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

#include <ForwardDeclarations.h>

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

typedef SolverBase< FullPivLU >	Base

typedef internal::plain_col_type< MatrixType, StorageIndex >::type	IntColVectorType

typedef internal::plain_row_type< MatrixType, StorageIndex >::type	IntRowVectorType

typedef _MatrixType	MatrixType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationPType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationQType

typedef MatrixType::PlainObject	PlainObject

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

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

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

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

internal::traits< MatrixType >::Scalar	determinant () const

Index	dimensionOfKernel () const

	FullPivLU ()
	Default Constructor. More...

template<typename MatrixType >
	FullPivLU ()

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

template<typename InputType >
	FullPivLU (EigenBase< InputType > &matrix)
	Constructs a LU factorization from a given matrix. More...

	FullPivLU (Index rows, Index cols)
	Default Constructor with memory preallocation. More...

template<typename MatrixType >
	FullPivLU (Index rows, Index cols)

const internal::image_retval< FullPivLU >	image (const MatrixType &originalMatrix) const

const Inverse< FullPivLU >	inverse () const

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

const internal::kernel_retval< FullPivLU >	kernel () const

const MatrixType &	matrixLU () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

const EIGEN_DEVICE_FUNC PermutationPType &	permutationP () const

const PermutationQType &	permutationQ () const

Index	rank () const

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

FullPivLU &	setThreshold (const RealScalar &threshold)

FullPivLU &	setThreshold (Default_t)

RealScalar	threshold () const

Protected Member Functions
void	computeInPlace ()

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
IntRowVectorType	m_colsTranspositions

signed char	m_det_pq

bool	m_isInitialized

RealScalar	m_l1_norm

MatrixType	m_lu

RealScalar	m_maxpivot

Index	m_nonzero_pivots

PermutationPType	m_p

RealScalar	m_prescribedThreshold

PermutationQType	m_q

IntColVectorType	m_rowsTranspositions

bool	m_usePrescribedThreshold

Friends
class	SolverBase< FullPivLU >

Detailed Description

LU decomposition of a matrix with complete 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 any matrix, with complete pivoting: the matrix A is decomposed as $A = P^{-1} L U Q^{-1}$ where L is unit-lower-triangular, U is upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any zeros are at the end.

This decomposition provides the generic approach to solving systems of linear equations, computing the rank, invertibility, inverse, kernel, and determinant.

This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, working with the SVD allows to select the smallest singular values of the matrix, something that the LU decomposition doesn't see.

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

As an example, here is how the original matrix can be retrieved:

typedef Matrix<double, 5, 3> Matrix5x3;
typedef Matrix<double, 5, 5> Matrix5x5;
Matrix5x3 m = Matrix5x3::Random();
cout << "Here is the matrix m:" << endl << m << endl;
Eigen::FullPivLU<Matrix5x3> lu(m);
cout << "Here is, up to permutations, its LU decomposition matrix:"
     << endl << lu.matrixLU() << endl;
cout << "Here is the L part:" << endl;
Matrix5x5 l = Matrix5x5::Identity();
l.block<5,3>(0,0).triangularView<StrictlyLower>() = lu.matrixLU();
cout << l << endl;
cout << "Here is the U part:" << endl;
Matrix5x3 u = lu.matrixLU().triangularView<Upper>();
cout << u << endl;
cout << "Let us now reconstruct the original matrix m:" << endl;
cout << lu.permutationP().inverse() * l * u * lu.permutationQ().inverse() << endl;

Output:

This class supports the inplace decomposition mechanism.

See also: MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()

Definition at line 268 of file ForwardDeclarations.h.

Member Typedef Documentation

◆ Base

typedef SolverBase<FullPivLU> Eigen::FullPivLU::Base

Definition at line 65 of file FullPivLU.h.

◆ IntColVectorType

typedef internal::plain_col_type<MatrixType, StorageIndex>::type Eigen::FullPivLU::IntColVectorType

Definition at line 74 of file FullPivLU.h.

◆ IntRowVectorType

typedef internal::plain_row_type<MatrixType, StorageIndex>::type Eigen::FullPivLU::IntRowVectorType

Definition at line 73 of file FullPivLU.h.

◆ MatrixType

typedef _MatrixType Eigen::FullPivLU::MatrixType

Definition at line 64 of file FullPivLU.h.

◆ PermutationPType

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::FullPivLU::PermutationPType

Definition at line 76 of file FullPivLU.h.

◆ PermutationQType

typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::FullPivLU::PermutationQType

Definition at line 75 of file FullPivLU.h.

◆ PlainObject

typedef MatrixType::PlainObject Eigen::FullPivLU::PlainObject

Definition at line 77 of file FullPivLU.h.

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 69 of file FullPivLU.h.

Constructor & Destructor Documentation

◆ FullPivLU() [1/6]

Eigen::FullPivLU::FullPivLU ( )

Default Constructor.

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

◆ FullPivLU() [2/6]

Eigen::FullPivLU::FullPivLU	(	Index	rows,
		Index	cols
	)

Default Constructor with memory preallocation.

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

See also: FullPivLU()

◆ FullPivLU() [3/6]

template<typename InputType >

Eigen::FullPivLU::FullPivLU ( const EigenBase< InputType > & matrix )

explicit

Constructor.

Parameters

matrix the matrix of which to compute the LU decomposition. It is required to be nonzero.

Definition at line 461 of file FullPivLU.h.

◆ FullPivLU() [4/6]

template<typename InputType >

Eigen::FullPivLU::FullPivLU ( EigenBase< InputType > & matrix )

explicit

Constructs a LU factorization from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also: FullPivLU(const EigenBase&)

Definition at line 475 of file FullPivLU.h.

◆ FullPivLU() [5/6]

template<typename MatrixType >

Eigen::FullPivLU::FullPivLU ( )

Definition at line 442 of file FullPivLU.h.

◆ FullPivLU() [6/6]

template<typename MatrixType >

Eigen::FullPivLU::FullPivLU	(	Index	rows,
		Index	cols
	)

Definition at line 448 of file FullPivLU.h.

Member Function Documentation

◆ _solve_impl()

template<typename RhsType , typename DstType >

void Eigen::FullPivLU::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

Definition at line 745 of file FullPivLU.h.

◆ _solve_impl_transposed()

template<bool Conjugate, typename RhsType , typename DstType >

void Eigen::FullPivLU::_solve_impl_transposed	(	const RhsType &	rhs,
		DstType &	dst
	)		const

Definition at line 792 of file FullPivLU.h.

◆ check_template_parameters()

static void Eigen::FullPivLU::check_template_parameters ( )

inlinestaticprotected

Definition at line 422 of file FullPivLU.h.

◆ cols()

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::FullPivLU::cols ( ) const

inline

Definition at line 410 of file FullPivLU.h.

◆ compute()

template<typename InputType >

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

inline

Computes the LU decomposition of the given matrix.

Parameters

matrix the matrix of which to compute the LU decomposition. It is required to be nonzero.

Returns: a reference to *this

Definition at line 120 of file FullPivLU.h.

◆ computeInPlace()

void Eigen::FullPivLU::computeInPlace ( )

protected

Definition at line 488 of file FullPivLU.h.

◆ determinant()

internal::traits< MatrixType >::Scalar Eigen::FullPivLU::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: This is only for square matrices.; 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 581 of file FullPivLU.h.

◆ dimensionOfKernel()

Index Eigen::FullPivLU::dimensionOfKernel ( ) const

inline

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

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 347 of file FullPivLU.h.

◆ image()

const internal::image_retval<FullPivLU> Eigen::FullPivLU::image ( const MatrixType & originalMatrix ) const

inline

Returns: the image of the matrix, also called its column-space. The columns of the returned matrix will form a basis of the image (column-space).

Parameters

originalMatrix the original matrix, of which *this is the LU decomposition. The reason why it is needed to pass it here, is that this allows a large optimization, as otherwise this method would need to reconstruct it from the LU decomposition.

Note: If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.; This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Example:

Matrix3d m;
m << 1,1,0,
     1,3,2,
     0,1,1;
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Notice that the middle column is the sum of the two others, so the "
     << "columns are linearly dependent." << endl;
cout << "Here is a matrix whose columns have the same span but are linearly independent:"
     << endl << m.fullPivLu().image(m) << endl;

Output:

See also: kernel()

Definition at line 216 of file FullPivLU.h.

◆ inverse()

const Inverse<FullPivLU> Eigen::FullPivLU::inverse ( ) const

inline

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

Note: If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.

See also: MatrixBase::inverse()

Definition at line 398 of file FullPivLU.h.

◆ isInjective()

bool Eigen::FullPivLU::isInjective ( ) const

inline

Returns: true if the matrix of which *this is the LU decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 360 of file FullPivLU.h.

◆ isInvertible()

bool Eigen::FullPivLU::isInvertible ( ) const

inline

Returns: true if the matrix of which *this is the LU decomposition is invertible.

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 385 of file FullPivLU.h.

◆ isSurjective()

bool Eigen::FullPivLU::isSurjective ( ) const

inline

Returns: true if the matrix of which *this is the LU decomposition represents a surjective linear map; false otherwise.

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 373 of file FullPivLU.h.

◆ kernel()

const internal::kernel_retval<FullPivLU> Eigen::FullPivLU::kernel ( ) const

inline

Returns: the kernel of the matrix, also called its null-space. The columns of the returned matrix will form a basis of the kernel.

Note: If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.; This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Example:

MatrixXf m = MatrixXf::Random(3,5);
cout << "Here is the matrix m:" << endl << m << endl;
MatrixXf ker = m.fullPivLu().kernel();
cout << "Here is a matrix whose columns form a basis of the kernel of m:"
     << endl << ker << endl;
cout << "By definition of the kernel, m*ker is zero:"
     << endl << m*ker << endl;

Output:

See also: image()

Definition at line 190 of file FullPivLU.h.

◆ matrixLU()

const MatrixType& Eigen::FullPivLU::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 132 of file FullPivLU.h.

◆ maxPivot()

RealScalar Eigen::FullPivLU::maxPivot ( ) const

inline

Returns: the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.

Definition at line 154 of file FullPivLU.h.

◆ nonzeroPivots()

Index Eigen::FullPivLU::nonzeroPivots ( ) const

inline

Returns: the number of nonzero pivots in the LU decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.

See also: rank()

Definition at line 145 of file FullPivLU.h.

◆ permutationP()

const EIGEN_DEVICE_FUNC PermutationPType& Eigen::FullPivLU::permutationP ( ) const

inline

Returns: the permutation matrix P

See also: permutationQ()

Definition at line 160 of file FullPivLU.h.

◆ permutationQ()

const PermutationQType& Eigen::FullPivLU::permutationQ ( ) const

inline

Returns: the permutation matrix Q

See also: permutationP()

Definition at line 170 of file FullPivLU.h.

◆ rank()

Index Eigen::FullPivLU::rank ( ) const

inline

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

Note: This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 330 of file FullPivLU.h.

◆ rcond()

RealScalar Eigen::FullPivLU::rcond ( ) const

inline

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

Definition at line 250 of file FullPivLU.h.

◆ reconstructedMatrix()

MatrixType Eigen::FullPivLU::reconstructedMatrix ( ) const

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

Definition at line 592 of file FullPivLU.h.

◆ rows()

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::FullPivLU::rows ( ) const

inline

Definition at line 408 of file FullPivLU.h.

◆ setThreshold() [1/2]

FullPivLU& Eigen::FullPivLU::setThreshold ( const RealScalar & threshold )

inline

Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. This is not used for the LU decomposition itself.

When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.

Parameters

threshold The new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than $\vert pivot \vert \leqslant threshold \times \vert maxpivot \vert$ where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

Definition at line 290 of file FullPivLU.h.

◆ setThreshold() [2/2]

FullPivLU& Eigen::FullPivLU::setThreshold ( Default_t )

inline

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

lu.setThreshold(Eigen::Default);

See the documentation of setThreshold(const RealScalar&).

Definition at line 305 of file FullPivLU.h.

◆ threshold()

RealScalar Eigen::FullPivLU::threshold ( ) const

inline

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

Definition at line 315 of file FullPivLU.h.

Friends And Related Function Documentation

◆ SolverBase< FullPivLU >

friend class SolverBase< FullPivLU >

friend

Definition at line 66 of file FullPivLU.h.

Member Data Documentation

◆ m_colsTranspositions

IntRowVectorType Eigen::FullPivLU::m_colsTranspositions

protected

Definition at line 433 of file FullPivLU.h.

◆ m_det_pq

signed char Eigen::FullPivLU::m_det_pq

protected

Definition at line 437 of file FullPivLU.h.

◆ m_isInitialized

bool Eigen::FullPivLU::m_isInitialized

protected

Definition at line 438 of file FullPivLU.h.

◆ m_l1_norm

RealScalar Eigen::FullPivLU::m_l1_norm

protected

Definition at line 435 of file FullPivLU.h.

◆ m_lu

MatrixType Eigen::FullPivLU::m_lu

protected

Definition at line 429 of file FullPivLU.h.

◆ m_maxpivot

RealScalar Eigen::FullPivLU::m_maxpivot

protected

Definition at line 436 of file FullPivLU.h.

◆ m_nonzero_pivots

Index Eigen::FullPivLU::m_nonzero_pivots

protected

Definition at line 434 of file FullPivLU.h.

◆ m_p

PermutationPType Eigen::FullPivLU::m_p

protected

Definition at line 430 of file FullPivLU.h.

◆ m_prescribedThreshold

RealScalar Eigen::FullPivLU::m_prescribedThreshold

protected

Definition at line 436 of file FullPivLU.h.

◆ m_q

PermutationQType Eigen::FullPivLU::m_q

protected

Definition at line 431 of file FullPivLU.h.

◆ m_rowsTranspositions

IntColVectorType Eigen::FullPivLU::m_rowsTranspositions

protected

Definition at line 432 of file FullPivLU.h.

◆ m_usePrescribedThreshold

bool Eigen::FullPivLU::m_usePrescribedThreshold

protected

Definition at line 438 of file FullPivLU.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

Friends

Detailed Description

Member Typedef Documentation

◆ Base

◆ IntColVectorType

◆ IntRowVectorType

◆ MatrixType

◆ PermutationPType

◆ PermutationQType

◆ PlainObject

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ FullPivLU() [1/6]

◆ FullPivLU() [2/6]

◆ FullPivLU() [3/6]

◆ FullPivLU() [4/6]

◆ FullPivLU() [5/6]

◆ FullPivLU() [6/6]

Member Function Documentation

◆ _solve_impl()

◆ _solve_impl_transposed()

◆ check_template_parameters()

◆ cols()

◆ compute()

◆ computeInPlace()

◆ determinant()

◆ dimensionOfKernel()

◆ image()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ kernel()

◆ matrixLU()

◆ maxPivot()

◆ nonzeroPivots()

◆ permutationP()

◆ permutationQ()

◆ rank()

◆ rcond()

◆ reconstructedMatrix()

◆ rows()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ threshold()

Friends And Related Function Documentation

◆ SolverBase< FullPivLU >

Member Data Documentation

◆ m_colsTranspositions

◆ m_det_pq

◆ m_isInitialized

◆ m_l1_norm

◆ m_lu

◆ m_maxpivot

◆ m_nonzero_pivots

◆ m_p

◆ m_prescribedThreshold

◆ m_q

◆ m_rowsTranspositions

◆ m_usePrescribedThreshold