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

#include <ColPivHouseholderQR.h>

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

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

typedef MatrixType_	MatrixType

typedef SolverBase< ColPivHouseholderQR >	Base

typedef PermutationIndex_	PermutationIndex

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

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationType

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

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

typedef internal::plain_row_type< MatrixType, RealScalar >::type	RealRowVectorType

typedef HouseholderSequence< MatrixType, internal::remove_all_t< typename HCoeffsType::ConjugateReturnType > >	HouseholderSequenceType

typedef MatrixType::PlainObject	PlainObject

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

typedef EigenBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >	Base

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

typedef Scalar	CoeffReturnType

typedef Transpose< const ColPivHouseholderQR< 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
	ColPivHouseholderQR ()
	Default Constructor. More...

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

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

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

HouseholderSequenceType	householderQ () const

HouseholderSequenceType	matrixQ () const

const MatrixType &	matrixQR () const

const MatrixType &	matrixR () const

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

const PermutationType &	colsPermutation () const

MatrixType::Scalar	determinant () const

MatrixType::RealScalar	absDeterminant () const

MatrixType::RealScalar	logAbsDeterminant () const

MatrixType::Scalar	signDeterminant () const

Index	rank () const

Index	dimensionOfKernel () const

bool	isInjective () const

bool	isSurjective () const

bool	isInvertible () const

const Inverse< ColPivHouseholderQR >	inverse () const

Index	rows () const

Index	cols () const

const HCoeffsType &	hCoeffs () const

ColPivHouseholderQR &	setThreshold (const RealScalar &threshold)

ColPivHouseholderQR &	setThreshold (Default_t)

RealScalar	threshold () const

Index	nonzeroPivots () const

RealScalar	maxPivot () const

ComputationInfo	info () const
	Reports whether the QR factorization was successful. More...

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

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

Public Member Functions inherited from Eigen::SolverBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >
	SolverBase ()

	~SolverBase ()

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

const ConstTransposeReturnType	transpose () const

const AdjointReturnType	adjoint () const

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

constexpr EIGEN_DEVICE_FUNC const ColPivHouseholderQR< 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< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >
void	_check_solve_assertion (const Rhs &b) const

Private Member Functions
void	init (Index rows, Index cols)

Friends
class	SolverBase< ColPivHouseholderQR >

class	CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >

Detailed Description

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

Householder rank-revealing QR decomposition of a matrix with column-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, Q and R such that

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

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

This decomposition performs column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::colPivHouseholderQr()

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ HCoeffsType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ HouseholderSequenceType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef HouseholderSequence<MatrixType, internal::remove_all_t<typename HCoeffsType::ConjugateReturnType> > Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::HouseholderSequenceType

◆ IntRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PermutationType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ RealRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_row_type<MatrixType, RealScalar>::type Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::RealRowVectorType

◆ RowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

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

Constructor & Destructor Documentation

◆ ColPivHouseholderQR() [1/4]

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Default Constructor.

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

       : m_qr(),
         m_hCoeffs(),
         m_colsPermutation(),
         m_colsTranspositions(),
         m_temp(),
         m_colNormsUpdated(),
         m_colNormsDirect(),
         m_isInitialized(false),
         m_usePrescribedThreshold(false) {}

◆ ColPivHouseholderQR() [2/4]

template<typename MatrixType_ , typename PermutationIndex_ >

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

112 : m_qr(rows, cols) { init(rows, cols); }

Eigen::ColPivHouseholderQR::rows

Index rows() const

Definition: ColPivHouseholderQR.h:326

Eigen::ColPivHouseholderQR::cols

Index cols() const

Definition: ColPivHouseholderQR.h:327

Eigen::ColPivHouseholderQR::init

void init(Index rows, Index cols)

Definition: ColPivHouseholderQR.h:76

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::cols(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::rows().

◆ ColPivHouseholderQR() [3/4]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColPivHouseholderQR ( 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:

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

qr.compute(matrix);

qr

HouseholderQR< MatrixXf > qr(A)

matrix

Eigen::Map< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic, Eigen::ColMajor >, 0, Eigen::OuterStride<> > matrix(T *data, int rows, int cols, int stride)

Definition: common.h:85

See also: compute()

                                                                    : m_qr(matrix.rows(), matrix.cols()) {
     init(matrix.rows(), matrix.cols());
     compute(matrix.derived());
   }

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

◆ ColPivHouseholderQR() [4/4]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColPivHouseholderQR ( 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: ColPivHouseholderQR(const EigenBase&)

                                                              : m_qr(matrix.derived()) {
     init(matrix.rows(), matrix.cols());
     computeInPlace();
   }

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init(), and matrix().

Member Function Documentation

◆ _solve_impl()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename RhsType , typename DstType >

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

                                                                                                             {
   const Index nonzero_pivots = nonzeroPivots();
  
   if (nonzero_pivots == 0) {
     dst.setZero();
     return;
   }
  
   typename RhsType::PlainObject c(rhs);
  
   c.applyOnTheLeft(householderQ().setLength(nonzero_pivots).adjoint());
  
   m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
       .template triangularView<Upper>()
       .solveInPlace(c.topRows(nonzero_pivots));
  
   for (Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
   for (Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
 }

References Eigen::SolverBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >::adjoint(), calibrate::c, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::cols(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::householderQ(), i, Eigen::PermutationMatrix< SizeAtCompileTime, MaxSizeAtCompileTime, StorageIndex_ >::indices(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_colsPermutation, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::nonzeroPivots(), and setZero().

◆ _solve_impl_transposed()

template<typename MatrixType_ , typename PermutationIndex_ >

template<bool Conjugate, typename RhsType , typename DstType >

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

                                                                                                      {
   const Index nonzero_pivots = nonzeroPivots();
  
   if (nonzero_pivots == 0) {
     dst.setZero();
     return;
   }
  
   typename RhsType::PlainObject c(m_colsPermutation.transpose() * rhs);
  
   m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
       .template triangularView<Upper>()
       .transpose()
       .template conjugateIf<Conjugate>()
       .solveInPlace(c.topRows(nonzero_pivots));
  
   dst.topRows(nonzero_pivots) = c.topRows(nonzero_pivots);
   dst.bottomRows(rows() - nonzero_pivots).setZero();
  
   dst.applyOnTheLeft(householderQ().setLength(nonzero_pivots).template conjugateIf<!Conjugate>());
 }

References calibrate::c, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::householderQ(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_colsPermutation, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::nonzeroPivots(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::rows(), and Eigen::PermutationBase< Derived >::transpose().

◆ absDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::RealScalar Eigen::ColPivHouseholderQR< 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()

                                                                                                       {
   using std::abs;
   eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   return isInjective() ? abs(m_qr.diagonal().prod()) : RealScalar(0);
 }

References abs(), eigen_assert, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr.

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

327 { return m_qr.cols(); }

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr.

◆ colsPermutation()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Returns: a const reference to the column permutation matrix

                                                  {
     eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
     return m_colsPermutation;
   }

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

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::colsPermutation(), and Eigen::LevenbergMarquardt< FunctorType_ >::minimizeOptimumStorageOneStep().

◆ compute() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

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

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColPivHouseholderQR(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::compute(), and ctms_decompositions().

◆ compute() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

ColPivHouseholderQR<MatrixType, PermutationIndex>& Eigen::ColPivHouseholderQR< 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 ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)

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

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr, and matrix().

◆ computeInPlace()

template<typename MatrixType , typename PermutationIndex >

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

protected

                                                                        {
   eigen_assert(m_qr.cols() <= NumTraits<PermutationIndex>::highest());
  
   using std::abs;
  
   Index rows = m_qr.rows();
   Index cols = m_qr.cols();
   Index size = m_qr.diagonalSize();
  
   m_hCoeffs.resize(size);
  
   m_temp.resize(cols);
  
   m_colsTranspositions.resize(m_qr.cols());
   Index number_of_transpositions = 0;
  
   m_colNormsUpdated.resize(cols);
   m_colNormsDirect.resize(cols);
   for (Index k = 0; k < cols; ++k) {
     // colNormsDirect(k) caches the most recent directly computed norm of
     // column k.
     m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
     m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
   }
  
   RealScalar threshold_helper =
       numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
   RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());
  
   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 look up in our table m_colNormsUpdated which column has the biggest norm
     Index biggest_col_index;
     RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols - k).maxCoeff(&biggest_col_index));
     biggest_col_index += k;
  
     // Track the number of meaningful pivots but do not stop the decomposition to make
     // sure that the initial matrix is properly reproduced. See bug 941.
     if (m_nonzero_pivots == size && biggest_col_sq_norm < threshold_helper * RealScalar(rows - k)) m_nonzero_pivots = k;
  
     // apply the transposition to the columns
     m_colsTranspositions.coeffRef(k) = static_cast<PermutationIndex>(biggest_col_index);
     if (k != biggest_col_index) {
       m_qr.col(k).swap(m_qr.col(biggest_col_index));
       std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
       std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
       ++number_of_transpositions;
     }
  
     // generate the householder vector, store it below the diagonal
     RealScalar beta;
     m_qr.col(k).tail(rows - k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
  
     // apply the householder transformation to the diagonal coefficient
     m_qr.coeffRef(k, k) = beta;
  
     // remember the maximum absolute value of diagonal coefficients
     if (abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
  
     // apply the householder transformation
     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));
  
     // update our table of norms of the columns
     for (Index j = k + 1; j < cols; ++j) {
       // The following implements the stable norm downgrade step discussed in
       // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
       // and used in LAPACK routines xGEQPF and xGEQP3.
       // See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
       if (!numext::is_exactly_zero(m_colNormsUpdated.coeffRef(j))) {
         RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
         temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
         temp = temp < RealScalar(0) ? RealScalar(0) : temp;
         RealScalar temp2 =
             temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) / m_colNormsDirect.coeffRef(j));
         if (temp2 <= norm_downdate_threshold) {
           // The updated norm has become too inaccurate so re-compute the column
           // norm directly.
           m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
           m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
         } else {
           m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
         }
       }
     }
   }
  
   m_colsPermutation.setIdentity(cols);
   for (Index k = 0; k < size /*m_nonzero_pivots*/; ++k)
     m_colsPermutation.applyTranspositionOnTheRight(k, static_cast<Index>(m_colsTranspositions.coeff(k)));
  
   m_det_p = (number_of_transpositions % 2) ? -1 : 1;
   m_isInitialized = true;
 }

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

◆ determinant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::Scalar Eigen::ColPivHouseholderQR< 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()

                                                                                                {
   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 isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().prod() : Scalar(0);
 }

References eigen_assert, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_det_p, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_hCoeffs, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr, and Eigen::run().

◆ dimensionOfKernel()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::ColPivHouseholderQR< 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&).

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

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

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::dimensionOfKernel().

◆ hCoeffs()

template<typename MatrixType_ , typename PermutationIndex_ >

const HCoeffsType& Eigen::ColPivHouseholderQR< 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.

333 { return m_hCoeffs; }

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_hCoeffs.

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::hCoeffs(), and Eigen::LevenbergMarquardt< FunctorType_ >::minimizeOptimumStorageOneStep().

◆ householderQ()

template<typename MatrixType , typename PermutationIndex >

ColPivHouseholderQR< MatrixType, PermutationIndex >::HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::householderQ

Returns: the matrix Q as a sequence of householder transformations. You can extract the meaningful part only by using:
qr.householderQ().setLength(qr.nonzeroPivots())

                                                                       {
   eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
   return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
 }

References eigen_assert, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_hCoeffs, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr.

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl_transposed(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixQ(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::matrixQ().

◆ info()

template<typename MatrixType_ , typename PermutationIndex_ >

ComputationInfo Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::info ( ) const

inline

Reports whether the QR factorization was successful.

Note: This function always returns Success. It is provided for compatibility with other factorization routines.

Returns: Success

                                {
     eigen_assert(m_isInitialized && "Decomposition is not initialized.");
     return Success;
   }

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

◆ init()

template<typename MatrixType_ , typename PermutationIndex_ >

void Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init	(	Index	rows,
		Index	cols
	)

inlineprivate

                                     {
     Index diag = numext::mini(rows, cols);
     m_hCoeffs.resize(diag);
     m_colsPermutation.resize(cols);
     m_colsTranspositions.resize(cols);
     m_temp.resize(cols);
     m_colNormsUpdated.resize(cols);
     m_colNormsDirect.resize(cols);
     m_isInitialized = false;
     m_usePrescribedThreshold = false;
   }

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

◆ inverse()

template<typename MatrixType_ , typename PermutationIndex_ >

const Inverse<ColPivHouseholderQR> Eigen::ColPivHouseholderQR< 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.

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

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

◆ isInjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::ColPivHouseholderQR< 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&).

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

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

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::absDeterminant(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::determinant(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInvertible(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::logAbsDeterminant(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::signDeterminant().

◆ isInvertible()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::ColPivHouseholderQR< 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&).

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

References eigen_assert, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isSurjective(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized.

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

◆ isSurjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::ColPivHouseholderQR< 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&).

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

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

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInvertible(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::isSurjective().

◆ logAbsDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::RealScalar Eigen::ColPivHouseholderQR< 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()

                                                                                                          {
   eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
   return isInjective() ? m_qr.diagonal().cwiseAbs().array().log().sum() : -NumTraits<RealScalar>::infinity();
 }

References eigen_assert, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr.

◆ matrixQ()

template<typename MatrixType_ , typename PermutationIndex_ >

HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixQ ( ) const

inline

165 { return householderQ(); }

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::householderQ().

◆ matrixQR()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

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

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

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

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::matrixQTZ(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::matrixT(), and Eigen::LevenbergMarquardt< FunctorType_ >::minimizeOptimumStorageOneStep().

◆ matrixR()

template<typename MatrixType_ , typename PermutationIndex_ >

const MatrixType& Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixR ( ) const

inline

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

Warning: The strict lower part of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixR().template triangularView<Upper>()

Eigen::ColPivHouseholderQR::matrixR
const MatrixType & matrixR() const
Definition: ColPivHouseholderQR.h:183

For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

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

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

◆ maxPivot()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

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

398 { return m_maxpivot; }

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

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::maxPivot().

◆ nonzeroPivots()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::ColPivHouseholderQR< 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()

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

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

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl_transposed(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::nonzeroPivots().

◆ rank()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::ColPivHouseholderQR< 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&).

                             {
     using std::abs;
     eigen_assert(m_isInitialized && "ColPivHouseholderQR 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;
   }

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

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::dimensionOfKernel(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isSurjective(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::rank(), and test_sparseqr_scalar().

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

326 { return m_qr.rows(); }

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr.

Referenced by gdb.printers._MatrixEntryIterator::__next__(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl_transposed(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::ColPivHouseholderQR(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isSurjective(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::rows(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

◆ setThreshold() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

ColPivHouseholderQR& Eigen::ColPivHouseholderQR< 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)

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

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

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::setThreshold().

◆ setThreshold() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

ColPivHouseholderQR& Eigen::ColPivHouseholderQR< 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:361

See the documentation of setThreshold(const RealScalar&).

                                                {
     m_usePrescribedThreshold = false;
     return *this;
   }

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

◆ signDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::Scalar Eigen::ColPivHouseholderQR< MatrixType, PermutationIndex >::signDeterminant

Returns: the sign 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(), logAbsDeterminant(), MatrixBase::determinant()

                                                                                                    {
   eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().array().sign().prod() : Scalar(0);
 }

References eigen_assert, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_det_p, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_hCoeffs, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_qr, and Eigen::run().

◆ threshold()

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::ColPivHouseholderQR< 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_qr.diagonalSize());
   }

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

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::rank(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::setThreshold(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::threshold().

Friends And Related Function Documentation

◆ CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >

template<typename MatrixType_ , typename PermutationIndex_ >

friend class CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >

friend

◆ SolverBase< ColPivHouseholderQR >

template<typename MatrixType_ , typename PermutationIndex_ >

friend class SolverBase< ColPivHouseholderQR >

friend

Member Data Documentation

◆ m_colNormsDirect

template<typename MatrixType_ , typename PermutationIndex_ >

RealRowVectorType Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_colNormsDirect

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init().

◆ m_colNormsUpdated

template<typename MatrixType_ , typename PermutationIndex_ >

RealRowVectorType Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_colNormsUpdated

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init().

◆ m_colsPermutation

template<typename MatrixType_ , typename PermutationIndex_ >

PermutationType Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_colsPermutation

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::_solve_impl_transposed(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::colsPermutation(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init().

◆ m_colsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init().

◆ m_det_p

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::determinant(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::signDeterminant().

◆ m_hCoeffs

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::determinant(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::hCoeffs(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::householderQ(), Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::signDeterminant().

◆ m_isInitialized

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

◆ m_maxpivot

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

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

◆ m_nonzero_pivots

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

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

◆ m_prescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

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

◆ m_qr

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

◆ m_temp

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::computeInPlace(), and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::init().

◆ m_usePrescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

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

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

Protected Attributes
MatrixType	m_qr

HCoeffsType	m_hCoeffs

PermutationType	m_colsPermutation

IntRowVectorType	m_colsTranspositions

RowVectorType	m_temp

RealRowVectorType	m_colNormsUpdated

RealRowVectorType	m_colNormsDirect

bool	m_isInitialized

bool	m_usePrescribedThreshold

RealScalar	m_prescribedThreshold

RealScalar	m_maxpivot

Index	m_nonzero_pivots

Index	m_det_p

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Friends

Detailed Description

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

Member Typedef Documentation

◆ Base

◆ HCoeffsType

◆ HouseholderSequenceType

◆ IntRowVectorType

◆ MatrixType

◆ PermutationIndex

◆ PermutationType

◆ PlainObject

◆ RealRowVectorType

◆ RowVectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ ColPivHouseholderQR() [1/4]

◆ ColPivHouseholderQR() [2/4]

◆ ColPivHouseholderQR() [3/4]

◆ ColPivHouseholderQR() [4/4]

Member Function Documentation

◆ _solve_impl()

◆ _solve_impl_transposed()

◆ absDeterminant()

◆ cols()

◆ colsPermutation()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeInPlace()

◆ determinant()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ householderQ()

◆ info()

◆ init()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQ()

◆ matrixQR()

◆ matrixR()

◆ maxPivot()

◆ nonzeroPivots()

◆ rank()

◆ rows()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ signDeterminant()

◆ threshold()

Friends And Related Function Documentation

◆ CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >

◆ SolverBase< ColPivHouseholderQR >

Member Data Documentation

◆ m_colNormsDirect

◆ m_colNormsUpdated

◆ m_colsPermutation

◆ m_colsTranspositions

◆ m_det_p

◆ m_hCoeffs

◆ m_isInitialized

◆ m_maxpivot

◆ m_nonzero_pivots

◆ m_prescribedThreshold

◆ m_qr

◆ m_temp

◆ m_usePrescribedThreshold

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