Householder rank-revealing QR decomposition of a matrix with full pivoting. More...

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

Public Types
enum	{ MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef SolverBase< FullPivHouseholderQR >	Base

typedef internal::plain_col_type< MatrixType >::type	ColVectorType

typedef internal::plain_diag_type< MatrixType >::type	HCoeffsType

typedef Matrix< PermutationIndex, 1, internal::min_size_prefer_dynamic(ColsAtCompileTime, RowsAtCompileTime), RowMajor, 1, internal::min_size_prefer_fixed(MaxColsAtCompileTime, MaxRowsAtCompileTime)>	IntDiagSizeVectorType

typedef internal::FullPivHouseholderQRMatrixQReturnType< MatrixType, PermutationIndex >	MatrixQReturnType

typedef MatrixType_	MatrixType

typedef PermutationIndex_	PermutationIndex

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationType

typedef MatrixType::PlainObject	PlainObject

typedef internal::plain_row_type< MatrixType >::type	RowVectorType

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

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

typedef EigenBase< FullPivHouseholderQR< MatrixType_, PermutationIndex_ > >	Base

typedef Scalar	CoeffReturnType

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

typedef internal::traits< FullPivHouseholderQR< 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
MatrixType::RealScalar	absDeterminant () const

Index	cols () const

const PermutationType &	colsPermutation () const

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

template<typename InputType >
FullPivHouseholderQR< MatrixType, PermutationIndex > &	compute (const EigenBase< InputType > &matrix)

MatrixType::Scalar	determinant () const

Index	dimensionOfKernel () const

	FullPivHouseholderQR ()
	Default Constructor. More...

template<typename InputType >
	FullPivHouseholderQR (const EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix. More...

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

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

const HCoeffsType &	hCoeffs () const

const Inverse< FullPivHouseholderQR >	inverse () const

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

MatrixType::RealScalar	logAbsDeterminant () const

MatrixQReturnType	matrixQ (void) const

const MatrixType &	matrixQR () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

Index	rank () const

Index	rows () const

const IntDiagSizeVectorType &	rowsTranspositions () const

FullPivHouseholderQR &	setThreshold (const RealScalar &threshold)

FullPivHouseholderQR &	setThreshold (Default_t)

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

RealScalar	threshold () const

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

FullPivHouseholderQR< MatrixType_, PermutationIndex_ > &	derived ()

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

const Solve< FullPivHouseholderQR< 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< FullPivHouseholderQR< MatrixType_, PermutationIndex_ > >
void	_check_solve_assertion (const Rhs &b) const

Protected Attributes
PermutationType	m_cols_permutation

IntDiagSizeVectorType	m_cols_transpositions

Index	m_det_p

HCoeffsType	m_hCoeffs

bool	m_isInitialized

RealScalar	m_maxpivot

Index	m_nonzero_pivots

RealScalar	m_precision

RealScalar	m_prescribedThreshold

MatrixType	m_qr

IntDiagSizeVectorType	m_rows_transpositions

RowVectorType	m_temp

bool	m_usePrescribedThreshold

Detailed Description

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

Householder rank-revealing QR decomposition of a matrix with full pivoting.

Template Parameters

MatrixType_ the type of the matrix of which we are computing the QR decomposition

This class performs a rank-revealing QR decomposition of a matrix A into matrices P, P', Q and R such that

\[ \mathbf{P} \, \mathbf{A} \, \mathbf{P}' = \mathbf{Q} \, \mathbf{R} \]

by using Householder transformations. Here, P and P' are permutation matrices, Q a unitary matrix and R an upper triangular matrix.

This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::fullPivHouseholderQr()

Definition at line 62 of file FullPivHouseholderQR.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 68 of file FullPivHouseholderQR.h.

◆ ColVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_col_type<MatrixType>::type Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColVectorType

Definition at line 84 of file FullPivHouseholderQR.h.

◆ HCoeffsType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_diag_type<MatrixType>::type Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::HCoeffsType

Definition at line 78 of file FullPivHouseholderQR.h.

◆ IntDiagSizeVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef Matrix<PermutationIndex, 1, internal::min_size_prefer_dynamic(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1, internal::min_size_prefer_fixed(MaxColsAtCompileTime, MaxRowsAtCompileTime)> Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::IntDiagSizeVectorType

Definition at line 81 of file FullPivHouseholderQR.h.

◆ MatrixQReturnType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType, PermutationIndex> Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::MatrixQReturnType

Definition at line 77 of file FullPivHouseholderQR.h.

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 67 of file FullPivHouseholderQR.h.

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationIndex_ Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::PermutationIndex

Definition at line 70 of file FullPivHouseholderQR.h.

◆ PermutationType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex> Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::PermutationType

Definition at line 82 of file FullPivHouseholderQR.h.

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 85 of file FullPivHouseholderQR.h.

◆ RowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_row_type<MatrixType>::type Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::RowVectorType

Definition at line 83 of file FullPivHouseholderQR.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 73 of file FullPivHouseholderQR.h.

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

Constructor & Destructor Documentation

◆ FullPivHouseholderQR() [1/4]

template<typename MatrixType_ , typename PermutationIndex_ >

Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::FullPivHouseholderQR ( )

inline

Default Constructor.

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

Definition at line 92 of file FullPivHouseholderQR.h.

       : m_qr(),
         m_hCoeffs(),
         m_rows_transpositions(),
         m_cols_transpositions(),
         m_cols_permutation(),
         m_temp(),
         m_isInitialized(false),
         m_usePrescribedThreshold(false) {}

◆ FullPivHouseholderQR() [2/4]

template<typename MatrixType_ , typename PermutationIndex_ >

Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::FullPivHouseholderQR	(	Index	rows,
		Index	cols
	)

inline

Default Constructor with memory preallocation.

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

See also: FullPivHouseholderQR()

Definition at line 108 of file FullPivHouseholderQR.h.

       : m_qr(rows, cols),
         m_hCoeffs((std::min)(rows,cols)),
         m_rows_transpositions((std::min)(rows,cols)),
         m_cols_transpositions((std::min)(rows,cols)),
         m_cols_permutation(cols),
         m_temp(cols),
         m_isInitialized(false),
         m_usePrescribedThreshold(false) {}

◆ FullPivHouseholderQR() [3/4]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::FullPivHouseholderQR ( const EigenBase< InputType > & matrix )

inlineexplicit

Constructs a QR factorization from a given matrix.

This constructor computes the QR factorization of the matrix matrix by calling the method compute(). It is a short cut for:

FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());

qr.compute(matrix);

qr

HouseholderQR< MatrixXf > qr(A)

See also: compute()

Definition at line 131 of file FullPivHouseholderQR.h.

       : m_qr(matrix.rows(), matrix.cols()),
         m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
         m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
         m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
         m_cols_permutation(matrix.cols()),
         m_temp(matrix.cols()),
         m_isInitialized(false),
         m_usePrescribedThreshold(false)
     {
       compute(matrix.derived());
     }

◆ FullPivHouseholderQR() [4/4]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::FullPivHouseholderQR ( EigenBase< InputType > & matrix )

inlineexplicit

Constructs a QR factorization from a given matrix.

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

See also: FullPivHouseholderQR(const EigenBase&)

Definition at line 151 of file FullPivHouseholderQR.h.

       : m_qr(matrix.derived()),
         m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
         m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
         m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
         m_cols_permutation(matrix.cols()),
         m_temp(matrix.cols()),
         m_isInitialized(false),
         m_usePrescribedThreshold(false)
     {
       computeInPlace();
     }

Member Function Documentation

◆ absDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType, PermutationIndex >::absDeterminant

Returns: the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.

Note: This is only for square matrices.

Warning: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.

See also: determinant(), logAbsDeterminant(), MatrixBase::determinant()

Definition at line 452 of file FullPivHouseholderQR.h.

 {
   using std::abs;
   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   return abs(m_qr.diagonal().prod());
 }

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 337 of file FullPivHouseholderQR.h.

337 { return m_qr.cols(); }

◆ colsPermutation()

template<typename MatrixType_ , typename PermutationIndex_ >

const PermutationType& Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation matrix

Definition at line 201 of file FullPivHouseholderQR.h.

     {
       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
       return m_cols_permutation;
     }

◆ compute() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

FullPivHouseholderQR& Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::compute ( const EigenBase< InputType > & matrix )

◆ compute() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

FullPivHouseholderQR<MatrixType, PermutationIndex>& Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::compute ( const EigenBase< InputType > & matrix )

Performs the QR factorization of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See also: class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)

Definition at line 476 of file FullPivHouseholderQR.h.

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

◆ computeInPlace()

template<typename MatrixType , typename PermutationIndex >

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

protected

Definition at line 484 of file FullPivHouseholderQR.h.

 {
   eigen_assert(m_qr.cols() <= NumTraits<PermutationIndex>::highest());
   using std::abs;
   Index rows = m_qr.rows();
   Index cols = m_qr.cols();
   Index size = (std::min)(rows,cols);
  
   
   m_hCoeffs.resize(size);
  
   m_temp.resize(cols);
  
   m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);
  
   m_rows_transpositions.resize(size);
   m_cols_transpositions.resize(size);
   Index number_of_transpositions = 0;
  
   RealScalar biggest(0);
  
   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)
   {
     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 score = m_qr.bottomRightCorner(rows-k, cols-k)
                       .unaryExpr(Scoring())
                       .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
     row_of_biggest_in_corner += k;
     col_of_biggest_in_corner += k;
     RealScalar biggest_in_corner = internal::abs_knowing_score<Scalar>()(m_qr(row_of_biggest_in_corner, col_of_biggest_in_corner), score);
     if(k==0) biggest = biggest_in_corner;
  
     // if the corner is negligible, then we have less than full rank, and we can finish early
     if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
     {
       m_nonzero_pivots = k;
       for(Index i = k; i < size; i++)
       {
         m_rows_transpositions.coeffRef(i) = internal::convert_index<PermutationIndex>(i);
         m_cols_transpositions.coeffRef(i) = internal::convert_index<PermutationIndex>(i);
         m_hCoeffs.coeffRef(i) = Scalar(0);
       }
       break;
     }
  
     m_rows_transpositions.coeffRef(k) = internal::convert_index<PermutationIndex>(row_of_biggest_in_corner);
     m_cols_transpositions.coeffRef(k) = internal::convert_index<PermutationIndex>(col_of_biggest_in_corner);
     if(k != row_of_biggest_in_corner) {
       m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
       ++number_of_transpositions;
     }
     if(k != col_of_biggest_in_corner) {
       m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
       ++number_of_transpositions;
     }
  
     RealScalar beta;
     m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
     m_qr.coeffRef(k,k) = beta;
  
     // remember the maximum absolute value of diagonal coefficients
     if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
  
     m_qr.bottomRightCorner(rows-k, cols-k-1)
         .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
   }
  
   m_cols_permutation.setIdentity(cols);
   for(Index k = 0; k < size; ++k)
     m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));
  
   m_det_p = (number_of_transpositions%2) ? -1 : 1;
   m_isInitialized = true;
 }

◆ determinant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::Scalar Eigen::FullPivHouseholderQR< MatrixType, PermutationIndex >::determinant

Returns: the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.

Note: This is only for square matrices.

Warning: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.

See also: absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()

Definition at line 442 of file FullPivHouseholderQR.h.

 {
   eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   Scalar detQ;
   internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
   return m_qr.diagonal().prod() * detQ * Scalar(m_det_p);
 }

◆ dimensionOfKernel()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::dimensionOfKernel ( ) const

inline

Returns: the dimension of the kernel of the matrix of which *this is the QR 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 281 of file FullPivHouseholderQR.h.

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

◆ hCoeffs()

template<typename MatrixType_ , typename PermutationIndex_ >

const HCoeffsType& Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::hCoeffs ( ) const

inline

Returns: a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

Definition at line 343 of file FullPivHouseholderQR.h.

343 { return m_hCoeffs; }

◆ inverse()

template<typename MatrixType_ , typename PermutationIndex_ >

const Inverse<FullPivHouseholderQR> Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::inverse ( ) const

inline

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

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

Definition at line 330 of file FullPivHouseholderQR.h.

     {
       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
       return Inverse<FullPivHouseholderQR>(*this);
     }

◆ isInjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective ( ) const

inline

Returns: true if the matrix of which *this is the QR 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 294 of file FullPivHouseholderQR.h.

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

◆ isInvertible()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInvertible ( ) const

inline

Returns: true if the matrix of which *this is the QR 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 319 of file FullPivHouseholderQR.h.

     {
       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
       return isInjective() && isSurjective();
     }

◆ isSurjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::isSurjective ( ) const

inline

Returns: true if the matrix of which *this is the QR 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 307 of file FullPivHouseholderQR.h.

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

◆ logAbsDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType, PermutationIndex >::logAbsDeterminant

Returns: the natural log of the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.

Note: This is only for square matrices.; This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.

See also: determinant(), absDeterminant(), MatrixBase::determinant()

Definition at line 461 of file FullPivHouseholderQR.h.

 {
   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   return m_qr.diagonal().cwiseAbs().array().log().sum();
 }

◆ matrixQ()

template<typename MatrixType , typename PermutationIndex >

FullPivHouseholderQR< MatrixType, PermutationIndex >::MatrixQReturnType Eigen::FullPivHouseholderQR< MatrixType, PermutationIndex >::matrixQ ( void ) const

inline

Returns: Expression object representing the matrix Q

Definition at line 718 of file FullPivHouseholderQR.h.

 {
   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
   return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
 }

◆ matrixQR()

template<typename MatrixType_ , typename PermutationIndex_ >

const MatrixType& Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixQR ( ) const

inline

Returns: a reference to the matrix where the Householder QR decomposition is stored

Definition at line 191 of file FullPivHouseholderQR.h.

     {
       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
       return m_qr;
     }

◆ maxPivot()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

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

Definition at line 412 of file FullPivHouseholderQR.h.

412 { return m_maxpivot; }

◆ nonzeroPivots()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::nonzeroPivots ( ) const

inline

Returns: the number of nonzero pivots in the QR 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 403 of file FullPivHouseholderQR.h.

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

◆ rank()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::rank ( ) const

inline

Returns: the rank of the matrix of which *this is the QR 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 264 of file FullPivHouseholderQR.h.

     {
       using std::abs;
       eigen_assert(m_isInitialized && "FullPivHouseholderQR 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_qr.coeff(i,i)) > premultiplied_threshold);
       return result;
     }

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 336 of file FullPivHouseholderQR.h.

336 { return m_qr.rows(); }

◆ rowsTranspositions()

template<typename MatrixType_ , typename PermutationIndex_ >

const IntDiagSizeVectorType& Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::rowsTranspositions ( ) const

inline

Returns: a const reference to the vector of indices representing the rows transpositions

Definition at line 208 of file FullPivHouseholderQR.h.

     {
       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
       return m_rows_transpositions;
     }

◆ setThreshold() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

FullPivHouseholderQR& Eigen::FullPivHouseholderQR< 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 QR 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 362 of file FullPivHouseholderQR.h.

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

◆ setThreshold() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

FullPivHouseholderQR& Eigen::FullPivHouseholderQR< 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.

qr.setThreshold(Eigen::Default);

Eigen::Default

@ Default

Definition: Constants.h:364

See the documentation of setThreshold(const RealScalar&).

Definition at line 377 of file FullPivHouseholderQR.h.

     {
       m_usePrescribedThreshold = false;
       return *this;
     }

◆ solve()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename Rhs >

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

inline

This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition.

Parameters

b	the right-hand-side of the equation to solve.

Returns: the exact or least-square solution if the rank is greater or equal to the number of columns of A, and an arbitrary solution otherwise.

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.

Example:

Matrix3f m = Matrix3f::Random();
Matrix3f y = Matrix3f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix3f x;
x = m.fullPivHouseholderQr().solve(y);
assert(y.isApprox(m*x));
cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;

Output:

Here is the matrix m:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Here is the matrix y:
  0.108   -0.27   0.832
-0.0452  0.0268   0.271
  0.258   0.904   0.435
Here is a solution x to the equation mx=y:
 0.609   2.68   1.67
-0.231  -1.57 0.0713
  0.51   3.51   1.05

◆ threshold()

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::FullPivHouseholderQR< 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 387 of file FullPivHouseholderQR.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_qr.diagonalSize());
     }

Member Data Documentation

◆ m_cols_permutation

template<typename MatrixType_ , typename PermutationIndex_ >

PermutationType Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_cols_permutation

protected

Definition at line 432 of file FullPivHouseholderQR.h.

◆ m_cols_transpositions

template<typename MatrixType_ , typename PermutationIndex_ >

IntDiagSizeVectorType Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_cols_transpositions

protected

Definition at line 431 of file FullPivHouseholderQR.h.

◆ m_det_p

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_det_p

protected

Definition at line 438 of file FullPivHouseholderQR.h.

◆ m_hCoeffs

template<typename MatrixType_ , typename PermutationIndex_ >

HCoeffsType Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_hCoeffs

protected

Definition at line 429 of file FullPivHouseholderQR.h.

◆ m_isInitialized

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 434 of file FullPivHouseholderQR.h.

◆ m_maxpivot

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 435 of file FullPivHouseholderQR.h.

◆ m_nonzero_pivots

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 436 of file FullPivHouseholderQR.h.

◆ m_precision

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_precision

protected

Definition at line 437 of file FullPivHouseholderQR.h.

◆ m_prescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 435 of file FullPivHouseholderQR.h.

◆ m_qr

template<typename MatrixType_ , typename PermutationIndex_ >

MatrixType Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr

protected

Definition at line 428 of file FullPivHouseholderQR.h.

◆ m_rows_transpositions

template<typename MatrixType_ , typename PermutationIndex_ >

IntDiagSizeVectorType Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_rows_transpositions

protected

Definition at line 430 of file FullPivHouseholderQR.h.

◆ m_temp

template<typename MatrixType_ , typename PermutationIndex_ >

RowVectorType Eigen::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_temp

protected

Definition at line 433 of file FullPivHouseholderQR.h.

◆ m_usePrescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 434 of file FullPivHouseholderQR.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::FullPivHouseholderQR< MatrixType_, PermutationIndex_ >

Member Typedef Documentation

◆ Base

◆ ColVectorType

◆ HCoeffsType

◆ IntDiagSizeVectorType

◆ MatrixQReturnType

◆ MatrixType

◆ PermutationIndex

◆ PermutationType

◆ PlainObject

◆ RowVectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ FullPivHouseholderQR() [1/4]

◆ FullPivHouseholderQR() [2/4]

◆ FullPivHouseholderQR() [3/4]

◆ FullPivHouseholderQR() [4/4]

Member Function Documentation

◆ absDeterminant()

◆ cols()

◆ colsPermutation()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeInPlace()

◆ determinant()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQ()

◆ matrixQR()

◆ maxPivot()

◆ nonzeroPivots()

◆ rank()

◆ rows()

◆ rowsTranspositions()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ solve()

◆ threshold()

Member Data Documentation

◆ m_cols_permutation

◆ m_cols_transpositions

◆ m_det_p

◆ m_hCoeffs

◆ m_isInitialized

◆ m_maxpivot

◆ m_nonzero_pivots

◆ m_precision

◆ m_prescribedThreshold

◆ m_qr

◆ m_rows_transpositions

◆ m_temp

◆ m_usePrescribedThreshold

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