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

#include <ForwardDeclarations.h>

Inheritance diagram for Eigen::PartialPivLU< _MatrixType >:

Public Types
enum	{ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef SolverBase< PartialPivLU >	Base

typedef _MatrixType	MatrixType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType

typedef MatrixType::PlainObject	PlainObject

typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType

Public Types inherited from Eigen::SolverBase< PartialPivLU< _MatrixType > >
enum

typedef internal::conditional< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, ConstTransposeReturnType >, ConstTransposeReturnType >::type	AdjointReturnType

typedef EigenBase< PartialPivLU< _MatrixType > >	Base

typedef Scalar	CoeffReturnType

typedef internal::add_const< Transpose< const PartialPivLU< _MatrixType > > >::type	ConstTransposeReturnType

typedef internal::traits< PartialPivLU< _MatrixType > >::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
template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const

template<bool Conjugate, typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl_transposed (const RhsType &rhs, DstType &dst) const

Index	cols () const

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...

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

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

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

const PermutationType &	permutationP () const

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

Index	rows () const

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

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

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

	SolverBase ()

ConstTransposeReturnType	transpose () const

	~SolverBase ()

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

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

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

EIGEN_DEVICE_FUNC Index	cols () const

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	const_derived () const

EIGEN_DEVICE_FUNC Derived &	derived ()

EIGEN_DEVICE_FUNC const Derived &	derived () const

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

EIGEN_DEVICE_FUNC Index	rows () const

EIGEN_DEVICE_FUNC Index	size () const

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

Protected Member Functions
void	compute ()

Static Protected Member Functions
static void	check_template_parameters ()

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>
class Eigen::PartialPivLU< _MatrixType >

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 250 of file ForwardDeclarations.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef SolverBase<PartialPivLU> Eigen::PartialPivLU< _MatrixType >::Base

Definition at line 81 of file PartialPivLU.h.

template<typename _MatrixType>

typedef _MatrixType Eigen::PartialPivLU< _MatrixType >::MatrixType

Definition at line 80 of file PartialPivLU.h.

template<typename _MatrixType>

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::PartialPivLU< _MatrixType >::PermutationType

Definition at line 88 of file PartialPivLU.h.

template<typename _MatrixType>

typedef MatrixType::PlainObject Eigen::PartialPivLU< _MatrixType >::PlainObject

Definition at line 90 of file PartialPivLU.h.

template<typename _MatrixType>

typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::PartialPivLU< _MatrixType >::TranspositionType

Definition at line 89 of file PartialPivLU.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 84 of file PartialPivLU.h.

Constructor & Destructor Documentation

template<typename MatrixType >

Eigen::PartialPivLU< MatrixType >::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 292 of file PartialPivLU.h.

template<typename MatrixType >

Eigen::PartialPivLU< MatrixType >::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 303 of file PartialPivLU.h.

template<typename MatrixType >

template<typename InputType >

Eigen::PartialPivLU< MatrixType >::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 315 of file PartialPivLU.h.

template<typename MatrixType >

template<typename InputType >

Eigen::PartialPivLU< MatrixType >::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 328 of file PartialPivLU.h.

Member Function Documentation

template<typename _MatrixType>

template<typename RhsType , typename DstType >

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

inline

Definition at line 226 of file PartialPivLU.h.

template<typename _MatrixType>

template<bool Conjugate, typename RhsType , typename DstType >

EIGEN_DEVICE_FUNC void Eigen::PartialPivLU< _MatrixType >::_solve_impl_transposed	(	const RhsType &	rhs,
		DstType &	dst
	)		const

inline

Definition at line 248 of file PartialPivLU.h.

template<typename _MatrixType>

static void Eigen::PartialPivLU< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

Definition at line 276 of file PartialPivLU.h.

template<typename _MatrixType>

Index Eigen::PartialPivLU< _MatrixType >::cols ( void ) const

inline

Definition at line 221 of file PartialPivLU.h.

template<typename _MatrixType>

template<typename InputType >

PartialPivLU& Eigen::PartialPivLU< _MatrixType >::compute ( const EigenBase< InputType > & matrix )

inline

Definition at line 129 of file PartialPivLU.h.

template<typename MatrixType >

void Eigen::PartialPivLU< MatrixType >::compute ( )

protected

Definition at line 515 of file PartialPivLU.h.

template<typename MatrixType >

PartialPivLU< MatrixType >::Scalar Eigen::PartialPivLU< MatrixType >::determinant ( ) const

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 539 of file PartialPivLU.h.

template<typename _MatrixType>

const Inverse<PartialPivLU> Eigen::PartialPivLU< _MatrixType >::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 197 of file PartialPivLU.h.

template<typename _MatrixType>

const MatrixType& Eigen::PartialPivLU< _MatrixType >::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 141 of file PartialPivLU.h.

template<typename _MatrixType>

const PermutationType& Eigen::PartialPivLU< _MatrixType >::permutationP ( ) const

inline

Returns: the permutation matrix P.

Definition at line 149 of file PartialPivLU.h.

template<typename _MatrixType>

RealScalar Eigen::PartialPivLU< _MatrixType >::rcond ( ) const

inline

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

Definition at line 184 of file PartialPivLU.h.

template<typename MatrixType >

MatrixType Eigen::PartialPivLU< MatrixType >::reconstructedMatrix ( ) const

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 549 of file PartialPivLU.h.

template<typename _MatrixType>

Index Eigen::PartialPivLU< _MatrixType >::rows ( void ) const

inline

Definition at line 220 of file PartialPivLU.h.

template<typename _MatrixType>

template<typename Rhs >

const Solve<PartialPivLU, Rhs> Eigen::PartialPivLU< _MatrixType >::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:

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()

Definition at line 175 of file PartialPivLU.h.

Member Data Documentation

template<typename _MatrixType>

signed char Eigen::PartialPivLU< _MatrixType >::m_det_p

protected

Definition at line 287 of file PartialPivLU.h.

template<typename _MatrixType>

bool Eigen::PartialPivLU< _MatrixType >::m_isInitialized

protected

Definition at line 288 of file PartialPivLU.h.

template<typename _MatrixType>

RealScalar Eigen::PartialPivLU< _MatrixType >::m_l1_norm

protected

Definition at line 286 of file PartialPivLU.h.

template<typename _MatrixType>

MatrixType Eigen::PartialPivLU< _MatrixType >::m_lu

protected

Definition at line 283 of file PartialPivLU.h.

template<typename _MatrixType>

PermutationType Eigen::PartialPivLU< _MatrixType >::m_p

protected

Definition at line 284 of file PartialPivLU.h.

template<typename _MatrixType>

TranspositionType Eigen::PartialPivLU< _MatrixType >::m_rowsTranspositions

protected

Definition at line 285 of file PartialPivLU.h.

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

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

template<typename _MatrixType> class Eigen::PartialPivLU< _MatrixType >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename _MatrixType>
class Eigen::PartialPivLU< _MatrixType >