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

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

Public Types
enum	{ MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef SolverBase< FullPivLU >	Base

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

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

typedef MatrixType_	MatrixType

using	PermutationIndex = PermutationIndex_

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex >	PermutationPType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationQType

typedef MatrixType::PlainObject	PlainObject

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

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

typedef EigenBase< FullPivLU< MatrixType_, PermutationIndex_ > >	Base

typedef Scalar	CoeffReturnType

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

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

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

typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

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

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

Index	dimensionOfKernel () const

	FullPivLU ()
	Default Constructor. 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...

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

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 PermutationPType &	permutationP () const

const PermutationQType &	permutationQ () const

Index	rank () const

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

FullPivLU &	setThreshold (const RealScalar &threshold)

FullPivLU &	setThreshold (Default_t)

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

RealScalar	threshold () const

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

FullPivLU< MatrixType_, PermutationIndex_ > &	derived ()

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

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

	SolverBase ()

const ConstTransposeReturnType	transpose () const

	~SolverBase ()

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

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

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

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

Derived &	const_cast_derived () const

const Derived &	const_derived () const

Derived &	derived ()

const Derived &	derived () const

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

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

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

Protected Member Functions
void	computeInPlace ()

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

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

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:

Here is the matrix m:
   0.68  -0.605 -0.0452
 -0.211   -0.33   0.258
  0.566   0.536   -0.27
  0.597  -0.444  0.0268
  0.823   0.108   0.904
Here is, up to permutations, its LU decomposition matrix:
 0.904  0.823  0.108
-0.299  0.812  0.569
 -0.05  0.888   -1.1
0.0296  0.705  0.768
 0.285 -0.549 0.0436
Here is the L part:
     1      0      0      0      0
-0.299      1      0      0      0
 -0.05  0.888      1      0      0
0.0296  0.705  0.768      1      0
 0.285 -0.549 0.0436      0      1
Here is the U part:
0.904 0.823 0.108
    0 0.812 0.569
    0     0  -1.1
    0     0     0
    0     0     0
Let us now reconstruct the original matrix m:
   0.68  -0.605 -0.0452
 -0.211   -0.33   0.258
  0.566   0.536   -0.27
  0.597  -0.444  0.0268
  0.823   0.108   0.904

This class supports the inplace decomposition mechanism.

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

Definition at line 62 of file FullPivLU.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 67 of file FullPivLU.h.

◆ IntColVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 77 of file FullPivLU.h.

◆ IntRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 76 of file FullPivLU.h.

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 66 of file FullPivLU.h.

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 75 of file FullPivLU.h.

◆ PermutationPType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 79 of file FullPivLU.h.

◆ PermutationQType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 78 of file FullPivLU.h.

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 80 of file FullPivLU.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 71 of file FullPivLU.h.

          {
       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&).

Definition at line 442 of file FullPivLU.h.

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

◆ 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()

Definition at line 448 of file FullPivLU.h.

   : 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.

Definition at line 461 of file FullPivLU.h.

   : 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());
 }

◆ 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&)

Definition at line 475 of file FullPivLU.h.

   : 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();
 }

Member Function Documentation

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 413 of file FullPivLU.h.

413 { return m_lu.cols(); }

◆ 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

Definition at line 123 of file FullPivLU.h.

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

◆ computeInPlace()

template<typename MatrixType , typename PermutationIndex >

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

protected

Definition at line 488 of file FullPivLU.h.

 {
   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;
 }

◆ 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()

Definition at line 578 of file FullPivLU.h.

 {
   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());
 }

◆ 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&).

Definition at line 350 of file FullPivLU.h.

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

Here is the matrix m:
1 1 0
1 3 2
0 1 1
Notice that the middle column is the sum of the two others, so the columns are linearly dependent.
Here is a matrix whose columns have the same span but are linearly independent:
1 1
3 1
1 0

See also: kernel()

Definition at line 219 of file FullPivLU.h.

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

◆ 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()

Definition at line 401 of file FullPivLU.h.

     {
       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);
     }

◆ 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&).

Definition at line 363 of file FullPivLU.h.

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

◆ 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&).

Definition at line 388 of file FullPivLU.h.

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

◆ 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&).

Definition at line 376 of file FullPivLU.h.

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

Here is the matrix m:
   0.68   0.597   -0.33   0.108   -0.27
 -0.211   0.823   0.536 -0.0452  0.0268
  0.566  -0.605  -0.444   0.258   0.904
Here is a matrix whose columns form a basis of the kernel of m:
 -0.219   0.763
0.00335  -0.447
      0       1
      1       0
 -0.145  -0.285
By definition of the kernel, m*ker is zero:
 7.45e-09  1.49e-08
-1.86e-09 -4.05e-08
        0 -2.98e-08

See also: image()

Definition at line 193 of file FullPivLU.h.

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

◆ 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()

Definition at line 135 of file FullPivLU.h.

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

Definition at line 157 of file FullPivLU.h.

157 { return 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()

Definition at line 148 of file FullPivLU.h.

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

◆ permutationP()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Returns: the permutation matrix P

See also: permutationQ()

Definition at line 163 of file FullPivLU.h.

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

Definition at line 173 of file FullPivLU.h.

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

Definition at line 333 of file FullPivLU.h.

     {
       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;
     }

◆ 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.

Definition at line 253 of file FullPivLU.h.

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

◆ reconstructedMatrix()

template<typename MatrixType , typename PermutationIndex >

MatrixType Eigen::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.

Definition at line 589 of file FullPivLU.h.

 {
   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;
 }

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 411 of file FullPivLU.h.

411 { return m_lu.rows(); }

◆ 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)

Definition at line 293 of file FullPivLU.h.

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

◆ 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:364

See the documentation of setThreshold(const RealScalar&).

Definition at line 308 of file FullPivLU.h.

     {
       m_usePrescribedThreshold = false;
       return *this;
     }

◆ solve()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename Rhs >

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

inline

Returns: a 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: a solution.

This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:

bool a_solution_exists = (A*result).isApprox(b, precision);

b

Array< int, 3, 1 > b

Definition: Array_variadic_ctor_cxx11.cpp:2

A

MatrixXcf A

Definition: ComplexEigenSolver_compute.cpp:1

Eigen::internal::isApprox

bool isApprox(const Scalar &x, const Scalar &y, const typename NumTraits< Scalar >::Real &precision=NumTraits< Scalar >::dummy_precision())

Definition: MathFunctions.h:1882

This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get inf or nan values.

If there exists more than one solution, this method will arbitrarily choose one. If you need a complete analysis of the space of solutions, take the one solution obtained by this method and add to it elements of the kernel, as determined by kernel().

Example:

Matrix<float,2,3> m = Matrix<float,2,3>::Random();
Matrix2f y = Matrix2f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix<float,3,2> x = m.fullPivLu().solve(y);
if((m*x).isApprox(y))
{
  cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;
}
else
  cout << "The equation mx=y does not have any solution." << endl;

Output:

Here is the matrix m:
  0.68  0.566  0.823
-0.211  0.597 -0.605
Here is the matrix y:
 -0.33 -0.444
 0.536  0.108
Here is a solution x to the equation mx=y:
     0      0
 0.291 -0.216
  -0.6 -0.391

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

◆ 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&).

Definition at line 318 of file FullPivLU.h.

     {
       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());
     }

Member Data Documentation

◆ m_colsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 433 of file FullPivLU.h.

◆ m_det_pq

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 437 of file FullPivLU.h.

◆ m_isInitialized

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 438 of file FullPivLU.h.

◆ m_l1_norm

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 435 of file FullPivLU.h.

◆ m_lu

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 429 of file FullPivLU.h.

◆ m_maxpivot

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 436 of file FullPivLU.h.

◆ m_nonzero_pivots

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 434 of file FullPivLU.h.

◆ m_p

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 430 of file FullPivLU.h.

◆ m_prescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 436 of file FullPivLU.h.

◆ m_q

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 431 of file FullPivLU.h.

◆ m_rowsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 432 of file FullPivLU.h.

◆ m_usePrescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

Protected Attributes

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

◆ 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]

◆ solve()

◆ threshold()

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_ >