relative error: 3.28e-16

Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution theoretically exists and is unique regardless of b.

See also: TriangularView::solve(), inverse(), computeInverse()

Member Data Documentation

◆ m_det_p

template<typename MatrixType_ , typename PermutationIndex_ >

signed char Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::m_det_p

protected

Definition at line 279 of file PartialPivLU.h.

◆ m_isInitialized

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized

protected

Definition at line 280 of file PartialPivLU.h.

◆ m_l1_norm

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::m_l1_norm

protected

Definition at line 278 of file PartialPivLU.h.

◆ m_lu

template<typename MatrixType_ , typename PermutationIndex_ >

MatrixType Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::m_lu

protected

Definition at line 275 of file PartialPivLU.h.

◆ m_p

template<typename MatrixType_ , typename PermutationIndex_ >

PermutationType Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::m_p

protected

Definition at line 276 of file PartialPivLU.h.

◆ m_rowsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

TranspositionType Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >::m_rowsTranspositions

protected

Definition at line 277 of file PartialPivLU.h.

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

template<typename MatrixType_, typename PermutationIndex_> class Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >

Member Typedef Documentation

◆ Base

◆ MatrixType

◆ PermutationIndex

◆ PermutationType

◆ PlainObject

◆ TranspositionType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ PartialPivLU() [1/4]

◆ PartialPivLU() [2/4]

◆ PartialPivLU() [3/4]

◆ PartialPivLU() [4/4]

Member Function Documentation

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ determinant()

◆ inverse()

◆ matrixLU()

◆ permutationP()

◆ rcond()

◆ reconstructedMatrix()

◆ rows()

◆ solve()

Member Data Documentation

◆ m_det_p

◆ m_isInitialized

◆ m_l1_norm

◆ m_lu

◆ m_p

◆ m_rowsTranspositions

template<typename MatrixType_, typename PermutationIndex_>
class Eigen::PartialPivLU< MatrixType_, PermutationIndex_ >