Public Types | Public Member Functions | Protected Attributes | List of all members
HouseholderQR< _MatrixType > Class Template Reference

Householder QR decomposition of a matrix. More...

#include <HouseholderQR.h>

Public Types

enum  {
  RowsAtCompileTime,
  ColsAtCompileTime,
  Options,
  MaxRowsAtCompileTime,
  MaxColsAtCompileTime
}
typedef
internal::plain_diag_type
< MatrixType >::type 
HCoeffsType
typedef HouseholderSequence
< MatrixType, HCoeffsType >
::ConjugateReturnType 
HouseholderSequenceType
typedef MatrixType::Index Index
typedef Matrix< Scalar,
RowsAtCompileTime,
RowsAtCompileTime,(MatrixType::Flags
&RowMajorBit)?RowMajor:ColMajor,
MaxRowsAtCompileTime,
MaxRowsAtCompileTime
MatrixQType
typedef _MatrixType MatrixType
typedef MatrixType::RealScalar RealScalar
typedef
internal::plain_row_type
< MatrixType >::type 
RowVectorType
typedef MatrixType::Scalar Scalar

Public Member Functions

MatrixType::RealScalar absDeterminant () const
Index cols () const
HouseholderQRcompute (const MatrixType &matrix)
const HCoeffsTypehCoeffs () const
HouseholderSequenceType householderQ () const
 HouseholderQR ()
 Default Constructor.
 HouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 HouseholderQR (const MatrixType &matrix)
MatrixType::RealScalar logAbsDeterminant () const
const MatrixTypematrixQR () const
Index rows () const
template<typename Rhs >
const internal::solve_retval
< HouseholderQR, Rhs > 
solve (const MatrixBase< 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.

Parameters
MatrixTypethe 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.

See Also
MatrixBase::householderQr()

Member Typedef Documentation

typedef internal::plain_diag_type<MatrixType>::type HCoeffsType
typedef MatrixType::Index Index
typedef _MatrixType MatrixType
typedef MatrixType::RealScalar RealScalar
typedef internal::plain_row_type<MatrixType>::type RowVectorType
typedef MatrixType::Scalar Scalar

Member Enumeration Documentation

anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Constructor & Destructor Documentation

HouseholderQR ( )
inline

Default Constructor.

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

Member Function Documentation

MatrixType::RealScalar absDeterminant ( ) const
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()

References abs(), and eigen_assert.

Index cols ( void  ) const
inline
HouseholderQR< MatrixType > & compute ( const MatrixType matrix)
const HCoeffsType& hCoeffs ( ) const
inline
HouseholderSequenceType householderQ ( void  ) const
inline
MatrixType::RealScalar logAbsDeterminant ( ) const
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()

References eigen_assert.

const MatrixType& matrixQR ( ) const
inline
Returns
a reference to the matrix where the Householder QR decomposition is stored in a LAPACK-compatible way.

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

Index rows ( void  ) const
inline
const internal::solve_retval<HouseholderQR, Rhs> 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
bthe right-hand-side of the equation to solve.
Returns
a solution.
Note
The case where b is a matrix is not yet implemented. Also, this code is space inefficient.

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();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
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

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

Member Data Documentation

HCoeffsType m_hCoeffs
protected
bool m_isInitialized
protected
MatrixType m_qr
protected
RowVectorType m_temp
protected

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