Householder QR decomposition of a matrix. More...

#include <src/eigen/Eigen/src/QR/HouseholderQR.h>

Inheritance diagram for Eigen::HouseholderQR< _MatrixType >:

Collaboration diagram for Eigen::HouseholderQR< _MatrixType >:

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

typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::StorageIndex	StorageIndex

typedef Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime,(MatrixType::Flags &RowMajorBit) ? RowMajor :ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixQType

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

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

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

Public Member Functions
	HouseholderQR ()
	Default Constructor.

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

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

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

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

HouseholderSequenceType	householderQ () const

const MatrixType &	matrixQR () const

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

MatrixType::RealScalar	absDeterminant () const

MatrixType::RealScalar	logAbsDeterminant () const

Index	rows () const

Index	cols () const

const HCoeffsType &	hCoeffs () const

template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const

template<typename RhsType , typename DstType >
void	_solve_impl (const RhsType &rhs, DstType &dst) const

Protected Member Functions
void	computeInPlace ()

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
MatrixType	m_qr

HCoeffsType	m_hCoeffs

RowVectorType	m_temp

bool	m_isInitialized

Detailed Description

template<typename _MatrixType>
class Eigen::HouseholderQR< _MatrixType >

Householder QR decomposition of a matrix.

Template Parameters

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

This class performs a QR decomposition of a matrix A into matrices Q and R such that

$\mathbf{A} = \mathbf{Q} \, \mathbf{R}$

by using Householder transformations. Here, Q a unitary matrix and R an upper triangular matrix. The result is stored in a compact way compatible with LAPACK.

Note that no pivoting is performed. This is not a rank-revealing decomposition. If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.

This Householder QR decomposition is faster, but less numerically stable and less feature-full than FullPivHouseholderQR or ColPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::householderQr()

Member Typedef Documentation

◆ HCoeffsType

template<typename _MatrixType >

typedef internal::plain_diag_type<MatrixType>::type Eigen::HouseholderQR< _MatrixType >::HCoeffsType

◆ HouseholderSequenceType

template<typename _MatrixType >

typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> Eigen::HouseholderQR< _MatrixType >::HouseholderSequenceType

◆ MatrixQType

template<typename _MatrixType >

typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> Eigen::HouseholderQR< _MatrixType >::MatrixQType

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::HouseholderQR< _MatrixType >::MatrixType

◆ RealScalar

template<typename _MatrixType >

typedef MatrixType::RealScalar Eigen::HouseholderQR< _MatrixType >::RealScalar

◆ RowVectorType

template<typename _MatrixType >

typedef internal::plain_row_type<MatrixType>::type Eigen::HouseholderQR< _MatrixType >::RowVectorType

◆ Scalar

template<typename _MatrixType >

typedef MatrixType::Scalar Eigen::HouseholderQR< _MatrixType >::Scalar

◆ StorageIndex

template<typename _MatrixType >

typedef MatrixType::StorageIndex Eigen::HouseholderQR< _MatrixType >::StorageIndex

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime

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

Constructor & Destructor Documentation

◆ HouseholderQR() [1/4]

template<typename _MatrixType >

Eigen::HouseholderQR< _MatrixType >::HouseholderQR ( )

inline

Default Constructor.

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

70: m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}

Eigen::HouseholderQR::m_temp

RowVectorType m_temp

Definition HouseholderQR.h:231

Eigen::HouseholderQR::m_isInitialized

bool m_isInitialized

Definition HouseholderQR.h:232

Eigen::HouseholderQR::m_hCoeffs

HCoeffsType m_hCoeffs

Definition HouseholderQR.h:230

Eigen::HouseholderQR::m_qr

MatrixType m_qr

Definition HouseholderQR.h:229

◆ HouseholderQR() [2/4]

template<typename _MatrixType >

Eigen::HouseholderQR< _MatrixType >::HouseholderQR	(	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: HouseholderQR()

      : m_qr(rows, cols),
        m_hCoeffs((std::min)(rows,cols)),
        m_temp(cols),
        m_isInitialized(false) {}

◆ HouseholderQR() [3/4]

template<typename _MatrixType >

template<typename InputType >

Eigen::HouseholderQR< _MatrixType >::HouseholderQR ( 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:

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

qr.compute(matrix);

Eigen::HouseholderQR

Householder QR decomposition of a matrix.

Definition HouseholderQR.h:45

See also: compute()

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

References Eigen::HouseholderQR< _MatrixType >::compute(), and Eigen::EigenBase< Derived >::derived().

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◆ HouseholderQR() [4/4]

template<typename _MatrixType >

template<typename InputType >

Eigen::HouseholderQR< _MatrixType >::HouseholderQR ( 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: HouseholderQR(const EigenBase&)

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

References Eigen::HouseholderQR< _MatrixType >::computeInPlace().

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Member Function Documentation

◆ _solve_impl() [1/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

EIGEN_DEVICE_FUNC void Eigen::HouseholderQR< _MatrixType >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

◆ _solve_impl() [2/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

void Eigen::HouseholderQR< _MatrixType >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

{
  const Index rank = (std::min)(rows(), cols());
  eigen_assert(rhs.rows() == rows());
 
  typename RhsType::PlainObject c(rhs);
 
  // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
  c.applyOnTheLeft(householderSequence(
    m_qr.leftCols(rank),
    m_hCoeffs.head(rank)).transpose()
  );
 
  m_qr.topLeftCorner(rank, rank)
      .template triangularView<Upper>()
      .solveInPlace(c.topRows(rank));
 
  dst.topRows(rank) = c.topRows(rank);
  dst.bottomRows(cols()-rank).setZero();
}

◆ absDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::HouseholderQR< MatrixType >::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: logAbsDeterminant(), MatrixBase::determinant()

{
  using std::abs;
  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!");
  return abs(m_qr.diagonal().prod());
}

References eigen_assert.

◆ check_template_parameters()

template<typename _MatrixType >

static void Eigen::HouseholderQR< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ cols()

template<typename _MatrixType >

Index Eigen::HouseholderQR< _MatrixType >::cols ( ) const

inline

206{ return m_qr.cols(); }

References Eigen::HouseholderQR< _MatrixType >::m_qr.

◆ compute()

template<typename _MatrixType >

template<typename InputType >

HouseholderQR & Eigen::HouseholderQR< _MatrixType >::compute ( const EigenBase< InputType > & matrix )

inline

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

References Eigen::HouseholderQR< _MatrixType >::computeInPlace(), Eigen::EigenBase< Derived >::derived(), and Eigen::HouseholderQR< _MatrixType >::m_qr.

Referenced by Eigen::HouseholderQR< _MatrixType >::HouseholderQR().

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◆ computeInPlace()

template<typename MatrixType >

void Eigen::HouseholderQR< MatrixType >::computeInPlace

protected

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 HouseholderQR, HouseholderQR(const MatrixType&)

{
  check_template_parameters();
  
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = (std::min)(rows,cols);
 
  m_hCoeffs.resize(size);
 
  m_temp.resize(cols);
 
  internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());
 
  m_isInitialized = true;
}

References Eigen::internal::householder_qr_inplace_blocked< MatrixQR, HCoeffs, MatrixQRScalar, InnerStrideIsOne >::run().

Referenced by Eigen::HouseholderQR< _MatrixType >::HouseholderQR(), and Eigen::HouseholderQR< _MatrixType >::compute().

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◆ hCoeffs()

template<typename _MatrixType >

const HCoeffsType & Eigen::HouseholderQR< _MatrixType >::hCoeffs ( ) const

inline

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

For advanced uses only.

212{ return m_hCoeffs; }

References Eigen::HouseholderQR< _MatrixType >::m_hCoeffs.

◆ householderQ()

template<typename _MatrixType >

HouseholderSequenceType Eigen::HouseholderQR< _MatrixType >::householderQ ( ) const

inline

This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.

The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object. Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:

Example:

Output:

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

References eigen_assert, Eigen::HouseholderQR< _MatrixType >::m_hCoeffs, Eigen::HouseholderQR< _MatrixType >::m_isInitialized, and Eigen::HouseholderQR< _MatrixType >::m_qr.

Referenced by Eigen::RealQZ< _MatrixType >::hessenbergTriangular(), and igl::per_vertex_point_to_plane_quadrics().

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◆ logAbsDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::HouseholderQR< MatrixType >::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: absDeterminant(), 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!");
  return m_qr.diagonal().cwiseAbs().array().log().sum();
}

References eigen_assert.

◆ matrixQR()

template<typename _MatrixType >

const MatrixType & Eigen::HouseholderQR< _MatrixType >::matrixQR ( ) const

inline

Returns: a reference to the matrix where the Householder QR decomposition is stored in a LAPACK-compatible way.

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

References eigen_assert, Eigen::HouseholderQR< _MatrixType >::m_isInitialized, and Eigen::HouseholderQR< _MatrixType >::m_qr.

Referenced by Eigen::RealQZ< _MatrixType >::hessenbergTriangular().

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◆ rows()

template<typename _MatrixType >

Index Eigen::HouseholderQR< _MatrixType >::rows ( ) const

inline

205{ return m_qr.rows(); }

References Eigen::HouseholderQR< _MatrixType >::m_qr.

◆ solve()

template<typename _MatrixType >

template<typename Rhs >

const Solve< HouseholderQR, Rhs > Eigen::HouseholderQR< _MatrixType >::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.

\note_about_checking_solutions

\note_about_arbitrary_choice_of_solution

Example:

Output:

    {
      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
      return Solve<HouseholderQR, Rhs>(*this, b.derived());
    }

References eigen_assert, and Eigen::HouseholderQR< _MatrixType >::m_isInitialized.

Member Data Documentation

◆ m_hCoeffs

template<typename _MatrixType >

HCoeffsType Eigen::HouseholderQR< _MatrixType >::m_hCoeffs

protected

Referenced by Eigen::HouseholderQR< _MatrixType >::hCoeffs(), and Eigen::HouseholderQR< _MatrixType >::householderQ().

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::HouseholderQR< _MatrixType >::m_isInitialized

protected

Referenced by Eigen::HouseholderQR< _MatrixType >::householderQ(), Eigen::HouseholderQR< _MatrixType >::matrixQR(), and Eigen::HouseholderQR< _MatrixType >::solve().

◆ m_qr

template<typename _MatrixType >

MatrixType Eigen::HouseholderQR< _MatrixType >::m_qr

protected

Referenced by Eigen::HouseholderQR< _MatrixType >::cols(), Eigen::HouseholderQR< _MatrixType >::compute(), Eigen::HouseholderQR< _MatrixType >::householderQ(), Eigen::HouseholderQR< _MatrixType >::matrixQR(), and Eigen::HouseholderQR< _MatrixType >::rows().

◆ m_temp

template<typename _MatrixType >

RowVectorType Eigen::HouseholderQR< _MatrixType >::m_temp

protected

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

src/eigen/Eigen/src/Core/util/ForwardDeclarations.h
src/eigen/Eigen/src/QR/HouseholderQR.h

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ HCoeffsType

◆ HouseholderSequenceType

◆ MatrixQType

◆ MatrixType

◆ RealScalar

◆ RowVectorType

◆ Scalar

◆ StorageIndex

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ HouseholderQR() [1/4]

◆ HouseholderQR() [2/4]

◆ HouseholderQR() [3/4]

◆ HouseholderQR() [4/4]

Member Function Documentation

◆ _solve_impl() [1/2]

◆ _solve_impl() [2/2]

◆ absDeterminant()

◆ check_template_parameters()

◆ cols()

◆ compute()

◆ computeInPlace()

◆ hCoeffs()

◆ householderQ()

◆ logAbsDeterminant()

◆ matrixQR()

◆ rows()

◆ solve()

Member Data Documentation

◆ m_hCoeffs

◆ m_isInitialized

◆ m_qr

◆ m_temp