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

#include <FullPivLU.h>

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

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

typedef MatrixType_	MatrixType

typedef SolverBase< FullPivLU >	Base

using	PermutationIndex = PermutationIndex_

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

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

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationQType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex >	PermutationPType

typedef MatrixType::PlainObject	PlainObject

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

typedef EigenBase< FullPivLU< MatrixType_, PermutationIndex_ > >	Base

typedef internal::traits< FullPivLU< MatrixType_, PermutationIndex_ > >::Scalar	Scalar

typedef Scalar	CoeffReturnType

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

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

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
	FullPivLU ()
	Default Constructor. More...

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

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

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)

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)

FullPivLU &	setThreshold (Default_t)

RealScalar	threshold () const

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_, PermutationIndex_ > >
	SolverBase ()

	~SolverBase ()

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

const ConstTransposeReturnType	transpose () const

const AdjointReturnType	adjoint () const

constexpr EIGEN_DEVICE_FUNC FullPivLU< MatrixType_, PermutationIndex_ > &	derived ()

constexpr EIGEN_DEVICE_FUNC const FullPivLU< MatrixType_, PermutationIndex_ > &	derived () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
constexpr EIGEN_DEVICE_FUNC Derived &	derived ()

constexpr EIGEN_DEVICE_FUNC const Derived &	derived () const

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	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

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

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

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

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

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

template<typename Device >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE DeviceWrapper< Derived, Device >	device (Device &device)

template<typename Device >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE DeviceWrapper< const Derived, Device >	device (Device &device) const

Protected Member Functions
void	computeInPlace ()

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

Friends
class	SolverBase< FullPivLU >

Detailed Description

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

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()

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ IntColVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_col_type<MatrixType, PermutationIndex>::type Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::IntColVectorType

◆ IntRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_row_type<MatrixType, PermutationIndex>::type Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::IntRowVectorType

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PermutationPType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex> Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::PermutationPType

◆ PermutationQType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex> Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::PermutationQType

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

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

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

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

Constructor & Destructor Documentation

◆ FullPivLU() [1/4]

template<typename MatrixType , typename PermutationIndex >

Eigen::FullPivLU< MatrixType, PermutationIndex >::FullPivLU

Default Constructor.

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

420 : m_isInitialized(false), m_usePrescribedThreshold(false) {}

Eigen::FullPivLU::m_usePrescribedThreshold

bool m_usePrescribedThreshold

Definition: FullPivLU.h:416

Eigen::FullPivLU::m_isInitialized

bool m_isInitialized

Definition: FullPivLU.h:416

◆ FullPivLU() [2/4]

template<typename MatrixType , typename PermutationIndex >

Eigen::FullPivLU< MatrixType, PermutationIndex >::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()

     : m_lu(rows, cols),
       m_p(rows),
       m_q(cols),
       m_rowsTranspositions(rows),
       m_colsTranspositions(cols),
       m_isInitialized(false),
       m_usePrescribedThreshold(false) {}

◆ FullPivLU() [3/4]

template<typename MatrixType , typename PermutationIndex >

template<typename InputType >

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

explicit

Constructor.

Parameters

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

     : m_lu(matrix.rows(), matrix.cols()),
       m_p(matrix.rows()),
       m_q(matrix.cols()),
       m_rowsTranspositions(matrix.rows()),
       m_colsTranspositions(matrix.cols()),
       m_isInitialized(false),
       m_usePrescribedThreshold(false) {
   compute(matrix.derived());
 }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::compute(), and matrix().

◆ FullPivLU() [4/4]

template<typename MatrixType , typename PermutationIndex >

template<typename InputType >

Eigen::FullPivLU< MatrixType, PermutationIndex >::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&)

     : m_lu(matrix.derived()),
       m_p(matrix.rows()),
       m_q(matrix.cols()),
       m_rowsTranspositions(matrix.rows()),
       m_colsTranspositions(matrix.cols()),
       m_isInitialized(false),
       m_usePrescribedThreshold(false) {
   computeInPlace();
 }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::computeInPlace().

Member Function Documentation

◆ _solve_impl()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename RhsType , typename DstType >

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

                                                                                                   {
   /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
    * So we proceed as follows:
    * Step 1: compute c = P * rhs.
    * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
    * Step 3: replace c by the solution x to Ux = c. May or may not exist.
    * Step 4: result = Q * c;
    */
  
   const Index rows = this->rows(), cols = this->cols(), nonzero_pivots = this->rank();
   const Index smalldim = (std::min)(rows, cols);
  
   if (nonzero_pivots == 0) {
     dst.setZero();
     return;
   }
  
   typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
  
   // Step 1
   c = permutationP() * rhs;
  
   // Step 2
   m_lu.topLeftCorner(smalldim, smalldim).template triangularView<UnitLower>().solveInPlace(c.topRows(smalldim));
   if (rows > cols) c.bottomRows(rows - cols) -= m_lu.bottomRows(rows - cols) * c.topRows(cols);
  
   // Step 3
   m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
       .template triangularView<Upper>()
       .solveInPlace(c.topRows(nonzero_pivots));
  
   // Step 4
   for (Index i = 0; i < nonzero_pivots; ++i) dst.row(permutationQ().indices().coeff(i)) = c.row(i);
   for (Index i = nonzero_pivots; i < m_lu.cols(); ++i) dst.row(permutationQ().indices().coeff(i)).setZero();
 }

References calibrate::c, cols, i, min, and rows.

◆ _solve_impl_transposed()

template<typename MatrixType_ , typename PermutationIndex_ >

template<bool Conjugate, typename RhsType , typename DstType >

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

                                                                                                              {
   /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1},
    * and since permutations are real and unitary, we can write this
    * as   A^T = Q U^T L^T P,
    * So we proceed as follows:
    * Step 1: compute c = Q^T rhs.
    * Step 2: replace c by the solution x to U^T x = c. May or may not exist.
    * Step 3: replace c by the solution x to L^T x = c.
    * Step 4: result = P^T c.
    * If Conjugate is true, replace "^T" by "^*" above.
    */
  
   const Index rows = this->rows(), cols = this->cols(), nonzero_pivots = this->rank();
   const Index smalldim = (std::min)(rows, cols);
  
   if (nonzero_pivots == 0) {
     dst.setZero();
     return;
   }
  
   typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
  
   // Step 1
   c = permutationQ().inverse() * rhs;
  
   // Step 2
   m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
       .template triangularView<Upper>()
       .transpose()
       .template conjugateIf<Conjugate>()
       .solveInPlace(c.topRows(nonzero_pivots));
  
   // Step 3
   m_lu.topLeftCorner(smalldim, smalldim)
       .template triangularView<UnitLower>()
       .transpose()
       .template conjugateIf<Conjugate>()
       .solveInPlace(c.topRows(smalldim));
  
   // Step 4
   PermutationPType invp = permutationP().inverse().eval();
   for (Index i = 0; i < smalldim; ++i) dst.row(invp.indices().coeff(i)) = c.row(i);
   for (Index i = smalldim; i < rows; ++i) dst.row(invp.indices().coeff(i)).setZero();
 }

References calibrate::c, cols, i, Eigen::PermutationMatrix< SizeAtCompileTime, MaxSizeAtCompileTime, StorageIndex_ >::indices(), Eigen::PermutationBase< Derived >::inverse(), min, and rows.

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::cols ( ) const

inline

392 { return m_lu.cols(); }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu.

Referenced by gdb.printers._MatrixEntryIterator::__next__(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::dimensionOfKernel(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isInjective(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

◆ compute()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

FullPivLU& Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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

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

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu, and matrix().

Referenced by ctms_decompositions(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::FullPivLU(), Eigen::internal::idrs(), and Eigen::internal::idrstabl().

◆ computeInPlace()

template<typename MatrixType , typename PermutationIndex >

void Eigen::FullPivLU< MatrixType, PermutationIndex >::computeInPlace

protected

                                                              {
   eigen_assert(m_lu.rows() <= NumTraits<PermutationIndex>::highest() &&
                m_lu.cols() <= NumTraits<PermutationIndex>::highest());
  
   m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
  
   const Index size = m_lu.diagonalSize();
   const Index rows = m_lu.rows();
   const Index cols = m_lu.cols();
  
   // will store the transpositions, before we accumulate them at the end.
   // can't accumulate on-the-fly because that will be done in reverse order for the rows.
   m_rowsTranspositions.resize(m_lu.rows());
   m_colsTranspositions.resize(m_lu.cols());
   Index number_of_transpositions = 0;  // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i
  
   m_nonzero_pivots = size;  // the generic case is that in which all pivots are nonzero (invertible case)
   m_maxpivot = RealScalar(0);
  
   for (Index k = 0; k < size; ++k) {
     // First, we need to find the pivot.
  
     // biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
     Index row_of_biggest_in_corner, col_of_biggest_in_corner;
     typedef internal::scalar_score_coeff_op<Scalar> Scoring;
     typedef typename Scoring::result_type Score;
     Score biggest_in_corner;
     biggest_in_corner = m_lu.bottomRightCorner(rows - k, cols - k)
                             .unaryExpr(Scoring())
                             .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
     row_of_biggest_in_corner += k;  // correct the values! since they were computed in the corner,
     col_of_biggest_in_corner += k;  // need to add k to them.
  
     if (numext::is_exactly_zero(biggest_in_corner)) {
       // before exiting, make sure to initialize the still uninitialized transpositions
       // in a sane state without destroying what we already have.
       m_nonzero_pivots = k;
       for (Index i = k; i < size; ++i) {
         m_rowsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
         m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
       }
       break;
     }
  
     RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(
         m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner);
     if (abs_pivot > m_maxpivot) m_maxpivot = abs_pivot;
  
     // Now that we've found the pivot, we need to apply the row/col swaps to
     // bring it to the location (k,k).
  
     m_rowsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner);
     m_colsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(col_of_biggest_in_corner);
     if (k != row_of_biggest_in_corner) {
       m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
       ++number_of_transpositions;
     }
     if (k != col_of_biggest_in_corner) {
       m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
       ++number_of_transpositions;
     }
  
     // Now that the pivot is at the right location, we update the remaining
     // bottom-right corner by Gaussian elimination.
  
     if (k < rows - 1) m_lu.col(k).tail(rows - k - 1) /= m_lu.coeff(k, k);
     if (k < size - 1)
       m_lu.block(k + 1, k + 1, rows - k - 1, cols - k - 1).noalias() -=
           m_lu.col(k).tail(rows - k - 1) * m_lu.row(k).tail(cols - k - 1);
   }
  
   // the main loop is over, we still have to accumulate the transpositions to find the
   // permutations P and Q
  
   m_p.setIdentity(rows);
   for (Index k = size - 1; k >= 0; --k) m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
  
   m_q.setIdentity(cols);
   for (Index k = 0; k < size; ++k) m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
  
   m_det_pq = (number_of_transpositions % 2) ? -1 : 1;
  
   m_isInitialized = true;
 }

References cols, eigen_assert, i, Eigen::numext::is_exactly_zero(), k, rows, and size.

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::compute(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::FullPivLU().

◆ determinant()

template<typename MatrixType , typename PermutationIndex >

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

                                                                                                      {
   eigen_assert(m_isInitialized && "LU is not initialized.");
   eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
   return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
 }

References eigen_assert.

◆ dimensionOfKernel()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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&).

                                          {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return cols() - rank();
   }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::cols(), eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank().

◆ image()

template<typename MatrixType_ , typename PermutationIndex_ >

const internal::image_retval<FullPivLU> Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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()

                                                                                              {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return internal::image_retval<FullPivLU>(*this, originalMatrix);
   }

References eigen_assert, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized.

Referenced by main().

◆ inverse()

template<typename MatrixType_ , typename PermutationIndex_ >

const Inverse<FullPivLU> Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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()

                                                   {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
     return Inverse<FullPivLU>(*this);
   }

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu.

◆ isInjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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&).

                                   {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return rank() == cols();
   }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::cols(), eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank().

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isInvertible().

◆ isInvertible()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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&).

                                    {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return isInjective() && (m_lu.rows() == m_lu.cols());
   }

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu.

Referenced by inverse_general_4x4().

◆ isSurjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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&).

                                    {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return rank() == rows();
   }

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rows().

◆ kernel()

template<typename MatrixType_ , typename PermutationIndex_ >

const internal::kernel_retval<FullPivLU> Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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()

                                                                {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return internal::kernel_retval<FullPivLU>(*this);
   }

References eigen_assert, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized.

Referenced by main().

◆ matrixLU()

template<typename MatrixType_ , typename PermutationIndex_ >

const MatrixType& Eigen::FullPivLU< 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()

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

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu.

◆ maxPivot()

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::maxPivot ( ) const

inline

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

155 { return m_maxpivot; }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_maxpivot.

◆ nonzeroPivots()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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()

                                      {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return m_nonzero_pivots;
   }

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_nonzero_pivots.

◆ permutationP()

template<typename MatrixType_ , typename PermutationIndex_ >

EIGEN_DEVICE_FUNC const PermutationPType& Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::permutationP ( ) const

inline

Returns: the permutation matrix P

See also: permutationQ()

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

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_p.

◆ permutationQ()

template<typename MatrixType_ , typename PermutationIndex_ >

const PermutationQType& Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::permutationQ ( ) const

inline

Returns: the permutation matrix Q

See also: permutationP()

                                                       {
     eigen_assert(m_isInitialized && "LU is not initialized.");
     return m_q;
   }

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_q.

◆ rank()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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&).

                             {
     using std::abs;
     eigen_assert(m_isInitialized && "LU is not initialized.");
     RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
     Index result = 0;
     for (Index i = 0; i < m_nonzero_pivots; ++i) result += (abs(m_lu.coeff(i, i)) > premultiplied_threshold);
     return result;
   }

References abs(), eigen_assert, i, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_maxpivot, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_nonzero_pivots, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::threshold().

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::dimensionOfKernel(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isSurjective(), and main().

◆ rcond()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

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

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

References eigen_assert, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_l1_norm, and Eigen::internal::rcond_estimate_helper().

◆ reconstructedMatrix()

template<typename MatrixType , typename PermutationIndex >

MatrixType Eigen::FullPivLU< MatrixType, PermutationIndex >::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.

                                                                               {
   eigen_assert(m_isInitialized && "LU is not initialized.");
   const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
   // LU
   MatrixType res(m_lu.rows(), m_lu.cols());
   // FIXME the .toDenseMatrix() should not be needed...
   res = m_lu.leftCols(smalldim).template triangularView<UnitLower>().toDenseMatrix() *
         m_lu.topRows(smalldim).template triangularView<Upper>().toDenseMatrix();
  
   // P^{-1}(LU)
   res = m_p.inverse() * res;
  
   // (P^{-1}LU)Q^{-1}
   res = res * m_q.inverse();
  
   return res;
 }

References eigen_assert, min, and res.

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rows ( ) const

inline

391 { return m_lu.rows(); }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu.

Referenced by gdb.printers._MatrixEntryIterator::__next__(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isSurjective(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

◆ setThreshold() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

FullPivLU& Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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)

                                                        {
     m_usePrescribedThreshold = true;
     m_prescribedThreshold = threshold;
     return *this;
   }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_prescribedThreshold, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_usePrescribedThreshold, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::threshold().

◆ setThreshold() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

FullPivLU& Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::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:361

See the documentation of setThreshold(const RealScalar&).

                                      {
     m_usePrescribedThreshold = false;
     return *this;
   }

References Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_usePrescribedThreshold.

◆ threshold()

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::threshold ( ) const

inline

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

See the documentation of setThreshold(const RealScalar&).

                                {
     eigen_assert(m_isInitialized || m_usePrescribedThreshold);
     return m_usePrescribedThreshold ? m_prescribedThreshold
                                     // this formula comes from experimenting (see "LU precision tuning" thread on the
                                     // list) and turns out to be identical to Higham's formula used already in LDLt.
                                     : NumTraits<Scalar>::epsilon() * RealScalar(m_lu.diagonalSize());
   }

References eigen_assert, oomph::SarahBL::epsilon, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_isInitialized, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_lu, Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_prescribedThreshold, and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_usePrescribedThreshold.

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::setThreshold().

Friends And Related Function Documentation

◆ SolverBase< FullPivLU >

template<typename MatrixType_ , typename PermutationIndex_ >

friend class SolverBase< FullPivLU >

friend

Member Data Documentation

◆ m_colsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

IntRowVectorType Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_colsTranspositions

protected

◆ m_det_pq

template<typename MatrixType_ , typename PermutationIndex_ >

signed char Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_det_pq

protected

◆ m_isInitialized

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

◆ m_l1_norm

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rcond().

◆ m_lu

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::cols(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::compute(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::inverse(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::isInvertible(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::matrixLU(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank(), Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rows(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::threshold().

◆ m_maxpivot

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_maxpivot

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::maxPivot(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank().

◆ m_nonzero_pivots

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_nonzero_pivots

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::nonzeroPivots(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::rank().

◆ m_p

template<typename MatrixType_ , typename PermutationIndex_ >

PermutationPType Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_p

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::permutationP().

◆ m_prescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_prescribedThreshold

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::setThreshold(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::threshold().

◆ m_q

template<typename MatrixType_ , typename PermutationIndex_ >

PermutationQType Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_q

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::permutationQ().

◆ m_rowsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

IntColVectorType Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_rowsTranspositions

protected

◆ m_usePrescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::m_usePrescribedThreshold

protected

Referenced by Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::setThreshold(), and Eigen::FullPivLU< MatrixType_, PermutationIndex_ >::threshold().

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

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Friends

Detailed Description

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

Member Typedef Documentation

◆ Base

◆ IntColVectorType

◆ IntRowVectorType

◆ MatrixType

◆ PermutationIndex

◆ 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()

◆ 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

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