Public Types | Public Member Functions | Protected Attributes
HouseholderQR< _MatrixType > Class Template Reference

Householder QR decomposition of a matrix. More...

#include <HouseholderQR.h>

List of all members.

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().

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

References 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();
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

References 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: