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

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

Public Types
enum	{ MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef SolverBase< ColPivHouseholderQR >	Base

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

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

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

typedef MatrixType_	MatrixType

typedef PermutationIndex_	PermutationIndex

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationType

typedef MatrixType::PlainObject	PlainObject

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

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

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

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

typedef EigenBase< ColPivHouseholderQR< MatrixType_, PermutationIndex_ > >	Base

typedef Scalar	CoeffReturnType

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

typedef internal::traits< ColPivHouseholderQR< 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

	ColPivHouseholderQR ()
	Default Constructor. 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...

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

Index	cols () const

const PermutationType &	colsPermutation () const

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

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

MatrixType::Scalar	determinant () const

Index	dimensionOfKernel () const

const HCoeffsType &	hCoeffs () const

HouseholderSequenceType	householderQ () const

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

const Inverse< ColPivHouseholderQR >	inverse () const

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

MatrixType::RealScalar	logAbsDeterminant () const

HouseholderSequenceType	matrixQ () const

const MatrixType &	matrixQR () const

const MatrixType &	matrixR () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

Index	rank () const

Index	rows () const

ColPivHouseholderQR &	setThreshold (const RealScalar &threshold)

ColPivHouseholderQR &	setThreshold (Default_t)

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

RealScalar	threshold () const

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

ColPivHouseholderQR< MatrixType_, PermutationIndex_ > &	derived ()

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

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

Protected Attributes
RealRowVectorType	m_colNormsDirect

RealRowVectorType	m_colNormsUpdated

PermutationType	m_colsPermutation

IntRowVectorType	m_colsTranspositions

Index	m_det_p

HCoeffsType	m_hCoeffs

bool	m_isInitialized

RealScalar	m_maxpivot

Index	m_nonzero_pivots

RealScalar	m_prescribedThreshold

MatrixType	m_qr

RowVectorType	m_temp

bool	m_usePrescribedThreshold

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

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()

Definition at line 53 of file ColPivHouseholderQR.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 59 of file ColPivHouseholderQR.h.

◆ HCoeffsType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 68 of file ColPivHouseholderQR.h.

◆ HouseholderSequenceType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 73 of file ColPivHouseholderQR.h.

◆ IntRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 70 of file ColPivHouseholderQR.h.

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 58 of file ColPivHouseholderQR.h.

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 61 of file ColPivHouseholderQR.h.

◆ PermutationType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 69 of file ColPivHouseholderQR.h.

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 74 of file ColPivHouseholderQR.h.

◆ RealRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 72 of file ColPivHouseholderQR.h.

◆ RowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Definition at line 71 of file ColPivHouseholderQR.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 64 of file ColPivHouseholderQR.h.

          {
       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&).

Definition at line 97 of file ColPivHouseholderQR.h.

       : 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()

Definition at line 114 of file ColPivHouseholderQR.h.

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

Eigen::ColPivHouseholderQR::rows

Index rows() const

Definition: ColPivHouseholderQR.h:327

Eigen::ColPivHouseholderQR::cols

Index cols() const

Definition: ColPivHouseholderQR.h:328

Eigen::ColPivHouseholderQR::init

void init(Index rows, Index cols)

Definition: ColPivHouseholderQR.h:77

◆ 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)

See also: compute()

Definition at line 129 of file ColPivHouseholderQR.h.

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

◆ 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&)

Definition at line 141 of file ColPivHouseholderQR.h.

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

Member Function Documentation

◆ 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()

Definition at line 457 of file ColPivHouseholderQR.h.

 {
   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 abs(m_qr.diagonal().prod());
 }

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 328 of file ColPivHouseholderQR.h.

328 { return m_qr.cols(); }

◆ colsPermutation()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Returns: a const reference to the column permutation matrix

Definition at line 199 of file ColPivHouseholderQR.h.

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

◆ compute() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

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

◆ 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&)

Definition at line 481 of file ColPivHouseholderQR.h.

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

◆ computeInPlace()

template<typename MatrixType , typename PermutationIndex >

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

protected

Definition at line 489 of file ColPivHouseholderQR.h.

 {
  
   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;
 }

◆ 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()

Definition at line 447 of file ColPivHouseholderQR.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::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&).

Definition at line 272 of file ColPivHouseholderQR.h.

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

◆ 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.

Definition at line 334 of file ColPivHouseholderQR.h.

334 { return m_hCoeffs; }

◆ 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())

Definition at line 660 of file ColPivHouseholderQR.h.

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

◆ 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

Definition at line 411 of file ColPivHouseholderQR.h.

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

◆ init()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inlineprivate

Definition at line 77 of file ColPivHouseholderQR.h.

                                       {
       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;
     }

◆ 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.

Definition at line 321 of file ColPivHouseholderQR.h.

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

◆ 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&).

Definition at line 285 of file ColPivHouseholderQR.h.

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

◆ 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&).

Definition at line 310 of file ColPivHouseholderQR.h.

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

◆ 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&).

Definition at line 298 of file ColPivHouseholderQR.h.

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

◆ 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()

Definition at line 466 of file ColPivHouseholderQR.h.

 {
   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 m_qr.diagonal().cwiseAbs().array().log().sum();
 }

◆ matrixQ()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 167 of file ColPivHouseholderQR.h.

     {
       return 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

Definition at line 174 of file ColPivHouseholderQR.h.

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

◆ 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:189

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

Definition at line 189 of file ColPivHouseholderQR.h.

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

Definition at line 403 of file ColPivHouseholderQR.h.

403 { return m_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()

Definition at line 394 of file ColPivHouseholderQR.h.

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

◆ 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&).

Definition at line 255 of file ColPivHouseholderQR.h.

     {
       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;
     }

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Definition at line 327 of file ColPivHouseholderQR.h.

327 { return m_qr.rows(); }

◆ 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)

Definition at line 353 of file ColPivHouseholderQR.h.

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

◆ 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:364

See the documentation of setThreshold(const RealScalar&).

Definition at line 368 of file ColPivHouseholderQR.h.

     {
       m_usePrescribedThreshold = false;
       return *this;
     }

◆ solve()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename Rhs >

const Solve<ColPivHouseholderQR, Rhs> Eigen::ColPivHouseholderQR< 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, if any exists.

Parameters

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

Returns: a solution.

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

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

b

Array< int, 3, 1 > b

Definition: Array_variadic_ctor_cxx11.cpp:2

A

MatrixXcf A

Definition: ComplexEigenSolver_compute.cpp:1

Eigen::internal::isApprox

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

Definition: MathFunctions.h:1882

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

If there exists more than one solution, this method will arbitrarily choose one.

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.colPivHouseholderQr().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::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&).

Definition at line 378 of file ColPivHouseholderQR.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_colNormsDirect

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 439 of file ColPivHouseholderQR.h.

◆ m_colNormsUpdated

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 438 of file ColPivHouseholderQR.h.

◆ m_colsPermutation

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 435 of file ColPivHouseholderQR.h.

◆ m_colsTranspositions

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 436 of file ColPivHouseholderQR.h.

◆ m_det_p

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 443 of file ColPivHouseholderQR.h.

◆ m_hCoeffs

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 434 of file ColPivHouseholderQR.h.

◆ m_isInitialized

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 440 of file ColPivHouseholderQR.h.

◆ m_maxpivot

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 441 of file ColPivHouseholderQR.h.

◆ m_nonzero_pivots

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 442 of file ColPivHouseholderQR.h.

◆ m_prescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 441 of file ColPivHouseholderQR.h.

◆ m_qr

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 433 of file ColPivHouseholderQR.h.

◆ m_temp

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 437 of file ColPivHouseholderQR.h.

◆ m_usePrescribedThreshold

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Definition at line 440 of file ColPivHouseholderQR.h.

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

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

◆ 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]

◆ solve()

◆ threshold()

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_ >