Eigen::HouseholderQR< MatrixType_ > Class Template Reference

Householder QR decomposition of a matrix. More...

Inheritance diagram for Eigen::HouseholderQR< MatrixType_ >:

Public Types
enum	{   MaxRowsAtCompileTime ,   MaxColsAtCompileTime }
typedef SolverBase< HouseholderQR >	Base
typedef internal::plain_diag_type< MatrixType >::type	HCoeffsType
typedef HouseholderSequence< MatrixType, internal::remove_all_t< typename HCoeffsType::ConjugateReturnType > >	HouseholderSequenceType
typedef Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime,(MatrixType::Flags &RowMajorBit) ? RowMajor :ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixQType
typedef MatrixType_	MatrixType
typedef internal::plain_row_type< MatrixType >::type	RowVectorType
Public Types inherited from Eigen::SolverBase< HouseholderQR< MatrixType_ > >
enum
typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType
typedef EigenBase< HouseholderQR< MatrixType_ > >	Base
typedef Scalar	CoeffReturnType
typedef Transpose< const HouseholderQR< MatrixType_ > >	ConstTransposeReturnType
typedef internal::traits< HouseholderQR< MatrixType_ > >::Scalar	Scalar
Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index	Index
	The interface type of indices. More...
typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
MatrixType::RealScalar	absDeterminant () const
Index	cols () const
template<typename InputType >
HouseholderQR &	compute (const EigenBase< InputType > &matrix)
MatrixType::Scalar	determinant () const
const HCoeffsType &	hCoeffs () const
HouseholderSequenceType	householderQ () const
	HouseholderQR ()
	Default Constructor. More...
template<typename InputType >
	HouseholderQR (const EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix. More...
template<typename InputType >
	HouseholderQR (EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix. More...
	HouseholderQR (Index rows, Index cols)
	Default Constructor with memory preallocation. More...
MatrixType::RealScalar	logAbsDeterminant () const
const MatrixType &	matrixQR () const
Index	rows () const
template<typename Rhs >
const Solve< HouseholderQR, Rhs >	solve (const MatrixBase< Rhs > &b) const
Public Member Functions inherited from Eigen::SolverBase< HouseholderQR< MatrixType_ > >
const AdjointReturnType	adjoint () const
HouseholderQR< MatrixType_ > &	derived ()
const HouseholderQR< MatrixType_ > &	derived () const
const Solve< HouseholderQR< MatrixType_ >, 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< HouseholderQR< MatrixType_ > >
void	_check_solve_assertion (const Rhs &b) const

Protected Attributes
HCoeffsType	m_hCoeffs
bool	m_isInitialized
MatrixType	m_qr
RowVectorType	m_temp

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()

Definition at line 58 of file HouseholderQR.h.

Member Typedef Documentation

Base

template<typename MatrixType_ >

typedef SolverBase<HouseholderQR> Eigen::HouseholderQR< MatrixType_ >::Base

Definition at line 64 of file HouseholderQR.h.

HCoeffsType

template<typename MatrixType_ >

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

Definition at line 73 of file HouseholderQR.h.

HouseholderSequenceType

template<typename MatrixType_ >

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

Definition at line 75 of file HouseholderQR.h.

MatrixQType

template<typename MatrixType_ >

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

Definition at line 72 of file HouseholderQR.h.

MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::HouseholderQR< MatrixType_ >::MatrixType

Definition at line 63 of file HouseholderQR.h.

RowVectorType

template<typename MatrixType_ >

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

Definition at line 74 of file HouseholderQR.h.

Member Enumeration Documentation

anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 68 of file HouseholderQR.h.

         {
      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&).

Definition at line 83 of file HouseholderQR.h.

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

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()

Definition at line 91 of file HouseholderQR.h.

      : 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);

See also

compute()

Definition at line 110 of file HouseholderQR.h.

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

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&)

Definition at line 128 of file HouseholderQR.h.

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

Member Function Documentation

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

determinant(), logAbsDeterminant(), MatrixBase::determinant()

Definition at line 312 of file HouseholderQR.h.

{
  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());
}

cols()

template<typename MatrixType_ >

Index Eigen::HouseholderQR< MatrixType_ >::cols ( void  ) const

inline

Definition at line 232 of file HouseholderQR.h.

{ return m_qr.cols(); }

compute()

template<typename MatrixType_ >

template<typename InputType >

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

inline

Definition at line 181 of file HouseholderQR.h.

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

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&)

Definition at line 506 of file HouseholderQR.h.

{
  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;
}

determinant()

template<typename MatrixType >

MatrixType::Scalar Eigen::HouseholderQR< MatrixType >::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 302 of file HouseholderQR.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;
}

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.

Definition at line 238 of file HouseholderQR.h.

{ return m_hCoeffs; }

householderQ()

template<typename MatrixType_ >

HouseholderSequenceType Eigen::HouseholderQR< MatrixType_ >::householderQ ( void  ) 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:

MatrixXf A(MatrixXf::Random(5,3)), thinQ(MatrixXf::Identity(5,3)), Q;
A.setRandom();
HouseholderQR<MatrixXf> qr(A);
Q = qr.householderQ();
thinQ = qr.householderQ() * thinQ;
std::cout << "The complete unitary matrix Q is:\n" << Q << "\n\n";
std::cout << "The thin matrix Q is:\n" << thinQ << "\n\n";

Output:

The complete unitary matrix Q is:
  -0.676   0.0793    0.713  -0.0788   -0.147
  -0.221   -0.322    -0.37   -0.366   -0.759
  -0.353   -0.345   -0.214    0.841  -0.0518
   0.582   -0.462    0.555    0.176   -0.329
  -0.174   -0.747 -0.00907   -0.348    0.539

The thin matrix Q is:
  -0.676   0.0793    0.713
  -0.221   -0.322    -0.37
  -0.353   -0.345   -0.214
   0.582   -0.462    0.555
  -0.174   -0.747 -0.00907

Definition at line 165 of file HouseholderQR.h.

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

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

determinant(), absDeterminant(), MatrixBase::determinant()

Definition at line 321 of file HouseholderQR.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!");
  return m_qr.diagonal().cwiseAbs().array().log().sum();
}

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.

Definition at line 174 of file HouseholderQR.h.

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

rows()

template<typename MatrixType_ >

Index Eigen::HouseholderQR< MatrixType_ >::rows ( void  ) const

inline

Definition at line 231 of file HouseholderQR.h.

{ return m_qr.rows(); }

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.

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); 

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:

typedef Matrix<float,3,3> Matrix3x3;
Matrix3x3 m = Matrix3x3::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.householderQr().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

Member Data Documentation

m_hCoeffs

template<typename MatrixType_ >

HCoeffsType Eigen::HouseholderQR< MatrixType_ >::m_hCoeffs

protected

Definition at line 255 of file HouseholderQR.h.

m_isInitialized

template<typename MatrixType_ >

bool Eigen::HouseholderQR< MatrixType_ >::m_isInitialized

protected

Definition at line 257 of file HouseholderQR.h.

m_qr

template<typename MatrixType_ >

MatrixType Eigen::HouseholderQR< MatrixType_ >::m_qr

protected

Definition at line 254 of file HouseholderQR.h.

m_temp

template<typename MatrixType_ >

RowVectorType Eigen::HouseholderQR< MatrixType_ >::m_temp

protected

Definition at line 256 of file HouseholderQR.h.

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

• ForwardDeclarations.h
• HouseholderQR.h