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

#include </home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/LU/FullPivLU.h>

Inheritance diagram for Eigen::FullPivLU< _MatrixType >:

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

typedef _MatrixType	MatrixType

typedef SolverBase< FullPivLU >	Base

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

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

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationQType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationPType

typedef MatrixType::PlainObject	PlainObject

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

typedef EigenBase< FullPivLU< _MatrixType > >	Base

typedef internal::traits< FullPivLU< _MatrixType > >::Scalar	Scalar

typedef Scalar	CoeffReturnType

typedef internal::add_const< Transpose< constDerived > >::type	ConstTransposeReturnType

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

Public Types inherited from Eigen::EigenBase< FullPivLU< _MatrixType > >
typedef Eigen::Index	Index
	The interface type of indices. More...

typedef internal::traits< FullPivLU< _MatrixType > >::StorageKind	StorageKind

Public Member Functions
	FullPivLU ()
	Default Constructor. More...

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

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

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

template<typename InputType >
FullPivLU &	compute (const EigenBase< InputType > &matrix)
	Computes the LU decomposition of the given matrix. More...

const MatrixType &	matrixLU () const

Index	nonzeroPivots () const

RealScalar	maxPivot () const

EIGEN_DEVICE_FUNC const PermutationPType &	permutationP () const

const PermutationQType &	permutationQ () const

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

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

RealScalar	rcond () const

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

FullPivLU &	setThreshold (const RealScalar &threshold)
	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. More...

FullPivLU &	setThreshold (Default_t)
	Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold. More...

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

Index	rank () const

Index	dimensionOfKernel () const

bool	isInjective () const

bool	isSurjective () const

bool	isInvertible () const

const Inverse< FullPivLU >	inverse () const

MatrixType	reconstructedMatrix () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

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

Public Member Functions inherited from Eigen::SolverBase< FullPivLU< _MatrixType > >
	SolverBase ()
	Default constructor. More...

	~SolverBase ()

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

ConstTransposeReturnType	transpose () const

AdjointReturnType	adjoint () const

EIGEN_DEVICE_FUNC FullPivLU< _MatrixType > &	derived ()

EIGEN_DEVICE_FUNC const FullPivLU< _MatrixType > &	derived () const

Public Member Functions inherited from Eigen::EigenBase< FullPivLU< _MatrixType > >
EIGEN_DEVICE_FUNC FullPivLU< _MatrixType > &	derived ()

EIGEN_DEVICE_FUNC const FullPivLU< _MatrixType > &	derived () const

EIGEN_DEVICE_FUNC FullPivLU< _MatrixType > &	const_cast_derived () const

EIGEN_DEVICE_FUNC const FullPivLU< _MatrixType > &	const_derived () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const

EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const

EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const

EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const

Protected Member Functions
void	computeInPlace ()

Protected Member Functions inherited from Eigen::SolverBase< FullPivLU< _MatrixType > >
void	_check_solve_assertion (const Rhs &b) const

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
MatrixType	m_lu

PermutationPType	m_p

PermutationQType	m_q

IntColVectorType	m_rowsTranspositions

IntRowVectorType	m_colsTranspositions

Index	m_nonzero_pivots

RealScalar	m_l1_norm

RealScalar	m_maxpivot

RealScalar	m_prescribedThreshold

signed char	m_det_pq

bool	m_isInitialized

bool	m_usePrescribedThreshold

Friends
class	SolverBase< FullPivLU >

Detailed Description

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

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:

Output:

This class supports the inplace decomposition mechanism.

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

Member Typedef Documentation

◆ Base

template<typename _MatrixType >

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

◆ IntColVectorType

template<typename _MatrixType >

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

◆ IntRowVectorType

template<typename _MatrixType >

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

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::FullPivLU< _MatrixType >::MatrixType

◆ PermutationPType

template<typename _MatrixType >

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

◆ PermutationQType

template<typename _MatrixType >

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

◆ PlainObject

template<typename _MatrixType >

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

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Constructor & Destructor Documentation

◆ FullPivLU() [1/4]

template<typename MatrixType >

Eigen::FullPivLU< MatrixType >::FullPivLU

Default Constructor.

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

◆ FullPivLU() [2/4]

template<typename MatrixType >

Eigen::FullPivLU< MatrixType >::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() [3/4]

template<typename MatrixType >

template<typename InputType >

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

explicit

Constructor.

Parameters

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

◆ FullPivLU() [4/4]

template<typename MatrixType >

template<typename InputType >

Eigen::FullPivLU< MatrixType >::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&)

Member Function Documentation

◆ _solve_impl()

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

◆ _solve_impl_transposed()

template<typename _MatrixType >

template<bool Conjugate, typename RhsType , typename DstType >

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

◆ check_template_parameters()

template<typename _MatrixType >

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

inlinestaticprotected

◆ cols()

template<typename _MatrixType >

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::FullPivLU< _MatrixType >::cols ( ) const

inline

◆ compute()

template<typename _MatrixType >

template<typename InputType >

FullPivLU & Eigen::FullPivLU< _MatrixType >::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

◆ computeInPlace()

template<typename MatrixType >

void Eigen::FullPivLU< MatrixType >::computeInPlace

protected

◆ determinant()

template<typename MatrixType >

internal::traits< MatrixType >::Scalar Eigen::FullPivLU< MatrixType >::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: 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()

◆ dimensionOfKernel()

template<typename _MatrixType >

Index Eigen::FullPivLU< _MatrixType >::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&).

◆ image()

template<typename _MatrixType >

const internal::image_retval< FullPivLU > Eigen::FullPivLU< _MatrixType >::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:

Output:

See also: kernel()

◆ inverse()

template<typename _MatrixType >

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

◆ isInjective()

template<typename _MatrixType >

bool Eigen::FullPivLU< _MatrixType >::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&).

◆ isInvertible()

template<typename _MatrixType >

bool Eigen::FullPivLU< _MatrixType >::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&).

◆ isSurjective()

template<typename _MatrixType >

bool Eigen::FullPivLU< _MatrixType >::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&).

◆ kernel()

template<typename _MatrixType >

const internal::kernel_retval< FullPivLU > Eigen::FullPivLU< _MatrixType >::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:

Output:

See also: image()

◆ matrixLU()

template<typename _MatrixType >

const MatrixType & Eigen::FullPivLU< _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()

◆ maxPivot()

template<typename _MatrixType >

RealScalar Eigen::FullPivLU< _MatrixType >::maxPivot ( ) const

inline

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

◆ nonzeroPivots()

template<typename _MatrixType >

Index Eigen::FullPivLU< _MatrixType >::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()

◆ permutationP()

template<typename _MatrixType >

EIGEN_DEVICE_FUNC const PermutationPType & Eigen::FullPivLU< _MatrixType >::permutationP ( ) const

inline

Returns: the permutation matrix P

See also: permutationQ()

◆ permutationQ()

template<typename _MatrixType >

const PermutationQType & Eigen::FullPivLU< _MatrixType >::permutationQ ( ) const

inline

Returns: the permutation matrix Q

See also: permutationP()

◆ rank()

template<typename _MatrixType >

Index Eigen::FullPivLU< _MatrixType >::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&).

◆ rcond()

template<typename _MatrixType >

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

inline

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

◆ reconstructedMatrix()

template<typename MatrixType >

MatrixType Eigen::FullPivLU< MatrixType >::reconstructedMatrix

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.

◆ rows()

template<typename _MatrixType >

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::FullPivLU< _MatrixType >::rows ( ) const

inline

◆ setThreshold() [1/2]

template<typename _MatrixType >

FullPivLU & Eigen::FullPivLU< _MatrixType >::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)

◆ setThreshold() [2/2]

template<typename _MatrixType >

FullPivLU & Eigen::FullPivLU< _MatrixType >::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);

Eigen::Default

@ Default

Definition: Constants.h:362

See the documentation of setThreshold(const RealScalar&).

◆ threshold()

template<typename _MatrixType >

RealScalar Eigen::FullPivLU< _MatrixType >::threshold ( ) const

inline

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

See the documentation of setThreshold(const RealScalar&).

Friends And Related Function Documentation

◆ SolverBase< FullPivLU >

template<typename _MatrixType >

friend class SolverBase< FullPivLU >

friend

Member Data Documentation

◆ m_colsTranspositions

template<typename _MatrixType >

IntRowVectorType Eigen::FullPivLU< _MatrixType >::m_colsTranspositions

protected

◆ m_det_pq

template<typename _MatrixType >

signed char Eigen::FullPivLU< _MatrixType >::m_det_pq

protected

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::FullPivLU< _MatrixType >::m_isInitialized

protected

◆ m_l1_norm

template<typename _MatrixType >

RealScalar Eigen::FullPivLU< _MatrixType >::m_l1_norm

protected

◆ m_lu

template<typename _MatrixType >

MatrixType Eigen::FullPivLU< _MatrixType >::m_lu

protected

◆ m_maxpivot

template<typename _MatrixType >

RealScalar Eigen::FullPivLU< _MatrixType >::m_maxpivot

protected

◆ m_nonzero_pivots

template<typename _MatrixType >

Index Eigen::FullPivLU< _MatrixType >::m_nonzero_pivots

protected

◆ m_p

template<typename _MatrixType >

PermutationPType Eigen::FullPivLU< _MatrixType >::m_p

protected

◆ m_prescribedThreshold

template<typename _MatrixType >

RealScalar Eigen::FullPivLU< _MatrixType >::m_prescribedThreshold

protected

◆ m_q

template<typename _MatrixType >

PermutationQType Eigen::FullPivLU< _MatrixType >::m_q

protected

◆ m_rowsTranspositions

template<typename _MatrixType >

IntColVectorType Eigen::FullPivLU< _MatrixType >::m_rowsTranspositions

protected

◆ m_usePrescribedThreshold

template<typename _MatrixType >

bool Eigen::FullPivLU< _MatrixType >::m_usePrescribedThreshold

protected

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

/home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/LU/FullPivLU.h

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() [1/4]

◆ FullPivLU() [2/4]

◆ FullPivLU() [3/4]

◆ FullPivLU() [4/4]

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